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MATHEMATICS 2º ESO
SECCIONES EUROPEAS
IES ANDRÉS DE VANDELVIRA
MANUEL VALERO LÓPEZ (MATEMÁTICAS)
ANTONIO MARTÍNEZ RESTA (INGLÉS)
OLGA LÓPEZ-GALIANO MORENO 2º ESO A
(Diseño de portada)
I.E.S. “Andrés de Vandelvira” - Sección Europea
Mathematics
INDEX
Unit 1 Numbers, operations, integers, divisibility
Remember how to read numbers
Reading powers
Calculations
Negative numbers
Order of operations
Multiples and factors
Exercises
Solutions
1-1
1-4
1-5
1-6
1-9
1-10
1-15
1-19
Unit 2 Decimal and sexagesimal system
Decimal numbers
Types of decimal numbers
Decimal numbers on the number line
Rounding Decimal Numbers
Sexagesimal system
Exercises
Solutions
2-1
2-1
2-2
2-2
2-4
2-7
2-9
Unit 3 Fractions
Fractions
Reading fractions
Equivalent fractions
Comparing and ordering fractions
Adding and subtracting fractions
Improper fractions, mixed numbers
Multiplying fractions
Multiplying a fraction by a whole number,
calculating a fraction of a quantity
Dividing fractions
Exercises
Powers
Rules for powers
Solutions
3-1
3-2
3-3
3-6
3-7
3-10
3-11
3-12
3-13
3-15
3-17
3-18
3-24
Unit 4 Proportions
Ratio
Proportions
Direct proportions. Exercises
Inverse proportions. Exercises
Solutions
4-1
4-2
4-4
4-7
4-11
Unit 5 Percentages
Percentage
Calculating a percentage of a quantity
Calculate the total from the percent
Percentage increase decrease
5-1
5-4
5-5
5-6
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Interest
Mixtures
Proportional division
Final exercises
Solutions
Mathematics
5-9
5-12
5-13
5-14
5-24
Unit 6 Algebra
Using letters for numbers
Mathematical language
Monomials
Addition and subtraction of monomials
Product of monomials
Quotient of monomials
Polynomials
Evaluating polynomials
Adding polynomials
Subtracting polynomials
Multiplying polynomials
Factorising
Three algebraic identities
Solutions
6-1
6-3
6-5
6-6
6-8
6-9
6-10
6-12
6-13
6-16
6-18
6-21
6-23
6-27
Unit 7 Equations
Definition
Linear equations language in equations
Solving easy equations, basic rules
Equations with denominators
Solving problems using linear equations
Quadratic equations, exercises
Solutions
7-1
7-1
7-2
7-6
7-10
7-17
7-28
Unit 8 Graphs
Coordinating the plane
Functions
Linear graphs
Exercises
Solutions
8-1
8-5
8-14
8-21
8-30
Unit 9 Statistics
Constructing a frequency table
Interpreting diagrams
Parameter statistics
Solutions
9-1
9-3
9-7
9-10
Unit 10 Similarity
Previous ideas
Similar shapes
Thales Theorem
Triangles put in the Thales position. Similar
10-1
10-1
10-3
10-5
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triangles
Similar triangles, similarity criteria
Exercises
Solutions
Mathematics
10-6
10-6
10-10
Unit 11 3-D Shapes
cuboid
Exercises
Prisms
Pyramids
Cylinders
Cones
Sphere
Solutions
11-1
11-2
11-4
11-7
11-9
11-11
11-13
11-17
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Mathematics
1 Numbers, operations, integers,
divisibility
1 Remember how to read numbers
Complete this table
Number
Cardinal
Ordinal
1
One
first (1 )
2
Two
second (2 )
3
Three
third (3 )
st
nd
rd
th
4
fourth (4 )
5
fifth
6
sixth
7
seventh
8
eighth
9
ninth
10
tenth
11
Eleven
eleventh
12
Twelve
13
Thirteen
14
Fourteen
15
Fifteen
16
Sixteen
17
Seventeen
18
Eighteen
19
Nineteen
20
Twenty
21
twenty-one
twenty-first
30
Thirty
thirtieth
40
Forty
22
23
24
25
26
27
28
29
50
60
70
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Mathematics
80
90
100
one hundred
hundredth
1,000
one thousand
100,000
1,000,000
hundred thousandth
one million
millionth
The names of the big numbers differ depending where you live. The places
are grouped by thousands in America, or France, by millions in Great Britain (not
always), Germany and Spain.
Name
million
American-French
1,000,000
1,000,000,000 (a
thousand millions)
1 with 12 zeros
billion
trillion
English-German-Spanish
1,000,000
1,000,000,000,000 (a million millions)
1 with 18 zeros
We will read the numbers in our own way although we must be capable of recognize
the correct meaning of numbers when the information comes to us from USA, for
example.
Decimals
Decimal fractions are said with each figure separate. We use a full stop (called
“point”), not a comma, before the fraction. Each place value has a value that is one
tenth of the value to the immediate left of it.
0.75 (nought) point seventy-five or zero point seven five
3.375 three point three seven five.
We will see all this more detailed in chapter 2
2 Fractions and percentages
Simple fractions are expressed by using “ordinal numbers” / (third, fourth, fifth..) with
some exceptions:
1/2
1/3
2/3
3/4
5/8
4/33
One half / a half
One third / a third
Two thirds
Three quarters
Five eighths
Four over thirty-three
Percentages:
We don’t use the article in front of the numeral
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10% of the people
Mathematics
Ten per cent of the people
We will see this more detailed in chapters 3 and 5
Exercise 1
a) Write in words the following numbers as in the example.
3 528: Three thousand, five hundred and twenty eight
86 424 ____________________________________________________________
987 _______________________________________________________________
3 270 ______________________________________________________________
30 001 _____________________________________________________________
1 487 070 ___________________________________________________________
320 569 ____________________________________________________________
20,890,300 _________________________________________________________
b) Read aloud the following numbers:
456
4 500
90 045 123
34 760 041
23 455 678
5 223 500 668 316
c) Write in words the following decimal and fractions:
0.056 ______________________________________________________________
23.45 cm ___________________________________________________________
₤1.20 ______________________________________________________________
3.77 _______________________________________________________________
5
_________________________________________________________________
4
7
_________________________________________________________________
2
12
________________________________________________________________
35
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Mathematics
37
_______________________________________________________________
100
3 Reading powers
6 5 Is read as
The fifth power of six
Six to the power of five
Six powered to five.
-
The most common is six to the power of five
6 is the base
5 is the index or exponent
Especial cases:
Squares and cubes (powers of two and three)
32 is read as:
- Three squared
- Three to the power of two.
53
Is read as:
- Five cubed
- Five to the power of three.
The most common is five cubed
Exercise 2 Calculate mentally and write in words the following powers:
a) 4 3 =
b) 6 2 =
2
c) (− 11) =
d) 2 5 =
e) 10 3 =
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Mathematics
f) 1000 2 =
3
g) (0.1) =
4 Calculations
Addition AND / PLUS
In small additions we say and for addition and is/are for the result
Example:
2+6 = 8 Is read as “two and six are eight” or “two and six is eight”
In larger additions and in more formal style (in maths) we use plus for +, and
equals or is for the result.
Example:
We read 234 + 25 = 259 like “two hundred and thirty four plus twenty five equals
/ is two hundred and fifty nine
Subtraction: MINUS/TAKE AWAY/FROM
Example:
9−5 = 4
In conversational style, with small numbers, people say:
- Five from nine leaves/is four
- Nine take away five leaves/is four
In a more formal style, or with larger numbers, we use minus and equals
510 - 302 = 208 Five hundred and ten minus three hundred and two equals /is two
hundred and eight
Multiplication
TIMES
MULTIPLIED BY
In small calculations we say:
3 x 4 = 12 three fours are twelve
6 x 7 = 42 six sevens are forty-two
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Mathematics
In larger calculations we can say
17 x 381 = 6,477
17 times 381 is/makes 6,477
In a more formal style:
17 multiplied by 381 equal 6,477
Division
270:3 = 90
DIVIDED BY
Two hundred and seventy divided by three equals ninety
But in smaller calculations like 8:2 = 4 we can say two into eight goes four (times)
5 Negative numbers
There are many situations in which you need to use numbers below zero, one of
these is temperature, others are money that you can deposit (positive) or withdraw
(negative) in a bank, steps that you can take forwards (positive) or backwards
(negative).
Positive integers are all the whole numbers greater than zero: 1, 2, 3, 4, 5, ...
Negative integers are all the opposites of these whole numbers: -1, -2, -3, -4, -5, …
The Number Line
The number line is a line labelled with the integers in increasing order from left to
right, that extends in both directions:
For any two different places on the number line, the integer on the right is
greater than the integer on the left.
Examples:
9 > 4 Is read: nine is ‘greater than’ four -7 < 9 Is read: minus seven is ‘less than’
nine.
5.1 Absolute Value of an integer
The number of units a number is from zero on the number line.
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Mathematics
If the number is positive, the absolute value is the same number.
If the number is negative, the absolute value is the opposite.
The absolute value of a number is always a positive number (or zero). We specify the
absolute value of a number n by writing n in between two vertical bars: |n|.
Exercise 3 Plot on the number line and after order them from less to great.
-2
+8
0
-5
3
5.2 Adding Integers
Rules for addition:
When adding integers with the same sign:
We add their absolute values, and give the result the same sign.
With the opposite signs:
We take their absolute values, subtract the smaller from the larger, and give the
result the sign of the integer with the larger absolute value.
5.3 Subtracting Integers
Rules for subtraction:
Subtracting an integer is the same as adding the opposite.
We convert the subtracted integer to its opposite, and add the two integers.
The result of subtracting two integers could be positive or negative.
Exercise 4 Calculate operating first the expressions into brackets.
a) 8 − (3 − 2 + 5)
b) 87 − 4 − 9) + (2 + 3 − 1)
c) (2 − 7) − (5 − 4) − (1 − 6)
d) (2 + 4 − 1) − (7 − 4 − 9)
e) − (1 − 2 + 3 − 7) + (3 − 4) − (2 − 3 − 5)
Exercise 5 Remove brackets and calculate:
a) 8 − (3 − 2 + 5)
b) 87 − 4 − 9) + (2 + 3 − 1)
c) (2 − 7) − (5 − 4) − (1 − 6)
d) (2 + 4 − 1) − (7 − 4 − 9)
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Mathematics
e) − (1 − 2 + 3 − 7) + (3 − 4) − (2 − 3 − 5)
Exercise 6 Calculate
a) 5 − 4 + 7 − 9 + 12 − 1 + 5 − 10
b) 4 − 3 + 21 − 32 − 5 + 6
c) 5 − [2 − (6 − 9)]
d) 5 − [5 + (3 − 2 − 9)] − (3 − 10 )
e) 1 − [(2 − 5) − 3 − (9 − 1)] − (2 − 3)
5.4 Multiplying Integers
To multiply a pair of integers:
If both numbers have the same sign (positive or negative)
Their product is the product of their absolute values (their product is positive)
If the numbers have opposite signs
Their product is the opposite of the product of their absolute values (their product is
negative).
If one or both of the integers is 0, the product is 0.
Look at the following chart below.
PRODUCT
+
-
+
POSITIVE NEGATIVE
-
NEGATIVE POSITIVE
To multiply any number of integers:
1. Count the number of negative numbers in the product.
2. Take the product of their absolute values.
If the number of negative integers counted in step 1 is even, the product is just the
product from step 2 (positive).
If the number of negative integers is odd, the product is the opposite of the product in
step 2 (give the product in step 2 a negative sign).
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Mathematics
If any of the integers in the product is 0, the product is 0.
5.5 Dividing Integers
To divide a pair of integers the rules are the same than for the product:
If both numbers have the same sign (positive or negative)
Divide the absolute values of the first integer by the absolute value of the second
integer (the result is positive)
If the numbers have opposite signs
Divide the absolute value of the first integer by the absolute value of the second
integer, and give this result a negative sign.
The chart is.
DIVISION
+
-
+
POSITIVE NEGATIVE
-
NEGATIVE POSITIVE
6 Order of operations
Do all operations in brackets first.
Then do multiplications and divisions in the order they appear.
Then do additions and subtractions in the order they occur
Easy way to remember them
Brackets
Exponents
Divisions
Multiplications
This gives you: BEDMAS.
Do one operation at a time.
Exercise 7 Calculate
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Additions
Subtractions
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Mathematics
a) ( −3)( −2)( −5)
b) 6 ⋅ ( −3) ⋅ ( −1)
c) 15 : ( −3) ⋅ ( −1)
d) (− 80 ) : [(− 8 ) ⋅ 2]
e) [( −80 ) : (− 8 )] ⋅ 2
f) [9 ⋅ (− 8 )] : [( −3) ⋅ ( −4)]
Exercise 8 Calculate
a) 3(5 − 8) + 2(8 − 12) − 3 + 3 ⋅ ( −5)
b) 17 − [2 ⋅ 3 + 5( −2 + 4)] ⋅ 3 − 6 ⋅ ( −5)
c) ( −2) 2 − ( −3) 2 + ( −1)3
d) ( +4) 2 + ( −4) 2 − ( −4) 2 − 4 2
e) 2 3 − 3 2 + 4 2 − 17
[
]
f) 5 2 ⋅ ( −3) 2 : ( −15)
Exercise 9 Calculate operating first the expressions into brackets
a) 3(2 − 6 + 5) + 7(3 − 2(5 − 12))
b) 2 ⋅ [ −3 ⋅ (5 ⋅ (3 − 4) + 1) − 2]
3
c) (− 2) ⋅ [ −2 + 3 ⋅ (1 − 7)] : (1 − 3)
d) 3 ⋅ (5 − 8) + 2 ⋅ (3 − 12) − 5 + 3 ⋅ 2
7 Multiples and factors
7.1 Multiples
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Mathematics
The products of a number with the natural numbers 1, 2, 3, 4, 5, ... are called
the multiples of the number.
The multiples of a number are obtained by multiplying the number by each of
the natural numbers.
7.2 Factors
A whole number that divides exactly into another whole number is called a
factor of that number.
If a number can be expressed as a product of two whole numbers, then the
whole numbers are called factors of that number.
7.3 Prime Numbers
If a number has only two different factors, 1 and itself, then the number is said
to be a prime number.
The Sieve of Eratosthenes
Have a look of the book of 1º ESO and you will see how to get a set of all the
prime numbers
Exercise 10 Write down all the prime numbers between 80 and 110 (you must
use the Sieve of Eratosthenes).
7.4 Tests of divisibility
One number is divisible by:
2 If the last digit is 0 or is divisible by 2, (0, 2, 4, 8).
3 If the sum of the digits is divisible by 3.
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Mathematics
4 If the last two digits are divisible by 4.
5 If the last digit is 0 or is divisible by 5, (0,5).
9 If the sum of the digits is divisible by 9.
8 If the half of it is divisible by 4.
6 If it is divisible by 2 and 3.
11 If the sum of the digits in the even position minus the sum of the digits in the
uneven position is 0 or divisible by 11.
Exercise 11 Write down four consecutive multiples of:
a) 7 greater than 100
b) 15 greater than 230
c) 9 greater than 1230
Exercise 12 Write down all the multiples of 6 between 92 and 109
Exercise 13 Write down all the multiples of 6 between 1200 and 1250
Exercise 14 Write down all the factors of
a) 18
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Mathematics
b) 90
c) 140
d) 80
e) 50
Exercise 15 Find out the missing figure so the number (there can be more than
one answer)
a) 3[ ]1 is a multiple of 3
b) 57[ ] is a multiple of 2
c) 23[ ] is a multiple of 5
d) 52[ ]3 is a multiple of 11
Exercise 16 Factorise:
a) 46
b) 180
c) 60
d) 1500
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Mathematics
7.5 Common Multiples
Multiples that are common to two or more numbers are said to be common
multiples.
Example:
Multiples of 2 are 2, 4, 6, 8, 10, 12, 14, 16, 18, …
Multiples of 3 are 3, 6, 9, 12, 15, 18, …
So, common multiples of 2 and 3 are 6, 12, 18, …
Lowest common multiple
The smallest common multiple of two or more numbers is called the lowest common
multiple (LCM).
Example:
Multiples of 8 are 8, 16, 24, 32, …
Multiples of 3 are 3, 6, 9, 12, 15, 18, 21, 24, …
LCM of 3 and 8 is 24
In general there are two methods for finding the lowest common multiple (LCM) of
two or more numbers:
Method I
For small numbers
List the multiples of the largest number and stop when you find a multiple of the other
number. This is the LCM.
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Mathematics
Method II
General
To find the lowest common multiple (LCM) of two or more higher numbers:
- Find the prime factor decomposition.
- Choose the non common factors and the common factors with the highest
exponents.
Example: Find the lowest common multiple of 18 and 24.
18 = 2 ⋅ 3 2
So, the LCM of 18 and 24 is LCM = 23 ⋅ 3 2 = 72 .
3
24 = 2 ⋅ 3
7.6 Common Factors
Factors that are common to two or more numbers are said to be common factors.
Highest Common Factor
The largest common factor of two or more numbers is called the highest common
factor (HCF).
Method I
For small numbers
For example:
8 = 1x 8 = 2 x 4
12 = 1x12 = 2 x 6 = 3 x 4
- Factors of 8 are 1,2, 4 and 8
- Factors of 12 are 1, 2, 3, 4, 6 and 12
So, the common factors of 8 and 12 are 1, 2 and 4
HCF is 4
Method II
General
To find the Highest Common Factor of two or more higher numbers:
- Find the prime factor decomposition.
- Choose only the common factors with the lowest exponents.
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Mathematics
Exercise 17 Factorise and then calculate the LCM and the HCF of these
numbers:
a) 360 and 300
b) 168 and 490
c) 12, 100 and 6
d) 14112, 1080 and 1008
e) 1600 and 1200
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f) 294, 1050 and 28
Exercise 18 We want to distribute 100 l of water in bottles which all have the
same capacity (a whole number of litres). Find out all the different solutions.
Indicate how many bottles we get in each case and the capacity of each one
Exercise 19 Sandra can pack her books in boxes of 5, 6 and 9. She has less
than 100 books. How many books has she got?
Exercise 20 We want to divide a rectangle of 600cm by 90 cm into equal
squares. Find out the length of the biggest square in cm. Calculate how many
squares we get.
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Mathematics
Exercise 21 Iberia has a flight from Madrid to Ankara every 8 days, British
Airways one every 12 days and Easy Jet one every 6 days; one day all three
have a flight to Ankara. After how many days will the three flights coincide
again?
Exercise 22 A group of students can be organized in lines of 5, 4 and 3
students and there are less than 100 students. How many are there?
Exercise 23 On a Christmas tree, there are two strings of lights, red lights flash
every 24 seconds and green lights every 36 seconds. They start flashing
simultaneously when we connect the tree. When will they flash together again?
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Mathematics
Solutions
Exercise 1
a) 86 424 Eighty six thousand, four hundred and twenty four.
987 Nine hundred and eighty seven
3 270 Three thousand, two hundred and seventy
30 001 Thirty thousand and one
1 487 070 One million, four hundred and eighty seven thousand and seventy
320 569 Three hundred and twenty thousand, five hundred and sixty nine
20,890,300 Twenty million, eight hundred and ninety thousand, three hundred
c) 0.056 Nought point zero, five, six
23.45 cm Twenty three point four, five cm
₤1.20 One pound, twenty or one twenty
3.77 Three point seven, seven.
5
Five fourths or five quarters or five over four
4
7
Seven halves
2
12
Twelve over thirty five
35
37
Thirty seven over one hundred or thirty seven hundredths
100
Exercise 2
a) 4 3 = 64 Four to the power of three is sixty four
b) 6 2 = 36 Six squared or to the power of two is thirty six
2
c) (− 11) = 121 Minus eleven squared or to the power of two is one hundred and
twenty one
d) 2 5 = 32 Two to the power of five is thirty two
e) 10 3 = 1000 Ten cubed or to the power of three is one thousand
f) 1000 2 = 1000000 A thousand squared (or to the power of two) is one million
3
g) (0.1) = 0.001 Zero point one cubed is zero/nought point zero, zero, one
Exercise 3
− 5 < −2 < 0 < 3 < 8
Exercise 4
a) 8 − (3 − 2 + 5) = 8 − 6 = 2 , b) (87 − 4 − 9) + (2 + 3 − 1) = 74 + 4 = 78
c) (2 − 7) − (5 − 4) − (1 − 6) = −5 − 1 + 5 = −1, d) (2 + 4 − 1) − (7 − 4 − 9) = 5 − ( −6) = 11
e) − (1 − 2 + 3 − 7) + (3 − 4) − (2 − 3 − 5) = −( −5) + ( −1) − ( −6) = 10
Exercise 5
a) 8 − (3 − 2 + 5) = 8 − 3 + 2 − 5 = 2 , b) (87 − 4 − 9) + (2 + 3 − 1) = 87 − 4 − 9 + 2 + 3 − 1
c) (2 − 7) − (5 − 4) − (1 − 6) = 2 − 7 − 5 + 4 − 1 + 6 = −1
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d) (2 + 4 − 1) − (7 − 4 − 9) = 2 + 4 − 1 − 7 + 4 + 9 = 11
e) − (1 − 2 + 3 − 7) + (3 − 4) − (2 − 3 − 5) = −1 + 2 − 3 + 7 + 3 − 4 − 2 + 3 + 5 = 10
Exercise 6
a) 5 − 4 + 7 − 9 + 12 − 1 + 5 − 10 = 5 , b) 4 − 3 + 21 − 32 − 5 + 6 = −9 , c) 5 − [2 − (6 − 9)] = 0 ,
d) 5 − [5 + (3 − 2 − 9)] − (3 − 10) = 15 , e) 1 − [(2 − 5) − 3 − (9 − 1)] − (2 − 3) = 16
Exercise 7
a) ( −3)( −2)( −5) = −30 , b) 6 ⋅ ( −3) ⋅ ( −1) = 18 , c) 15 : ( −3) ⋅ ( −1) = 5
d) (− 80 ) : [(− 8 ) ⋅ 2] = 5 , e) [( −80) : (− 8 )] ⋅ 2 = 20 , f) [9 ⋅ (− 8 )] : [( −3) ⋅ ( −4)] = −6
Exercise 8
a) 3(5 − 8) + 2(8 − 12) − 3 + 3 ⋅ ( −5) = −35 , b) 17 − [2 ⋅ 3 + 5( −2 + 4)] ⋅ 3 − 6 ⋅ ( −5) = −1
c) ( −2) 2 − ( −3) 2 + ( −1)3 = −6 , d) ( +4) 2 + ( −4) 2 − ( −4) 2 − 4 2 = 0 , e) 2 3 − 3 2 + 4 2 − 17 = 14
[
]
f) 5 2 ⋅ ( −3) 2 : ( −15) = −15
Exercise 9
a) 3(2 − 6 + 5) + 7(3 − 2(5 − 12)) = 122 , b) 2 ⋅ [ −3 ⋅ (5 ⋅ (3 − 4) + 1) − 2] = 20
3
c) (− 2) ⋅ [ −2 + 3 ⋅ (1 − 7)] : (1 − 3) = −80 , d) 3 ⋅ (5 − 8) + 2 ⋅ (3 − 12) − 5 + 3 ⋅ 2 = −26
Exercise 10
Exercise 11
a) 105, 112, 119, 126, b) 240, 255, 270, 285, c) 1233, 1242, 1251, 1260
Exercise 12 96, 102, 108
Exercise 13 1200, 1206, 1212, 1218, 1224, 1230, 1236, 1242, 1248
Exercise 14
a) 1, 2, 3, 6, 9, 18, b) 1, 2, 3, 5, 6, 9, 10, 16, 18, 30, 45, 90
c) 1, 2, 4, 5, 7, 10, 14, 20, 28, 35, 70, 140, d) 1, 2, 4, 5, 8, 10, 16, 20, 40, 80
e) 1, 2, 5, 10, 25, 50
Exercise 15
a) 2, 5, 8; b) 0, 2, 4, 6, 8; c) 0, 5; d) 1
Exercise 16
a) 2 ⋅ 23 , b) 2 2 ·3 2 ·5 , c) 2 2 ·3·5 , d) 2 2 ·3·5 3 , e) 3 3 ·5
Exercise 17
a) 2 2 ·3 2 ·5 , 2 2 ·3·5 2 , LCM 1800, HCF 60; b) 2 2 ·3·7 , 2·5·7 2 , LCM 5580, HCF 14
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c) 2 2 ·3 , 2 2 ·5 2 , 2·3 , LCM 300, HCF 2; d) 2 5 ·3 2 ·7 2 , 2 3 ·3 3 ·5 , 2 4 ·3 2 ·7 , LCM 211680,
HCF 72; e) 2 6 ·5 2 , 2 4 ·3·5 2 , LCM 4800, HCF 400; f) 2·3·7 2 , 2·3·5 2 ·7 , 2 2 ·7 , LCM
14700, HCF 14
Exercise 18
1 bottle of 100 l, 2 bottles of 50 l, 4 bottles of 25 l, 5 bottles of 20 l, 10 bottles of 10 l,
20 bottles of 5 l, 25 bottles of 4 l, 50 bottles of 2 l, 100 bottles of 1 l
Exercise 19
She has got 90 books.
Exercise 20
The sides of the biggest square are 30cm, we get 60 squares.
Exercise 21
The three flights will coincide again after 24 days.
Exercise 22
There are 60 students.
Exercise 23
They will flash together after 1 min and 12 seconds.
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Mathematics
2 Decimal and sexagesimal
system
Keywords
Regular numbers
irrational number
sexagesimal system
repeating decimals
rounding
period
number line
1 Decimal numbers
Remember that to express numbers that are not whole numbers we use
decimal numbers as 75.324 in which every digit has a value which is divided by
ten when we move to the right. So
7 is seventy units
5 is five units
3 is three tenths of a unit
2 two hundredths of the unit
4 is four thousandths of a unit
And we continue like that if there are more digits.
This is the decimal system that is commonly use nowadays except, sometimes,
for time and angles.
We read these numbers naming the whole part then “point” and then the
decimals digits one by one
Example The number 75.324 is read as seventy five, point, three, two, four
When the numbers express money or length can be read on a different way, for
example 5.24€ is read as five point twenty four euros or five euros and twenty
five cents or the number 5.36 m can be read as five point thirty six metres.
2 Types of decimal numbers
As a result of some operations we can get different types of decimal numbers:
Regular numbers: Are decimal numbers with a limited quantity of decimal
digits and from them all could be zeros.
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Example
Mathematics
14
= 2.8 we find an exact division
5
Repeating decimals: There is a group of digits that are repeated forever.
Examples
If we divide 4 by 3 we get 1.3333333…
Calculating
13
= 0.36363636...
33
The group of repeated decimal digits is called the period on the first case our
period is 3 on the second case the period is 36
)
The number 1.33333... must be written as 1.3 or 1. 3 and 0.36363636... as
∩
0. 36 or 0.36
Irrational numbers: They have an unlimited quantity of decimal digits but there
is not any period
Example: Calculating
2 we get 1.414213562… and we don’t find any
sequence on the digits we get.
3 Decimal numbers on the number line
Every decimal number has a place on the number line between two integers.
For example representing the numbers 2.3 or 1.4 we divide the units into ten
equal parts and we find a point to represent these numbers
There is a peculiar propriety with the decimals on the number lien that is that
between any two decimal numbers there is always a decimal number and it
doesn’t matter how close they are on the line.
Example between 2.4 and 2.5 we can find 2.44 and 2.4<2.44<2.5 or between
2.44 and 2.45 we can find 2.443 and 2.44<2.443<2.45, etc.
4 Rounding Decimal Numbers
Remember that to round a number to any decimal place value we look at the
digit to the right of the place we wish to round to and when the digit 5, 6, 7, 8, or
9, appears in that place, you must add one unit to the last digit; when the digit 0,
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1, 2, 3, or 4 appears in that place, you must cut the number.
Example 1.18 rounded to the nearest tenth is 1.2 while 3.453 rounded to the
nearest hundredth is 3.45
Exercises
1. Write in words
3.006
45.204
34.40 metres
2. Order from less to high
)
a) 5.1, 4.99, 5.21, 5.201, 4.9
b) –2, -1.78, 1.5, 1.8, 1.09
c) 0.0003, 0.1, 0.007, -1
3. Find two numbers between each pair of decimal numbers
a) 2.5 and 2.6
b) 7.07 and 7.08
c) –2.01 and –2
d) 5.203 and 5.21
e) - 0.001 and - 0.0009
f) 78.00003 and 78.00005
4. Plot on the number line the decimal numbers:
a) 1.3
b) 2.34
c) –2.5
d) 0.347
e) 79.45
(Don’t use the same number line for all the numbers!).
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5 Round:
2.36 to the nearest tenth
6.3757 to the nearest hundredth
0.8903 to the nearest thousandth
17.17 to the nearest unit
176.705 to the nearest hundred
67,756 to the nearest thousand
5 Sexagesimal system
There are some magnitudes as angles and time in which the decimal system is
not the only one that is used. Sexagesimal system is more frequently used; on
this system every unit is divided into 60
equal parts to get the subunit.
For the angles the unit is the degree.
The subunits of the degree are the
minute and the second.
One minute 1′ =
1
of a degree, that is
60
1º = 60′
One second 1′′ =
Inclinometer
1
of a minute, that is 1′ = 60′′
60
Using this system an angle “a” can be expressed for example a = 43º 43′ 20′′
and we need to operate angles expressed in this form.
For time the unit as you know is the hour and is divided into minutes and
seconds with the same relationship they have on the angles.
A period of time is expressed as 3 h 5 min 3 s for example.
We need to be able to operate in this system using these two magnitudes.
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Exercises
1. Convert into minutes
a) 7º
b) 21º
c) 0.3º
d) 3º 4’ 55’’
e) 5.67º
2. Convert into seconds
a) 2 min
b) 37 min
c) 1 h 12 min
d) 2.5 h
e) 3.47 min
3. Convert into hours
a) 345 min
b) 5 h 35 min
c) 7 h 70 min
d) 5400 s
4. Convert into degrees
a) 3700’’
b) 23º 34’ 57’’
c) 340’
5.1 Addition.
We need to add separately degrees or hours minutes and seconds and then
convert the seconds into minutes and the minutes into degrees/hours if we get
more than 60 subunits.
Example
Add 35º 23′ 30′′ + 25º 53′ 58′′
Adding separately we get 35º 23′ 30′′ + 25º 53′ 58′′ = 60º 76′ 88′′ and as
88′′ = 1′ 28′′ we add 1′ and get 77′ = 1º 17′ we add 1º and the solution is
61º 17′ 28′′
5.2 Subtraction
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We need to subtract separately degrees/hours minutes and seconds, if we do
not have enough seconds or minutes we convert one degree/hour into minutes
or a minute into seconds.
Example
Subtract 3 h 25 min 34 s and 1 h 46 min 50 s; we write 3 h 25 min 34 s as 2 h
84 min 94 s and
2 h 84 min 94 s
- 1 h 46 min 50 s
1 h 38 min 44 s
5.3 Multiplication by a whole number
We multiply separately degrees/hours minutes and seconds and then convert
the seconds into minutes and the minutes into degrees when we get more than
60 subunits.
Example
Multiply (12º 33′ 25′′) ⋅ 4
12º
4
33’
4
25”
4
48
133
13’
2º 13’
100
1’ 40”
50º
Solution 50º 20’ 40”
5.4 Division by a whole number
We divide the degrees/hours, and the remainder is converted into minutes that
must be added to the previous quantity that we had, divide the minutes and we
repeat the same that we have done before. The remainder is in seconds.
Example
Divide (34 h 25 min 55 s ) : 4
34 h
2 h x 60 =
13 min
120 min
25 s
4
8 h 33 min 21” quotient
133 min
1min x 60 =
60 s
85 s
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Exercises
1 Add:
a) 45º 55’ 31” + 56º 41’ 34”
b) 39 h 55 min 17 s+ 4 h 14 min 33 s
c) 233º 5’ 59” + 79º 48’ 40”
2 a) Subtract 56 h 24 min 16 s and 19 h 35 min 43 s
b) Calculate the complement of 33º 55’ 43”
c) Calculate the supplement of 102º 55’
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3 Given A = 52º 12’ 27” Calculate: a) 5·A
b) 4·A
c)
A
3
4 We want to divide the full turn into seven equal sectors. Which is the
angle of each sector?
5 A train arrives at a station at 17 h 35 min after a travel of 3 h 45 min, at
what time did the train departed?
6 A sportsman starts its training at 8 h 43 min he runs go and back and
spends 1 h 23 min 40 s going and 1 h 45 min 50 s coming back. At what
time does it finish its training?
7 I spend 1 h 34 min going from A to B riding my bicycle. Running I need
the double of the time and using my car I need the third. How long does it
take to me going from A to B running and how long driving?
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Solutions
Exercises I
1 3.006 Three point zero, zero, six.
45.204 Forty five point two, zero, four.
34. 40 Thirty four point forty metres
)
2 a) 4.99, 4.9 , 5.1, 5.21, 5.201
b) –2, -1.78, 1.09, 1.5, 1.8
c) -1, 0.0003, 0.007, 0.1
3 (Examples)
a) 2.5 < 2.55 < 2.56 < 2.6 b) 7.07 < 7.075 < 7.076 < 7.08
c) –2.01 < –2.002 <–2.003 < –2 d) 5.203 < 5.204 < 5.205 < 5.21
e) –0.001< – 0.00096 < – 0.00095 < –0.0009
f) 78.00003 < 78.00004 <78.000046 < 78.00005
4 We must decide the appropriate scale on each case
5
2.36 ≈ 2.4
6.3757 ≈ 6.38
0.8903 ≈ 0.890
17.17 ≈ 17
176.705 ≈ 176 .71
67756 ≈ 68000
Exercises II
)
1. a) 420′ ,b) 126 0′ , c) 18′ , d) 184.916′ , e) 340 .2′
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2. a) 120′′ , b) 222 0′′ , c) 432 0′′′ , d) 900 0′′ , e) 208 .2′′
)
)
3. a) 5.75h , b) 5.583h , c) 8.16h , d) 1.5h
)
)
4. a) 1.027º , b) 23.5825 º , c) 5.6º
Exercises III
1 a) 102º 37′ 5′′ , b) 44 h 9 min 50 s , c) 312º 54′ 38′′
2 a) 36 h 48 min 33 s , b) 56º 4′ 17′′ , c) 77º 5′
3 a) 261º 2′ 15′′ , b) 208º 49′ 48′′ , c) 17º 24′ 9′′
4 The angle of each sector is 51º 25′ 43.8′′
5 The train departed at 13 h 50 min.
6 The sportsman finishes its training at 11 h 52 min 30 s
7 Running I need 3 h 8 min. If I go by car I need 31min 20 s
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3 Fractions
Keywords
Fraction
numerator
denominator
simplest form
improper fractions
power
exponent
index
standard form
root
Remember:
1 Fractions
A fraction is a number that expresses part of a unit or a part of a quantity.
Fractions are written in the form
a
where a and b are whole numbers, and
b
the number b is not 0.
They can be written also in the form a/b
The number a is called the numerator, it is always an integer, and the
number b is called the denominator, it can be any whole number except
zero.
The denominator is the number, which indicates how many parts the unit is
divided into.
The numerator of a fraction indicates how many equal parts of the unit are
taken.
4
6
represents the shaded portion of the
rectangle
2 Reading fractions
We use the cardinals to name the numerator and the ordinals for the
denominator with two exceptions, when the denominators are 2 and 4, for
denominators larger than 10 we can say “over” and do not use ordinal, so we
read:
1
3
one half
three halves
2
2
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2
two thirds
3
1
a quarter or a fourth
4
12
twelve over fifteen or twelve fifteenths
15
3
three quarters or three fourths
4
6
six tenths
10
17
seventeen over thirty-two
32
Exercise1 write in words the following fractions
1
3
2
5
3
7
4
10
5
12
6
19
7
200
3
10
524
1000
3456
5461
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Exercise 2 express in figures:
Six sevenths
four elevenths
A half
three quarters
seventeen over three hundred and forty one
thirty two over five hundred and twenty two
sixty two over seventy one
3 Equivalent fractions
Equivalent fractions are different fractions
that name the same amount.
4
2
Example
and
are equivalent as can
6
3
be seen in the drawing on the right
The rule is:
The value of a fraction does not change
multiplying or dividing its numerator and
denominator by the same number
The process of dividing numerator and denominator by the same number is
called reduction
12
3
is equivalent to , because we have divided both the numerator and the
20
5
denominator by 4.
The fractions
1 2 3 100
512
, , ,
and
are all equivalent fractions.
2 4 6 200
1024
We can test if two fractions are equivalent by cross-multiplying their
numerators and denominators. This is also called taking the cross-product.
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a
b
So if we want to test if
c
d
12
24
and
are equivalent fractions
20
40
The first cross-product is the product of the first numerator and the second
denominator: 12 × 40 = 480.
The second cross-product is the product of the second numerator and the
first denominator: 24 × 20 = 480.
Since the cross-products are the same, the fractions are equivalent.
Simplest form
When numerator and denominator have no common factors the fraction is in
the simplest form or in its lowest terms
We know that 4/12 = 2/6 = 1/3
4 and 12 have a common factor (4), so 4/12 can be written as 1/3 (Divide the
top and the bottom by 4.)
2 and 6 have a common factor (2), so 2/6 can be written as 1/3 (Divide the
top and the bottom by 2.)
However, 1 and 3 have no common factors, so 1/3 is the simplest form of
these fractions.
There are two methods of reducing a fraction to the lowest terms.
Method 1:
Divide the numerator and denominator by their HCF.
12
12 12 : 6 22
. The HCF of 12 and 30 is 6 so
=
=
30
30 30 : 6 5
Method 2:
Divide the numerator and denominator by any common factor. Keep dividing
until there are no more common factors.
12 12 : 2 6
6:3 2
=
=
=
=
30 30 : 2 15 15 : 3 5
Exercise 3
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a) Write a sequence of equivalent fractions as in the example of the first
line
Start
1
2
2
5
3
7
1
−
3
2
4
3
6
Equivalent fractions
4
5
8
10
b) Express these fractions in the simplest form
20
44
26
52
29
58
70
119
84
119
42
238
20
48
56
84
22
66
34
68
52
117
54
117
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6
12
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39
234
205
445
72
272
324
720
720
864
272
425
510
561
3360
3528
3553
5643
7429
8303
c) Express as its simplest form a fraction that describes these
situations
1 I have 51 pens and 9 of them are black.
2 In our school 48 of the 84 teachers are women
3 Count how many boys and girls are in our class of 2º ESO and write
down the fraction of each compared with the total number of pupils.
4 Comparing and ordering fractions
1. To compare fractions with the same denominator, look at their numerators.
The largest fraction is the one with the largest numerator.
2. To compare fractions with different denominators, take the cross product.
Compare the cross products.
a. If the cross-products are equal, the fractions are equivalent.
b. If the first cross product is larger, the first fraction is larger.
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c. If the second cross product is larger, the second fraction is larger.
3. We can also compare fractions dividing the numerator by the denominator
5 Adding and subtracting fractions
1. If the fractions have the same denominator:
The numerator of the sum is found by simply adding the numerators over the
denominator.
Their difference is the difference of the numerators over the denominator.
We do not add or subtract the denominators!
Reduce always when possible.
2. If the fractions have different denominators for example
3 7
+
5 8
follow these steps:
1) Reduce them to a common denominator.
For this problem complete the following steps:
a) We find the LCM of the denominators as it is the smallest number that
both denominators divide into. For 5 and 8, it is easy to see that 40 is
the LCM. We look for equivalent fractions with 40 as common
denominator.
In our example
3 [ ]
=
5 40
and
7 [ ]
=
8 40
b) We divide every new denominator by the previous one and then we
multiply the result by the numerator.
So in
3 [ ]
=
5 40
we divide the new denominator 40 by the previous one that is
5, what gives us 8, we must multiply 3 by 8, so
3 24
=
.
5 40
Repeating the process with the second fraction:
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7 [ ] 35
=
=
8 40 40
We have multiplied the top and the bottom by 5
2) Add the numerators and do not change the denominator.
3 7 24 35 24 + 35 59
+ =
+
=
=
5 8 40 40
40
40
3) Reduce if possible.
Exercise 4
Calculate
a)
1 7 5
− +
2 9 6
b) 5 −
3 17
+
4 10
c)
2 5
8
− +
6 9 27
d)
5
7 25
− +
26 6 39
e) 3 −
f)
1
1 4
+ 1− −
3
5 15
2 3 13 5
−
−
+
5 10 35 14
Operate and reduce when necessary
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7 4 1
a) 5 −  +  −
 5 7  35
b)
5 1 3 1
+ − −  + 3
3 2 2 4
 3 11 
c) 1 − 
+ 
 10 6 
 7 2
d) 11 − 
− 
 10 5 
 3  1    3 1 
e)  − 1 −  − 2 +  + 
 5  3    5 9 
3 
1    3
1  2
f) 1 +  +  5 −  − 3 +  +
 −
4    5 10  15
2 
1 
1   3  2 1
7 
g) −  −  2 −  +  −  +
−

6  10  3 10 30 
5 
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6 Improper fractions, mixed numbers
Improper fractions have numerators that are larger than or equal to their
denominators.
For example
15 7
18
, , and
are improper fractions.
7 7
3
Mixed numbers have a whole number part and a fraction part.
3
1
For example 2 , 5 are mixed numbers, meaning
5
3
2
3
3
1
1
= 2 + and 5 = 5 + .
5
5
3
3
Converting improper fractions to mixed numbers
To change an improper fraction into a mixed number, divide the numerator by
the denominator. The quotient is the whole part and the remainder is the
numerator of the fractional part.
For example
 quotient 5
17
17
2
, 17 : 3 ⇒ 
so
=5
3
3
3
remainder 2
Converting mixed numbers to improper fractions
To change a mixed number into an improper fraction, multiply the whole
number by the denominator and add it to the numerator of the fractional part.
For example
7
2 7 ⋅ 3 + 2 23
=
=
3
3
3
Note that converting mixed numbers to improper fractions is the same as
adding whole numbers and fractions
Exercise 5
Convert the mixed numbers to improper fractions, operate and convert
the results to mixed numbers.
a) 5
1  2 4
2
− 3 +  + 4
3  3 5
15
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5 13  3 1 
5
b) 3 + − 1 −  + 3
3 2  2 4
12
c) 7
3
1 12
+2 −
9
3 27
7 Multiplying fractions
When two fractions are multiplied, the result is a fraction with a numerator
that is the product of the fraction’s numerators and a denominator that is the
product of the fraction’s denominators. Reduce when possible.
Examples:
7 2 7⋅2
7
7
⋅ =
=
=
We cancel the common factor of 2 in the top and
6 5 6 ⋅ 5 3 ⋅ 5 15
bottom of the product. Remember that like factors in the numerator and
denominator cancel out.
7 2 7⋅2
7
7
⋅ =
=
=
6 5 6 ⋅ 5 3 ⋅ 5 15
Exercise 6
Operate and reduce when possible
a)
7 1
⋅
3 14
b)
3 −5
⋅
5 13
c) −
d)
5  2 
⋅− 
6  15 
12
5
⋅3⋅
5
27
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 26   1   9 
e)  −
⋅− ⋅−

 3   2   39 
4  1 1

f) 1 −  ·  + 
7 3 2

5 25 
2

g)  − 2  · 1 − −

4 12 
7

8 Multiplying a fraction by a whole number,
calculating a fraction of a quantity
To multiply a fraction by a whole number you must choose one of the two
methods:
1. Write the whole number as an improper fraction with a denominator of
1, and then multiply as fractions.
2. Multiply the whole number by the numerator and do not change the
denominator.
Example:
a) 6 ⋅
2 6 2 12
= ⋅ =
7 1 7 7
Exercise 7
Calculate:
a)
7
of 108
12
b)
2
of 91
7
c)
7
of 112
8
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d)
10
of 1311
23
Calculate the unknown number in the following cases:
a)
3
of a number is 27, so the number is:
8
b)
2
of a number is 64, so the number is:
3
c)
17
of a number is 629 so the number is:
22
d)
2
of a number is 360 so the number is:
17
9 Dividing fractions
To divide fractions, multiply the first by the reciprocal of the second fraction.
The reciprocal of a fraction is obtained by switching its numerator and
denominator.
We can also take the cross product.
To divide a number by a fraction, multiply the number by the reciprocal of the
fraction.
Example:
4 7 4 ⋅ 11 44
4 7 4 11 44
:
= ⋅ =
or simply taking the cross product :
=
=
5 11 5 7 35
5 11 5 ⋅ 7 35
Exercise 8
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a)
5 3 7
: + 
4 2 4
1   1 1

b)  2 −
: + 
12   3 2 

 2 1  11
c)  −  :
 5 4  20
 1 1  1 1
d)  −  :  + 
3 5 3 5
e)
5 1 3 1
⋅ − :  : 2
3 2 2 4
 2  1    3 9  
f)  : 1 −  ⋅ 2 −  :

 5  5    5 20 
1
2
g)
3
2−
4
1+
1 3
−
7
h) 3 2 +
1 8
5+
3
3-14
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Exercises
3
of the passengers
5
get off. How many people get off the train? How many people are left on
the train?
9 There are 300 passengers on a train. At a station,
10 Allan has 120€. He decides to save
2
1
of this and to spend
on
5
6
books. How much does he save? How much does he spend on books?
How much is left?
11 In a magazine there are three adverts on the same
1
1
uses of the page, advert 2 uses and advert 3 uses
4
8
What fraction of the page do the three adverts use?
3
An advert uses
of the page, if the cost of an advert
16
1
of the page, How much does it cost?
32
3-15
page. Advert 1
1
of the page.
16
is 12€ for each
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12 A farmer owns 360 hectares of land. He plants potatoes on
3
of his
10
1
of the remainder. How many hectares are planted
6
with potatoes? How many hectares are planted with beans? How many
hectares are left?
land and beans on
13 A journey is 120 miles. Richard has driven
3
of this distance and in
5
5
of the rest. How much farther does he have to drive
6
to complete the journey?
a second stage
1
1
of her money and she spent
on
4
8
seeing a movie. Which part of her money did she spend?
14 Sue bought a record with
3-16
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3
of their original price and the sale price
5
is 35€. What was the original price?
15 At a sale shirts are sold by
3
3
of his salary on his own, gives
of the remainder to
8
5
his parents and saves 450€ what is his salary?
16 Joe spends
9 Powers
A power is a mathematical operation that indicates that many equal
numbers are multiplied by themselves, it is written as
base and n is the exponent or index.
The meaning is:
Examples:
53 = 5x5x5 = 125
3-17
an
where a is the
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(− 3)4 = (− 3) ⋅ (− 3) ⋅ (− 3) ⋅ (− 3) = +81
3
2
2 2 2  8 
  =  x x  =  
3
 3   3   3   27 
Remember how to name powers
5 4 Is read as
-
Five to the power of four
The forth power of five.
Five to the forth power
We will read it as five
to the power of four
There are two special cases with exponents 2 and 3 (squares and cubes), so
we read, for example, 7 2 as seven squared and 6 3 as six cubed.
10 Rules for powers
1 Product
When doing the product of two powers with the same base, the base
remains unchanged and the exponents are added
am ⋅ an = am +n
Example: a 3 ⋅ a 4 = (a ⋅ a ⋅ a ) ⋅ (a ⋅ a ⋅ a ⋅ a ) = a 3+ 4 = a 7
Exercise 17 Fill in the missing numbers.
a) 3 3 ⋅ 3 7 == 3 [ ]
b) 7 5 ⋅ 7 8 = 7 [
e)
[ ] ⋅ [ ]4 = 25
]
c)
[ ]2 ⋅ 6 7 = 6 [
f) 5 3 ⋅ 5 7 == 5 [ ]
2 Divisions
3-18
]
d) 2 5 ⋅ 2 [ ] = 2 9
g) 23 ⋅ [
][
]
= 29
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In the quotient of two powers with the same base, the base remains
unchanged and the exponents are subtracted
a m : a n = a m −n
Example:
Exercise 18 Fill in the missing numbers.
a)
715
7
12
= 7[ ]
d) 3 [ ] : [
2
]
b) 1213 : [
=3
7
e)
318
13
[ ]
]7 = 12[
][
:= [
]
c) 13 5 : [
]
f)
]3 = 13 2
[ ]2 : [ ]9 = 9 [
]
3 Power of a product
The power of a product is the product of the powers
(a ⋅ b )n = a n ⋅ b n
4
Example: (a ⋅ b ) = (a ⋅ b ) ⋅ (a ⋅ b ) ⋅ (a ⋅ b ) ⋅ (a ⋅ b ) = (a ⋅ a ⋅ a ⋅ a ) ⋅ (b ⋅ b ⋅ b ⋅ b ) = a 4 ⋅ b 4
4 Power of a quotient
The power of a quotient is the quotient of the powers
n
n
(a : b )
an
a
n
n
= a : b or   = n
b
b
4
4
a
 a   a   a   a  (a ⋅ a ⋅ a ⋅ a ) a
Example:   =   ⋅   ⋅   ⋅   =
= 4
b 
 b   b   b   b  (b ⋅ b ⋅ b ⋅ b ) b
5 Power of a power
When powering another power we multiply the exponents
m n
(a )
= a m ⋅n
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( )
Example: a 4
3
= a 4 ⋅ a 4 ⋅ a 4 = a 4 + 4 + 4 = a 4⋅3 = a 12
Exercise 19 Fill in the missing numbers.
( )
a) 4 2
d)
([
5
(
= 4[ ] = 2[ ]
]2 )
3
2
b) (− 3 )
( )[
= 56
e) 2 2
)[
]
]
= 38
=[
]8
(
c) 3 [ ]
)
2
= 38
6 Negative exponents
a −n =
1
an
Example: 2 −3 means
1
2
3
=
1
8
Exercise 20 Fill in the missing numbers.
a)
( )−3 =
c)
([
1
27
]−2 )
−3
−3
b) (7)
= 56
( )[
d) 2 2
1
1
= [ ]=
[ ]
7
]
=[
]−8
7 Scientific notation, standard form
There is a way of writing numbers that are too large and also for numbers
that are too small using powers of ten. Big numbers are written in the form:
a × 10 b , where b is a whole and positive number.
Example 70 000 = 7 x10 4 and 237 000 000 = 2.37 x10 8
Small numbers can be written as a × 10 b , where b is negative.
Example: 0.0037 =
3 .7
3.7
= 5 = 3.7 x10 −5
100000 10
Exercise 21 Express in standard form the following numbers:
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a) 34,000,000,000
b) 3.5 billions
c) 357,650,000
d) 0.034
e) 0.000000056
f) The number of seconds in a year (round appropriately)
g) The length of your class room in km
h) Calculate how many litres of water there are in a swimming pool
that measures 25m length, 8.5m width and 2 m depth
Exercise 22 Write using the ordinary decimal notation
a) 5.43 x10 8
b) 1.05 x10 2
3-21
c) 2.055 x10 13
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d) 7.26 x10 −2
e) 7 x10 −7
f) 90000 x10 −7
Exercise 23 Express as just one power
a) 7 3 ⋅ 7 −5 =
b) 412 ⋅ 4 −3 ⋅ 4 6 ⋅ 4 =
c) 9 2 ⋅ 9 7 ⋅ 9 −2 =
d) 5 7 : 5 3 =
57
e)
5
−3
( )
13
g)
2
13 ⋅ 13
3
i) 112 ⋅ 113
)
(
4
f) 14 2
=
6
[ (
(
=
)
=
h) 5 2 ⋅ 3 2 = [
=
 62
j)  2
3
k) 3 8 : 3 −2 ⋅ 3 3
)]
2
l)
=

 = [

57 ⋅ 53
(5 )
4 2
]2
]2
=
Exercise 24 Calculate
3
5
14
b) (− 1)
c) (− 2)
d) (− 1)
e) (− 3 )
f) − 3 4
g) 2 −3
h) 10 −5
i) 7 −2
j) 2 −3
a) 5 0
4
( )
3
8 Fractional indices roots
A power with a fractional exponent means
1
an
=na
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Examples:
1
2
a
= a
1
3
a
=3a
1
4
a
=4a
1
n
a
=na
Exercise 25 Convert into a root and calculate the value of the following:
1
1
a) 100 2
b) 0.0013
 1
c)  27 3 




2
2
d) 27 3
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Solutions
1
1
2
3
One third
two fifths
three sevenths
3
5
7
4
5
four tenths or four over ten
five twelfths or five over twelve
10
12
6
7
six nineteenths or six over nineteen
seven over two hundreds
19
200
3
524
three tenths or three over ten
five hundred and twenty four
10
1000
thousandths or five hundred and twenty four over one thousand
3456
three thousand four hundred and fifty six over five thousand four
5461
hundred and sixty one.
6
7
3
three quarters
4
2 Six sevenths
four elevenths
4
11
A half
1
2
seventeen over three hundred and forty one
thirty two over five hundred and twenty two
sixty two over seventy one
17
341
32
522
70
71
3 a)
Start
2
5
3
7
1
−
3
4
10
6
14
2
−
6
6
15
9
21
3
−
9
Equivalent fractions
8
10
12
20
25
30
12
15
18
28
35
42
4
5
6
−
−
−
12
15
18
14
35
21
49
7
−
21
20 5 26 1 29 1 70 10 84 12 42
3 20 5
=
,
= ,
= ,
=
,
=
,
=
,
=
44 11 52 2 58 2 119 17 119 17 238 17 48 12
56 2 22 1 34 1 52 4 54
6
39
1 205 41
= ,
= ,
= ,
= ,
=
,
= ,
=
,
84 3 66 3 68 2 117 9 117 13 234 6 445 89
72
9 324
9 720 5 272 16 510 10 3360 20
=
,
=
,
= ,
=
,
=
,
=
.
272 34 720 20 864 6 425 25 561 11 3528 21
3553 17
7429 17
=
[: 11 y 19] ,
=
[: 23 y 19]
5643 27
8303 19
3
4
c) 1)
of my pens are black. 2) of the teachers in our school are women
17
7
b)
3-24
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5
119
2
1
16
3
, b)
, c)
, d) − , e)
,f)
9
20
27
3
5
35
Operate and reduce when necessary
47
17
107
25
41
7
a) 3 , b)
, c) −
, e) −
, g)
, d)
, f)
12
15
10
9
12
5
1  2 4
2 16  11 4  62
5 a) 5 −  3 +  + 4
=
− + +
=5
3  3 5
15 3  3 5  15
5 13  3 1 
5 14 13  5 1  41 37
b) 3 + − 1 −  + 3
=
+ − −  +
=
3 2  2 4
12 3 2  2 4  12 3
3
1 12 66 7 12 83
c) 7 + 2 −
=
+ −
=
9
3 27
9 3 27 9
1
3
1
4
5
6 a) , b) −
, c) , d) , e) − 1 , f)
, g) 4
6
13
9
3
14
4 a)
7 a) 63, b) 26, c)98, d) 570
The unknown numbers are: a)72, b)96, c)814, d)3060
8 a)
5
23
3
1
31
1
6
21
, b)
, c)
, d) , e)
, f)
, g)
, h)
13
10
11
4
6
3
5
32
9 180 passengers get off the train 120 passengers are left on the train.
10 Allan saves 48€, he spends on books 20€, there are 52€ left
11 The three adverts use
7
of the page. The advert costs 72€,
16
12 180 hectares of potatoes, 60 of beans, 192 hectares are left
13 Richard drives 72 miles, in a second stage 40miles, he has to drive 8
miles to complete the journey.
14 Sue spends
3
of her money.
8
15 The original price was 58.33 €.
16 Joe’s salary is 1200 €.
17 a) 3 3 ⋅ 3 7 = 3 [10 ] , b) 7 5 ⋅ 7 8 = 7 [ 13 ] , c) [ 6
e) [2] ⋅ [2] = 2 5 , f) 5 3 ⋅ 5 7 == 5 [10 ] , g)
4
]2 ⋅ 6 7 = 6 [ 9 ] ,
[6 ]
23 ⋅ [2] = 29
3-25
d)
2 5 ⋅ 2 [4 ] = 2 9 ,
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715
7
= 7 [3 ] , b) 1213 : [2] = 12 [6 ] ,
12
7
18 a)
e)
318
:= [3
13
[3]
2
5
10
4
8
−3
20 a) (3)
3
2
][6] , f) [9]2 : [9]9 = 9 [−7] .
( ) = 4 [ ] = 2[
[ ]
e) (2 ) = [2]
a) 4 2
c) 13 5 : [ 13] = 13 2 , d) 3 [9 ] : [3] = 3 7 ,
=
20 ]
(
2
, b) (− 3 )
)[ ] = 3 , c) (3 [ ] )
4
8
4
2
1
1
1
−3
−2
, b) (7) = [3 ] =
, c) [5]
[343]
27
7
( )
2 3
= 3 8 , d) [5]
( )
−3
= 56 ,
[−4 ]
= 5 6 , d) (2 2 )
−8
= [2]
21 a) 3.4 ⋅ 1010 , b) 3.5 ⋅ 1012 , c) 3.5765 ⋅ 10 8 , d) 3.4 ⋅ 10 −2 , e) 5.6 ⋅ 10 −8 f)
2.94336 ⋅ 10 8 ≈ 2.94 ⋅ 10 8 s in a year, f)
g) 4.25 ⋅ 10 5 litres.
22 a) 543,000,000 , b) 105 , c) 20,550,000,000,000 , d) 0.0726 ,
e) 0.0000007 , f) 0.009
(
)
2
23 a) 7 −2
, b) 416 , c) 9 7 , d) 5 4 , e) 5 10 , f) 14 8 , g) 13 −4 , h) 5 2 ⋅ 3 2 = [15 ] ,
 62
i) 1130 , j)  2
3

 = [2]2 , k) 3 6 , l) 5 2

24 a) 1, b) − 1 , c) − 32 , d) 1, e) 81, f) − 81, g)
1
1
1
, h) 0.00001 , i)
, j)
8
49
512
2
 31 
25 a) 100 = 100 = 10 , b) 0.001 = 0.001 = 0.1, c)  27  = 2 27 2 = 9 ,




1
2
1
3
3
2
d) 27 3 = 2 27 2 = 9
3-26
4 Proportions
Keywords
Ratio
proportion
mean
extreme
1 Ratio
A ratio is like a fraction. If we want to compare two quantities we can
divide both numbers, then we can express it
a) As a fraction
b) As a decimal number
c) As a ratio
Example 1 Comparing the numbers 6 and 2, we write
that 6 contains the number 2 three times.
Comparing now the numbers 5 and 2
We can write the comparison as:
a) A fraction
5
2
b) A decimal number 2.5
c) A ratio 5:2 (read “5 to 2”)
4-1
6
= 3 which means
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The three possibilities have the same meaning, the difference between a
ratio and a fraction is that in fractions we only use whole numbers and in
ratios we can also use decimal numbers.
As fractions and ratios mean the same thing, the properties of the
fractions can also be used in ratios, one important property is:
We can multiply or divide both terms in a ratio by the same number.
A ratio in its most simple form is the fraction in its lowest terms (always
expressed with whole numbers).
Example 2 These ratios are equivalent
12:8 = 3:2 (3:2 is the ratio expressed in its simplest form)
4.5:9 = 9:18 = 1:2. (1:2 is the ratio expressed in its simplest form).
Exercise 1 Express the following ratios in their most simple form
a) 16 to 8
b) 8 to 20
c) 4.5:5
d) 4.5:3
e) 3/7:2
f) 15:21
2 Proportions
When two ratios are equal, the four terms are in proportion or are a
proportion, so
3 6
=
or 3:7 = 6:14 (is read “three is to seven as six is to
7 14
fourteen) are a proportion.
If we write
a c
=
b d
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b and c are called the means
a and d are called the extremes
A fundamental property of proportions is that the product of the
extremes is equal to the product of the means; this is very useful when we
know three of the numbers of a proportion and want to calculate the fourth.
If we have
a x
a⋅c
= , we say b ⋅ x = a ⋅ c and then x =
.
b c
b
The rule is one mean is the product of the extremes divided by the
other mean.
If we have
a b
a⋅c
= and x =
x c
b
The rule is one extreme is the product of the means divided by the
other extreme.
Exercise 2 Write three different proportions with these ratios:
7
5
a) 3:2
b)
c) 2.3:5
d) 7.2 to 1
3
4
Exercise 3 Calculate the unknown number in these proportions
a)
2 5
=
3 x
b)
5
x
=
2 .4 8
c)
x 12
=
9 24
d)
2 9
=
x 2
Exercise 4
a) Which number has the same ratio to 5 as 4 to 9?
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b) Find x if
4 12
=
(x is called third proportional to 4 and 12)
12 x
c) Find x if
18 7
= (x is the fourth proportional to 18, 5 and 7)
5
x
d) Find x if
4 x
=
x 5
3 Direct proportions
We say that there is a direct proportionality between two magnitudes if
an increase on one magnitude causes a proportional increase on the other
and a decrease on the first quantity causes a proportional decrease on the
second.
Note that a direct proportion is the same as proportions we have seen
on the previous point.
A direct proportion is also called simple proportion.
The best way to recognize if two magnitudes are in a direct proportion
is to see if when we double one the other also doubles and if we half the first,
the other also halves.
When we are trying to find a number in an exercise involving
magnitudes in a direct proportion there are two methods:
I -Unitary method
1. We convert the proportion in 1:n or n:1 (the most convenient)
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2. We multiply by the third quantity.
Example 3: A man walks 5200m in 2
1
hours. How much will he walk in 7 h
2
at the same speed?
1. If he walks 5200 m in 2.5 h, in 1 h he will walk
5200
= 2080 m .
2 .5
2. In 7 h he will walk 2080 ⋅ 7 = 14560 m
II –The fractional method or proportion.
You must work as we have seen before with proportions.
Example: With the same data as in example 3:
5200 m x m
5200 ⋅ 7
=
⇒x=
= 14560 m
2 .5 h
7h
2 .5
It is a good idea to write always the units so we can see that we are
organising the quantities correctly.
Exercise 5 Complete the table with the cost in pounds of a piece of silk
Metres of silk
3
Cost in £
5
7
12
1
9
Exercise 6 Complete the table if the magnitudes are directly
proportional
Metres of silk
1
Cost in £
4
5
2
50
7
Exercise 7 Complete the table of the cost in € and the litres of petrol
bought
Petrol in litres
Cost in €
1
5
40
6.20
4-5
7
50
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Exercise 8 A phone call costs €0.25 each 2 minutes or fraction rounded
to the seconds. Complete the table.
Call length in min.
7.5
Cost in €
13min 25 s
12
Exercise 9 Richard earns £17.5 for working 7 hours. How much will he
earn for working 9 hours.
Exercise 10 We have paid for 7 nights in “Hotel los Llanos” 364€. How
much will we pay for 3 nights? How much for 15 nights?
Exercise 11 For cooking a cake for 6 people the recipe says that we
need 3 eggs, 150g of flour and 50 g of sugar. Calculate how much of
each ingredient we need to cook a cake for 9 people.
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Exercise 12 Mara has exchanged $370 and has received 255€ without
any commission. How much in € will she receive for $20? How much in
$ will she receive for 60€?
Exercise 13 A 25 kg tin of paint covers 70 m2 of wall. How many kg
would be needed to cover 53 m2 of wall?
Exercise 14 My car uses 16 litres of petrol to travel 250 km.
a) How far can I travel with 55 litres?
b) How much petrol would I need to travel 180 km?
4 Inverse proportions
We say that there is an inverse proportionality between two
magnitudes if an increase in one magnitude causes a proportional decrease
in the other and a decrease in the first magnitude causes a proportional
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increase in the other. That is if one magnitude is multiplied by 2, 3, … this
causes in the second a division by 2, 3, … etc.
Example 4: If 18 men can do a job in 10 days, in how many days will 45 men
do the same job?
This is an inverse proportion because with double the men, half the days are
required. We will work in two steps.
1. Write the proportion (be careful! The same magnitude on each side)
18 men 10 days
=
45 men
x days
2. Make the inverse in one ratio
18 men
x days
=
45 men 10 days
3. Solve as we have done previously x =
18 ⋅ 10
= 4 days .
45
Exercise 15 Two pumps take 5 days to empty a pool. How long will 5
pumps take to empty the same pool?
Exercise 16 At 65 km/h a journey takes 5 h 25 min. how long will the
journey take at a speed of 75 km/h?
Exercise 17 It takes 12 hours for 3 bricklayers to
build a wall. How long will it take for 5 bricklayers?
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Exercise 18 At 130 km/h a train takes 1h and 23 min for its journey from
Albacete to Valencia. How long will the same journey take with the AVE
at a speed of 190km/h? How long with a speed of 250 km/h?
Exercise 19 A company needs 33 workers to pack its production in 25
days, if the total production needs to be packed in 15 days. How many
extra workers do they need?
Exercise 20 James can write 8 pages with 25 lines per page in one hour.
How many pages can he write if there are 20 lines on each page?
Exercise 21 If 2 eggs take 6 minutes to boil, how long will 5 eggs take?
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Exercise 22 There have been 13 winners in a lottery and each one will
receive 23000€, but there are 3 more winners, how much will each one
receive now?
Exercise 23 If i ride my bicycle at an average speed of 15 km/h I travel a
distance of 22 km in a certain period of time, if the speed is 17 km/h,
how far will I travel?
Exercise 24 Crash barriers are to be put on a stretch of motorway. We
need 56 pieces of material which are each 2.5 m long. How many pieces
of 3.5 m will we need for the same stretch?
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Solutions
Exercise 1 a) 2 to 1, b) 2 to 5, c) 9:10, d) 3:2, e) 3:14, f) 5:7.
Exercise 2 (examples) a) 3:2 as 6:4, b)
7 14
=
, c) 2.3:5 as 4.6 to 10
5 10
3
as 14.4 to 3.5.
4
)
)
Exercise 3 a) x = 7.5 , b) x = 16.6 , c) x = 4.5 , d) x = 0.4 .
d) 7.2 to 1
)
)
Exercise 4 a) x = 2.2 , b) x = 36 , c) x = 1.94 d) x = 20 ≈ 4.47 .
Exercise 5
Metres of silk
Cost in £
3
5
7
11.67
5.4
9
12
20
1
1.67
Exercise 6
Metres of silk
Cost in £
1
12.5
0.4
5
4
50
2
25
0.56
7
Exercise 7
Petrol in litres
Cost in €
1
1.24
32.26
40
5
6.20
7
8.68
40.32
50
96
12
13min 25 s
1.68
Exercise 8
Call length in min. 7.5
Cost in €
0.94
Exercise 9 £22.50, (it is a direct proportion).
Exercise 10 156€, 780€, (it is a direct proportion).
Exercise 11 4.5 eggs, 225g of flour and 75 g of sugar (it is a direct
proportionality).
Exercise 12 For $20 she will receive 13.78€, for 60€ she will receive 87.06$,
(it is a direct proportion).
Exercise 13 18.93 kg of paint, (it is a direct proportion).
Exercise 14 a) 859.4 km, 11.52 l. (it is a direct proportion).
Exercise 15 (Inverse proportion)
2 pumps → 5 days
x=
2·5
= 2 days
5
5 pumps → x days
65·5.42
Exercise 16 (It is an inverse proportionality) x =
= 4 h 42 min .
75
12·3
Exercise 17 (It is an inverse proportionality) x =
= 7 h 12 min .
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Exercise 18 x =
Mathematics
130·1.38
130·1.38
= 0.94 h = 56 min , x =
= 0.72 h = 42 min .
190
250
33·1.25
= 55 , so they need 22 extra workers.
15
25·8
Exercise 20 x =
= 10 pages .
20
Exercise 19 x =
Exercise 21 6 minutes, there is not any proportionality.
13·2300
= 18687.50 €.
16
17·22
= 24.93 km.
Exercise 23 (it is a direct proportionality), x =
15
Exercise 22 (It is an inverse proportionality), x =
Exercise 24 (It is an inverse proportionality), x =
4-12
56·2.5
= 40 pieces.
3 .5
5 Percentages
Keywords
Percentage
percent
increase
decrease
Interest
principal
rate interest
mixture
proportional division
1 Percentage
A percent is a ratio to 100.
A percentage can be considered as a fraction with denominator 100.
Percent and hundredths are basically equivalent. This makes
conversion between percent and decimals very easy.
To convert from a decimal to a percent, just move the decimal 2 places
to the right.
To convert a fraction into a percentage we multiply the numerator by
100 and we make the division.
There are some very easy cases of percentages that we can match
mentally to fractions such as 50% =
1
1
1
1
, 25% = , 20% = , 10% =
, etc.
2
4
5
10
Example 1
a) The ratio 3:8 or the fraction
3
3 ⋅ 100
can be expressed as
= 37.5% .
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Mathematics
10 10 ⋅ 100
=
= 200%
5
5
Summarising ideas we can write:
As a ratio or a fraction
As a decimal number
As a percentage
3:8
0.375
37.5%
10
5
2
200%
Making the division ⇒
Multiplying by 100 ⇒
Exercise 1 Complete the table doing the calculations in your notebook.
Ratio or fraction
Decimal number
Percentage
30%
70%
0.27
1.5
2
35
7
12
0.68
28.3%
2.8:5
Exercise 2 Convert the following percentages to fractions.
a)
35%
b) 3.4%
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c) 560%
d) 40%
e)
f)
45%
g) 15%
0.7%
h) 0.02%
Exercise 3 Convert the following to percentages.
a) 0.15
c)
1
6
b) 0.634
5
12
d)
e) 0.12
f) 1.34
g) 0.06
h) 0.75
i)
3
5
j)
1
16
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2 Calculating a percentage of a quantity
We know that for calculating a fraction of a quantity we must multiply
the fraction by the number
Example 2
5
5
of 200 is ⋅ 200 = 125 it is the same with percentages
8
8
To calculate the percentage of a quantity we must multiply it by the
percent and divide by 100.
Example 3
Calculate the 35% of 28 =
35
⋅ 28 = 9.8
100
Exercise 4 Calculate
a) 25% of 68
b) 32.5% of 500
c) 70% of 25 h 45 min
d) 1% & 3% of 50
e) 1% & 5% of 600
f) 1%, 0.5% & 6% of 500
1
g) 5 % of 800
2
h) 15% of 5€
i) 7% of 120$
Exercise 5 The population of a town is 652000 and 35% of them live in
the centre district. How many of them live in this district?
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Exercise 6 You have the possibility of choosing a prize:
Prize 1: 25% of 570€. Prize 2: 30% of 450€. Prize 3: 95% of 150€
Which one do you choose? Explain your answer.
3. Calculate the total from the percent.
Calculating the total from the percent and the part can be done using
direct proportion.
Example 4 In the class 13 students didn’t do their homework; this was 52% of
the class. How many students are in this class?
We can say 52 ------------ 100 total
13 ------------ x total
so x =
13 ⋅ 100
= 25 students.
52
Exercise 7 In a sale the price of a television set is 150€ which is 65% of
the usual price, what was the original price?
Exercise 8 The 6% of the population of Albacete are immigrants and
there are 9900 immigrants living in our city, what is the population of
Albacete?
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4 Percentage increase decrease
4.1 Calculate a number increased or decreased in a
percentage
It can be done in two steps:
1. Calculate the % of the quantity
2. Add or subtract the percentage to the original quantity.
Example 5 the population of a town is 63500 and last year it increased by
8%, what is the population now?
1. 8% of 63500 are
63500 ⋅ 8
= 5080
100
2. The current population is 63500 + 5080 = 68580 people
We can also use a formula, if a is the % increase, c the initial quantity and f
a 

the final quantity, then: f = 1 +
 ⋅ c , in the same example, we can say
 100 
8 

f = 1 +
 ⋅ 63500 = 1.08 ⋅ 63500 = 68580 people .
 100 
Example 6 The price of some clothes is 68€ and there is a discount of 7%,
what is the final price?
1. The discount is 7% of 68,
68 ⋅ 7
= 4.76€
100
2. The final price is 68 – 4.76 = 63.24€
We can use a similar formula, being a is the % decrease, c the initial
a 

quantity and f the final quantity, f = 1 −
⋅c .
 100 
4.2 Finding the original amount
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If we know the % increase or decrease and the value we can find the
total using proportions.
Example 7 The net salary of an employee is 1230€ after paying 18% of IRPF,
what is the gross of his/her salary?
82€ net --------- 100€ gross
1230€ net --------- x€ gross
and x =
1230 ⋅ 100
≈ 1500€
82
Example 8 I have bought a pair of jeans for €32, the IVA is 16%, what was
the price before IVA?
100€ net --------- 116€ plus IVA
x € net ----------- 32€ plus IVA and x =
32 ⋅ 100
≈ 28€
116
Exercise 9 For each quantity including 12% IVA calculate the original
cost excluding IVA.
a) 147.84 €
b) 65€
c) $2072
d) 500€
Exercise 10 Calculate the price of the following items before 15%
discount.
a) Jacket £200
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b) Car £15800
c) Shoes £80
Exercise 11 The price of an electric oven before taxes is 560€ plus 17%
IVA and the salesman offers a 12% discount, what is the final price?
Exercise 12 Find the current price for each item.
Item
Car
Coat
Book
Bus
Holiday Globes House
ticket
Old
25.000€ 350€
58€
2.5€
1800€
25€
250.000€
30%
12%
12%
6%
25%
3% incr.
disc.
disc.
incr.
incr.
disc.
price
Change 7%
incr.
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4.3 Finding the % increase or decrease
If we know the amount of the increase or decrease and the initial value
we can find the % increase or decrease using proportions.
Initial value ----------- increase/decrease
100 -------------------- x % increase/decrease
A formula can be used:
% increase =
increase
decrease
⋅ 100 or % decrease =
⋅ 100
initial value
initial value
Example 9 Last year there were 1560 employees in a company, this year 230
new people have been employed. What has been the % increase of the staff
in the company?
% increase =
230
⋅ 100 = 14.7%
1560
Example 10 I have paid 215€ for a coat and the original price was 230€.
What is the % discount?
% discount =
230 − 215
⋅ 100 = 6.52%
230
Exercise 13 Find % increase or decrease for each item.
Item
1
2
3
4
5
Old price €
750
430
12.3
34
57
New price €
776
370
19.4
60
13
% inc. or dec.
5. Interest
When money is invested in a bank or any other financial society or in
the case of a loan each year interest is directly proportional to the amount of
the deposit or the loan.
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If we consider that the annual interest is not added to the principal
(original amount), this is simple interest in this case the formula for the
interest or benefit after t years is:
I=
C⋅r ⋅ t
100
Where: I is the total interest, C is the principal, r is the % rate interest and t
is the number of periods, usually years.
Example 11 An amount of 1200€ at 4% per year produces after 5 years
I=
1200 ⋅ 4 ⋅ 5
= 240€
100
We can use also proportions and work in to steps
For one year 100€ principal -------- 4€ interest
1200€ principal -------- x€ interest
So x =
1200 ⋅ 4
= 48€
100
And in 5 years I = 48 ⋅ 5 = 240€
But you can see that it is not advisable.
Exercise 14 Calculate the interest for a deposit of 2450€ in the following
cases:
a) At 4% in 5 years
b) at 6% in 4 years
c) at 12% in 12 years
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Exercise 15 Complete the table
Principal € 300
2300
Annual
5%
2%
4%
16
1
3
5
130
500
1125
4.3%
5000
rate
Nº of years 7
Interest €
Calculations here
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6. Mixtures
These exercises are about how to find the price of a mixture of several
quantities of products with different prices in order not to have any profit or
loss when they are sold or, in other occasions, having a certain benefit.
Example 12 We mix 34 kg of tea at 2.5€ per kg with 25 kg of tea at 3.5€ per
kg and 71 kg of tea at 4.5€ per kg. At what price per kg must we sell the
mixture?
We must calculate the total price of the whole tea and divide it by the number
of kg of tea; it is very useful to organize the calculations as follows:
Item
weight
price per kg
total cost
Tea 1
34kg
2.5€
85€
Tea 2
25kg
3.5€
87.5€
Tea 3
71kg
4.5€
319.5€
Mixture
130kg
x€
492€
So each kg of the mixture must be sold at
492
= 3.78€ per kg
130
Example 13 A mixture consists of 15 kg of coffee purchased at 8€/kg and
chicory purchased at 2€/kg, how many kg of chicory do we need to mix if we
want to sell the mixture at 6€/kg?
We organize our calculations in a similar way
Item
difference of price
nº of kg
total
Coffee
8 – 6 = 2€
15
30€ loos
Chicory
6 – 2 = 4€
x
4x€ gain
As the loss must be equal to the gain 4 x = 30 ⇒ x =
5-12
30
= 7.5 kg
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Mathematics
Exercise 16 A merchant mixes 50l of spirit at 4€/l with 40l at 3€/l and 10l
of water costing nothing, he wants to have a profit of 12%, what price
should each l of the mixture be?
Exercise 17 12l of cologne at 25€/l have been mixed with 17l at 12€/l if
the mixture is sold at 18€/l calculate the % of gain or loss. Sol: 3.5%
gain
Exercise 18 How many kg of coffee at 7.5€/kg do we need to mix with
12kg of coffee at 10€/kg so the mixture can be sold at 8€ per kg without
any gain or loss? Sol: 48Kg
7. Proportional division
In this kind of problems something has to be shared in proportion to
certain data.
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It is very common in business; for example, that two or more people
form a partnership and they each invest a different amount of money, then
the profit is divided among them in proportion to the invested capital.
Example 14 Three partners invest 30,000, 40,000 and 50,000€ respectively
in a business. After a year the profit is 7500€, how much each one of them
receive?
The total invest is 30,000 + 40,000 + 50000 = 120,000€
So the profit for each one is:
B1 =
7500 ⋅ 30000
= 1875 €,
120000
B3 =
7500 ⋅ 50000
= 3125 €
120000
B2 =
7500 ⋅ 40000
= 2500 €
120000
and
Final exercises
Exercise 19 Convert the following percentages to fractions giving the
answer in the simplest form.
a) 12%
b) 15%
c) 82%
d) 17%
e) 120%
f) 50%
g) 55%
h) 80%
i) 60%
j) 78%
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k) 4%
l) 175%
Exercise 20 Convert the following decimals to percentages.
a) 0.17
b) 0.03
c) 0.15
d) 0.9
e) 1.5
f) 3.12
Exercise 21 Convert the following fractions to percentages
4
7
a)
2
5
c)
9
7
d)
5
10
e)
3
4
f)
1
3
g)
6
25
h)
i)
8
9
b)
j)
13
50
7
8
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Exercise 22 Calculate using mental method when appropriate
a) 12% of 515
b) 50% of 800
c) 7% of 3570
d) 112% of 300
e) 25% of 1000
f) 13.4% of 27
g) 5% of 80
h) 90% of 180
Exercise 23 Express each of the following as percentages.
a) 17 out of 50
b ) 5 out of 15
c) 6 out of 35
d) 2 out of 20
e) 7 out of 155
f) 67 out of 130
g) 86 out of 600
h) 70 out of 200
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Exercise 24 We make a lemon refresh with 230ml of pure lemon juice
and 340ml of water, what is the percentage of lemon in this beverage?
Exercise 25 A farmer sells 420 sheep of its 700 in total. What
percentage of sheep has he sold?
Exercise 26 A boy spends 6 euros of his pocket money on the cinema,
7.5€ on a sandwich and a coke and he still has 8€ left.
Which percentage of the whole has he left?
Which percentage has he spent on each thing?
Exercise 27 Joe’s mother has cooked 40 cookies; he has eaten 7, which
percentage of the cookies is left?
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Exercise 28 In our school there are 990 students and 91 teachers, which
is the percentage of teachers compared with the number of students?
Exercise 29 The value of a house is 120,000€ and the value of the
content is 25,000€ express the content value as a percentage of the
total.
Exercise 30 An 80ml bottle of perfume contains 25ml of extra free
perfume, what is the percentage of free perfume?
Exercise 31 A beverage contains 300ml of orange juice, 250ml of water
and 300ml of milk. Express the percentage of each component of the
drink.
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Exercise 32 Calculate the final amount:
a) £1500 plus 7.5% VAT
b) 2700€ plus 12% IVA
c) 3200€ of salary increased in 2.3%
d) 12€ with a 17% discount
e) 1530€ after an 8% discount
f) £170 reduced by 15%
Exercise 33 Peter earns 1200€ per month, but from this amount he
needs to pay 21% of IRPF, how much does he bring home?
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Exercise 34 The price of a bicycle is 230€ plus 6% IVA and there is a
special offer of 12% discount, which is the final price of the bicycle?
Exercise 35 I have paid 788.5€ for a computer in an offer of 17%
discount, which was the usual price of the computer?
Exercise 36 Only 9000 of the 20,000 different types of fish are caught,
what percentage of the types of fish do fishermen catch?
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Exercise 37 In a shop there is an offer as listed
SALE PRICE
DISCOUNT
LCD TV SET
509€
12%
COMPUTER
644€
5%
DVD
315€
10%
PDA
164€
18%
ITEM
Which were the usual prices of these items?
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Exercise 38 Three farmers have sold 350, 200 and 170 kg of pears to a
salesman and he has paid 1080€ in total. How much will each farmer
receive?
Exercise 39 A man gives 180€ to his three children aged 3, 5 and 8
years old, they must share the money proportionally to their ages. How
much does each one get?
Exercise 40 Divide 195 into three parts proportional to:
a) 2, 3 and 4
b)
1 1
1
, , and
2 3
4
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Exercise 41 A merchant of spirits mixes 50l of wine at 1.20€/l with 120l
of wine at 1.8€/l, what is the price of each l of the mixture?
Exercise 42 A grocer mixes two different types of oranges, 65kg costing
3€/kg with 25kg costing 4€/kg, he sells the oranges at 3.5€/kg what is
his gain percent? What should be the price of each kg so not to have
any gain or loss?
Exercise 43 How many kg of potatoes at 1.8€/kg must be mixed with
50kg of potatoes at 2.5€/kg so that the potatoes could be sold at 2€/kg?
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Solutions
1
Ratio / fraction
Decimal number
Percentage
0.3
30%
3
10
0.7
70%
7
10
0.27
27%
27
100
1.5
150%
3
2
0.286
28.6%
2
7
2.92
292%
35
12
0.68
68%
17
25
0.283
28.3%
283
1000
2.8:5
0.56
56%
7
17
28
2
9
7
3
1
2 a)
, b)
, c)
, d)
, e)
, f)
, g)
, h)
20
500
5
5
20
1000
20
5000
3 a) 15%, b) 63.4%, c) 16.67%, d) 41.67%, e) 12%, f) 134%, g) 6%
h) 75%, i) 60%, j) 6 . 2 5 % .
4 a) 17, b) 162.5, c) 18 h 1 min 30 s, d) 0.5 and 1.5, e) 6 and 30,
f) 5, 2.5 and 30, g) 44, h) 0.75 €, i) 8.4 $.
5 228200 people live in the centre district.
6 The prizes 1 or 3 are equal 142.5€, prize 2 is 135€, so, I will choose 1 or 3.
7 The original price was 230.77€.
8 With this data the population of Albacete would be 165000 people.
9 a) 132€, b) 58€, c) $1850, d) 446.4€.
10 a) Jacket £235.29, b) Car £18588.24, c) Shoes £94.12
11 The final price is 576.58€.
12
Car
Coat
26750€ 245€
Book
51.04€
Bus ticket Holliday Globes
2.8€
1908€ 18.75€
House
257500€
13
Item
1
2
3
4
5
% inc. or dec.
3.5%
13.95%
54.4%
76.47%
77.19%
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14 a) 490€, b) 588 €, c) 3528€.
15
Principal €
Annual rate
Nº of years
Interest €
300
4.3%
7
90.30
2300
5%
16
1840
16
Item
Spirit 1
Spirit 2
water
Mixture
Amount
50 l
40 l
10 l
100 l
price per l
4€
3€
0€
x€
x = 3.58 €/l
6500
2%
1
130
17
Item
Amount
price per l
Cologne 1 12 l
25€
Cologne 2 17 l
12€
Mixture
29 l
18€
522 − 504
Gain =
·100 = 3.57%
504
18
Item
difference of price
Coffee 1
8 – 7.5 = 0.5€
Coffee 2
10 – 8 = 2€
0.5 x = 24 ⇒ x = 48 kg
4166.67
4%
3
500
5000
4.5%
5
1125
total cost
200€
120€
0€
320€ + 12% = 358.4€
total cost
300€
204€ ⇒ 300+204 = 504
522
nº of kg
x
12
total
0.5x€ loos
24€ gain
3
3
41
17
6
1
11
4
3
39
, b)
, c)
, d)
, e) , f) , g)
, h) , i) , j)
,
25
20
50
100
5
2
20
5
5
50
1
7
k)
, l)
25
4
19 a)
20 a) 17%, b) 3%, c) 15%, d) 90%, e) 150%, f) 312%.
21 a) 40%, b) 57.1%, c) 128.6%, d) 50%, e) 75%, f) 33.3%, g) 24%, h) 26%,
i) 88.9%, j) 87.5%.
22 a) 61.8, b) 400, c) 249.9, d) 336, e) 250, f) 3.62, g) 4, h) 162.
23 a) 34%, b ) 33.3%, c) 17%, d) 10%, e) 4.5%, f) 51.5%, g) 14.3%, h) 35%
24 40.3% of lemon.
25 The farmer sells 60% of his sheeps.
26 The boy has 37% left, he spends 28% on the cinema and 35% on a
sandwich and a coke.
27 There is 82.5% of the cookies left.
28 The percentage of teachers compared with the students is 9.2%.
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29 The content value as a percentage of the total is
30 The percentage of free perfume is
25
·100 = 17.2% .
25 + 120
25
·100 = 31.3% .
80
31 It has 35.3% of orange juice, 29.4% of water and 35.3% of milk.
32 a) £1612.5, b) 3024€, c) 3273.6€, d) 9.96€, e) 1407.60€, f) £144.50
33 Peter earns 948€ per month, after taxes.
34 The final price of the bicycle is 214.5€.
35 The price of the computer was 950€.
36 45% of fish are caught.
37 The usual prices were: LCD TV Set 578.41€, Computer 677.89€,
DVD 350€, PDA 200€.
38 The farmers wil receive 525€, 300€ and 255€, respectively.
39 33.75€, 56.25€ and 90€
40 a) 43.33, 65 and 86.67; b) 90, 60 and 45
41
Item
Wine 1
wine 2
Mixture
Amount
50 l
120 l
170 l
price per l
1.20€
1.8€
x€
So each l of the mixture must be sold at
total cost
60€
216€
276€
276
= 1.62€ .
170
42
Item
Amount
price per kg
total cost
Oranges 1 65 kg
3€
195€
oranges 2 25 kg
4€
100€ ⇒ 195+100 = 295
Mixture
90 kg
3.5€
315
315 − 295
Gain =
·100 = 6.78% The price without any gain or loss should be
295
295
= 3.28€
90
43
Item
difference of price
nº of kg
total
Potatoes 1
2 – 1.8 = 0.2€
x
0.2x€ loos
Potatoes 2
2.5 – 2 = 0.5€
50
25€ gain
As the loss must be equal to the gain 0.2x = 25 ⇒ x = 125 kg .
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6 Algebra
Keywords
Algebra
equation
expression
Monomial
coefficient degree
formula
identity
variable
like term
polynomial
1 Using letters for numbers
Algebra is a branch of mathematics in which symbols, usually letters of
the alphabet, represent numbers or members of a specified set and are used
to represent quantities so that we can use letters for the arithmetical
operations such as +, −, ×, ÷ and the power.
What do you do when you want to refer to a number that you do not
know? Suppose you wanted to refer to the number of buildings in your town,
but haven't counted them yet. You could say 'blank' number of buildings, or
perhaps '?' number of buildings.
In mathematics, letters are often used to represent numbers that we do not
know - so you could say 'x' number of buildings, or 'q' number of buildings.
These are called variables.
Look at these examples:
•
The triple of a number:
3n
•
The triple of a number minus five units: 3n – 5
•
The following number to x:
x+1
•
The preceding number to y:
y–1
•
An even number:
2a
•
An odd number:
2z + 1 or 2z – 1
Exercise 1
Find the expression:
1. I start with x, double it and then subtract 6.
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2. I start with x, add 4 and then square the result.
3. I start with x, take away 5, double the result and then divide by 3.
4. I start with x, multiply by 4 and then subtract t.
5. I start with x, add y and then double the result.
6. I start with a, double it and then add b.
7. I start with n, square it and then subtract n.
8. I start with x, add 2 and then square the result.
9. A brick weighs x kg. How much do 6 bricks weigh? How much do
n bricks weigh?
10. A man shares x euros between n children. How much does each
child receive?
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2. Mathematical language.
In Algebra, you use letter symbols to represent unknowns in a variety of
situations:
EQUATIONS
In an equation the letters stand for one or more
particular numbers (the solutions of the equation), for
example, 2x + 1 = x – 2
EXPRESSIONS
In an expression there is no equals sign, for example,
3x² + 2x – 1.
IDENTITIES
In an identity there is an equals sign (sometimes written
≡) but the equality holds for all values of the unknown,
for example:
2( x + 1) = 2x + 2
In a formula letters stand for defined quantities or
variables, for example, d = s·t. (d is distance, s is speed
and t is time)
FORMULAE
Exercise 2
Separate
the
equations,
the
formulae,
expressions:
a) x ( x + 1 ) = x² + x
b) 7y + 10
c) V = I · R
d) x² - 3x + 10
e) 7x + 11 = x – 9
f) (x + 1)² = x² + 2x + 1
g) A = π · r²
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h) x² - 7x = 0
Exercise 3
Write an equation for the following statements:
a. If you multiply a number by 3 and then add 4, the answer is 13
b. The addition of a number and its consecutive is 81.
c. If you multiply the number by 2 and then subtract 5, the answer is
4.
d. If you subtract 11 from the number and then treble the result, the
answer is 20.
e. If you treble the number, take away 6 and then multiply the result
by 2, the answer is 18.
f. If you multiply the number by 5 and subtract 4, you get the same
answer as when you add 3 to the number and then double the
result.
g. The sum of four consecutive numbers is 90.
h. The sum of three consecutive odd numbers is 177.
i. When a number is doubled and then added to 13, the result is 38.
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3. Monomials.
A monomial is an algebraic expression consisting of only one term, which has
a known value (coefficient) multiplied by one or some unknown values
represented by letters with exponents that must be constant and positive
whole numbers (literal part). For example:
Degree
Coefficien
Literal part
If the literal part of a monomial has only one letter, then the degree is the
exponent of the letter.
If the literal part of a monomial has more than one letter, then the degree is
the sum of the exponents of every letter. For example:
•
The degree of − 2x 3 is 3
•
The degree of 3 x 5 y 2 is 5 + 2 = 7.
•
The degree of 7 x 3 yz is 3 + 1 + 1 = 5.
Exercise 4 Complete the table
Monomial
− 3x 2 y 3
Variables
x and y
Coefficient
-3
Literal part
x2y3
degree
5
7x 3 y z
−
4 3
x
5
32 x 3
5
Exercise 5 Name the variables, coefficient, literal part and degree of the
following monomials
a) 5 xy
b) − 2x
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c) 5 x −2 y 3
d) xy 2
e) − x 2 yz
f) − 23 x
g)
h)
5
7x5 y8
3
x
i) −
3 4 2 4
x y z t
7
j) 8
k) 0 x 2 y
4. Addition and subtraction of monomials.
You can add monomials only if they have the same literal part (they are also
called like terms). In this case, you sum the coefficients and leave the same
literal part.
Like terms use exactly the same literal part
3x and 6x are like terms
3x and 3xy are unlike terms
3x and 3x² are unlike terms
Look at these examples:
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4 xy 2 + 3 xy 2 = 7 xy 2
5 x 2 + 3 − 2x 2 + 1 = 3 x 2 + 4 , and we can not add the terms 3 x 2 and 4
Exercise 6
Collect like terms to simplify each expression:
a. x 2 + 3 x − x + 3 x 2 =
b. 5 y + 3 x + 2y + 4 x =
c.
(2x + 3 ) − (5x − 7) − (x − 1) =
d.
2 2 1
3
1
x + x − x2 − x + 2 =
3
2
2
5
e.
(x + 2) + (2x + 7 ) − (3 x + 4 )
f.
(x
2
+ 2x − 2 x 2 − x + 3 x 2 + 5 x =
g.
(x
2
+ y + 7x 2 − 3y − x 2 + 7y =
h.
[(3x
) (
) (
2
) (
) (
)
)
5 

+ 2x 2 − 5 x 2 − x +  8 x 2 + x  =
2 

) (
)]
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5. Product of monomials.
If you want to multiply two or more monomials, you just have to multiply the
coefficients, and add the exponents of the equal letters:
+
Product
Look at these examples:
a. ( 2xy² )·(-5x² y) = -10 x³ y³
b. 3a² · 2ab = 6a³ b
Exercise 7 Multiply
a) (5 x ) ⋅ (3 x )
( )
b) (2x ) ⋅ 3 x 2
( )
c) (− 2x ) ⋅ x 3
(
)(
d) 4 x y ⋅ 2x 2 y
(
)
)( )
e) − 7 x 2 y ⋅ xy 2
(
)
f) 5 x 3 y 2 ⋅ (xy )
2

g)  x 3 y  ⋅ (3 x )
3

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( )( )
h) (3 x ) ⋅ 2x 2 ⋅ 5 x 3
( )
( )
i) x 2 ⋅ (− 2x ) 3 x 3
( )
j) (3ab ) ⋅ 2a 2
(
)
k) 2a2b3 ⋅ a
(
)
l) 2x 2 z ⋅ (3zx )
6. Quotient of monomials.
If you want to divide two monomials, you just have to divide the coefficients,
and subtract the exponents of the equal letters. The quotient of monomials
gives an expression that is not always a monomial.
_
Division
You can also simplify the fractions that result from the division. Look at these
examples:
(
)
a. 10 x 3 : (2x ) = 5 x 2
b. 8 x 2 y : 6 y 3 =
8x 2 y 2 ⋅ 4x ⋅ x ⋅ y
4x 2
This expression is not a monomial
=
=
6y 3
2 ⋅ 3 ⋅ y ⋅ y ⋅ y 3y 2
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Exercise 8 Operate
(
)( )
a) 15 x 3 : 3 x 2
( )
b) 2x 4 : (3 x )
( )
c) (6 x ) : 2x 3
(
)
d) 12a 2b : 3a
e) 15a3b 2 : ab
f) 3 x 5 y 2 : x 2 y 2
(
)
( )(
)
g) (5 xy ) ⋅ 2x 2 y : 3 x
h) xy 3 ⋅ 3 x 2 y : 2x 2 y 2
[( )
]( )
i) 3 x 4 : (2x ) ⋅ 4 x 2
j)
(3ab ) : (2a2 )
k)
(2a b )⋅ 7a
l)
(12x yz ) : (3xz )
4
2
5
3
7. Polynomials.
A polynomial is the addition or subtraction of two or more monomials.
•
If there are two monomials, it is called a binomial, for example x² + x
•
If there are three monomials, it is called a trinomial, for example 2x² 3x + 1
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The following are NOT polynomials:
1
x
x2 − 4
x 2 + 3 x + 2 x −2
The degree of the entire polynomial is the degree of the highest-degree term
that it contains, so
x2 + 2x – 7 is a second-degree trinomial, and x4 – 7x3 is a fourthdegree binomial.
The polynomial that follows is a second-degree polynomial, and there are
three terms: 4x² is the leading term, and (-7) is the constant term.
Polynomials are usually written this way, with the terms written in
"decreasing" order; that is, with the highest exponent first, the next highest
next, and so forth, until you get down to the constant term.
Polynomials are also sometimes named for their degree:
•
a second-degree polynomial, such as 4x2, x2 – 9, or ax2 + bx + c, is
also called a "quadratic"
•
a third-degree polynomial, such as –6x3 or x3 – 27, is also called a
"cubic"
•
a fourth-degree polynomial, such as x4 or 2x4 – 3x2 + 9, is sometimes
called a "quartic"
Exercise 9 Name the variables, degree, principal term, constant term of
the following polynomials
a) 5 x 3 + 2x
b) x 2 − 4
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c) 3 x 5 − 2x 2 − 3 x − 1
d) 3 + 2x +
7 3
x
3
e) 5
f) −
g)
7
23 2
x +x+2
3
22 x 5 y 8 + 2x 3 y + 2xy − y − 3
h) 0 x 5 − 2x 3 − 4 x − x
8. Evaluating polynomials.
“Evaluating" a polynomial is the same as calculating its number value at a
given value of the variable: you substitute the given value of x, and calculate
the value of the polynomial. For instance:
Evaluate 2x3 – x2 – 4x + 2 at x = –3
Plug in –3 for x, remembering to be careful with brackets and negatives:
2(–3)3 – (–3)2 – 4(–3) + 2
= 2(–27) – (9) + 12 + 2
= –54 – 9 + 14
= –63 + 14
= –49
Always remember to be careful with the minus signs!
Exercise 10 Evaluate the polynomials:
x 3 − 2x 2 + 3 x − 4 , 5 x 3 + 2x , 3 x 5 − 2x 2 − 3 x − 1 and 3 + 2x +
values of x:
a) x = 0
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b) x = 1
c) x = 2
d) x = -1
9. Adding polynomials.
When adding polynomials you must add each like term of the polynomials,
that is, monomials that have the same literal part, you use what you know
about the addition of monomials.
There are two ways of doing it. The format you use, horizontal or vertical, is
a matter of preference (unless the instructions explicitly tell you otherwise).
Given a choice, you should use whichever format you're more comfortable
with.
Note that, for simple additions, horizontal addition (so you don't have to
rewrite the problem) is probably the simplest, but, once the polynomials get
complicated, vertical addition is probably the safest (so you don't "drop", or
lose, terms and minus signs). Here is an example:
Simplify (3x3 + 3x2 – 4x + 5) + (x3 – 2x2 + x – 4)
Horizontally:
(3x3 + 3x2 – 4x + 5) + (x3 – 2x2 + x – 4)
= 3 x3 + 3 x2 – 4 x + 5 + x3 – 2 x2 + x – 4
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= 3x3 + x3 + 3x2 – 2x2 – 4x + x + 5 – 4
= 4x3 + x2 – 3x + 1
Vertically: Cop
3 x3 + 3 x2 – 4 x + 5
x3 – 2 x2 + x – 4
4x3 +x2 – 3x + 1
Either way, I get the same answer: 4x3 + x2 – 3x + 1.
Exercise 11
Calculate the following additions of polynomials:
1. (14x + 5) + (10x + 5)
2. (10x + 12) + (6x + 20)
3. (19x2 + 12x + 12) + (7x2 + 10x + 13)
4. (17x2 + 20x + 11) + (15x2 + 11x + 17)
5. (-15x2 - 5x + 9) + (-6x2 - 19x - 16) + (-15x2 - 14x - 13) + (9x2 - 14x + 20)
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6. (-13x2 - 13x - 10) + (19x2 - 19x - 5)
7. (4x2 - 6x + 7) + (-19x2 - 15x - 18)
8. (-13x2 - 5x - 14) + (-14x2 - 20x + 8)
9. ( 9x5 - 14x4 + 18x) + (-6x5 - 12x4 - 9x)
10. (-20x2 + 13x - 4) + (11x2 - 13x - 10)
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10. Subtracting polynomials.
When subtracting polynomials you must realize that a subtraction is the
addition of the first term and the opposite of the second:
A – B = A + (- B)
Notice that running the negative through the brackets changes the sign on
each term inside the brackets. Look at this example:
Simplify (6x3 – 2x2 + 8x) – (4x3 – 11x + 10)
Horizontally:
(6x3 – 2x2 + 8x) – (4x3 – 11x + 10)
= (6x3 – 2x2 + 8x) – 1(4x3 – 11x + 10)
= 6x3 – 2x2 + 8x – 4x3 + 11x – 10
= 6x3 – 4x3 – 2x2 + 8x + 11x – 10
= 2x3 – 2x2 + 19x – 10
Vertically:
Write out the polynomials, leaving gaps when necessary, and change
the signs in the second line. Then add:
6 x3 – 2 x2 + 8 x
– 4 x3
+ 11x – 10
3
2
2x – 2x + 19x – 10
Either way, I get the same answer: 2x3 – 2x2 + 19x – 10
Exercise 12
Calculate the following subtractions:
1. (6x + 14) - (9x + 5)
2. (6x + 19) - (14x + 5)
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3. (14x2 + 13x + 12) - (7x2 + 20x + 4)
4. (19x2 + 9x + 16) - (5x2 + 12x + 7)
5. (15x2 - 9x + 9) - (13x2 + 15x + 5) - (-16x2 + 20x + 16) - (-7x2 + 10x - 10)
6. (-9x2 - 4x - 4) - (-9x2 - 11x + 12)
7. From 19x2 + 11x + 15 subtract -5x2 - 6x - 6
8. (-18x2 + 7x - 14) - (-20x2 + 17x - 12)
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9. Subtract -10x2 + 16x - 15 from -6x2 + 19x + 19
10. (17x5 + 19x2 + 15) - (9x7 - 11x - 17)
11. Multiplying polynomials.
•
The first step up in complexity is a monomial times a multi-term
polynomial. To do this, I have to distribute the monomial through the
brackets. For example:
–3x(4x2 – x + 10) = –3x(4x2) – 3x(-x) – 3x(10) = –12x3 + 3x2 – 30x
•
The second step in complexity is a multi - term polynomial times a
multi - term polynomial. Look at these examples:
1. Simplify 3(x + 2) = 3·x + 3·2 = 3x + 6
2. Simplify 3x2·(x + 5) = 3x2·x + 3x2·5 = 3x3 + 15x2
3. Simplify (x + 3)(x + 2)
The first way I can do this is "horizontally", where I distribute twice:
Cop
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2
2
= x(x) + 3(x) + x(2) + 3(2) = x + 3x + 2x + 6 = x + 5x + 6
The "vertical" method is much simpler, because it is similar to the
multiplications learnt at primary school:
4. Simplify (x – 4 )(x – 3 )
x–4
x–3
x2 – 4x
–3x + 12
x2 – 7x + 12
5. Simplify (4x2 – 4x – 7)(x + 3)
4x2 – 4x – 7
x+3
4x3 – 4x2 – 7x
12x2 –12x – 21
4x3 + 8 x2 – 19x – 21
Multiply out the brackets means the same as:
•
Remove the brackets
•
Expand the brackets (or the expression)
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Exercise 13 Expand
1.
2 (x + 5)
2.
5 (3x – 2)
3.
4 (2x – 3)
4.
-8 (x + 2)
5.
-4 (x – 5)
6.
9 (3x + 2)
7.
-6 (5x – 4)
8.
x (x +7)
9.
2x (3x + 2)
10. -3x2 (x + 2)
11. -2x (5x2 + 3y2)
12. 3x2y (2x + 3y)
13. -2x5 (7x2+3)
14. 7x2 (4x2 + 2x – 4)
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Exercise 14 Remove brackets or multiply:
1.
(11x + 11) (11x + 10)
2.
(9x + 9) (3x + 3)
3.
(8x + 11) (5x + 11)
4.
(9x + 7) (6x + 4)
5.
(6x – 2) (-7x + 3)
6.
(9x + 5) (2x – 2)
7.
8.
− x 2 − 5x + 4
3x − 8
9.
2x 2 − 3x − 1
− 2x − 3
10.
3x 2 + 2x − 7
2x 2 − x + 4
5x 2 − 4x + 1
−x−2
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11.
Mathematics
12. 11x 2 + 2x − 1
− 3x 2 − 10x + 4
2x 2 − 3
− 11x + 1
12. Factorising.
Factorising is the reverse process of multiplying out a bracket. The factorised
expression has a polynomial inside a bracket, and a term outside.
This term outside must be a common term (a number or a letter) . It means
that the number or the letter (s) can be found in every term of the expression.
The trick is to see what can be factored out of every term in the expression.
Just don't make the mistake of thinking that "factoring" means "dividing off
and making disappear". Nothing disappears when you factor; things just get
rearranged. Here are some examples of how to factor:
•
•
•
3x – 12 = 3(x – 4)
12y2 – 5y = y(12y – 5)
3x3 + 6x2 – 15x = 3x(x2 + 2x – 5)
Remember: when the term to be factored out coincides
with one of the addends, the unit always remains:
x 2 + x = x (x + 1)
Exercise 15 Factorise:
1. 12x2 + 3x
2. 2xy - 3x
3. 2x2 + 18x
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4. 6x3 – 2x2
5. 3x2y +2x
6. 5x+15
Exercise 16 Factorise:
1. -336x3 + 288x
2. 3x2 - 3x
3. -3x3 - 33x
4. -15x2 + 18x
5. 4x3 - 4x
6. 160x3 + 100x2 - 180x
7. 19x3 - 19x
8. -6x3 + 8x
9. 36x3 - 24x2 + 8x
10. -14x2 + 16x
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13. Three algebraic identities.
There are three formulas about operations with binomials that are very
common and it is useful to memorise.
1. Square of an addition.
2
We want to find a formula to work out (a + b ) where a and b can be numbers
or any kind of monomial.
(a + b)2 = (a + b)(a + b ) Let’s do it carefully
2
So (a + b ) = (a + b )(a + b ) = a2 + 2ab + b 2
a+b
a+b
a2 + ab
ab + b2
2
a + 2ab + b2
2
Note that (a + b ) is not the same as a 2 + b 2 so don’t forget the term 2ab
2. Square of a subtraction.
2
Now we want a formula to expand (a − b )
(a − b)2 = (a − b)(a − b) = a2 − 2ab + b2
a–b
a–b
a2 – ab
2
– ab + b
2
a – 2ab + b2
2
Note again that (a − b ) is not the same as a2 − b2 but something absolutely
different
3. Difference of two squares
What happens if we multiply (a + b )(a − b ) ?
The terms ab and – ab cancel out, so
(a + b)(a − b ) = a2 − b2
a+b
a–b
a2 – ab
2
+ab – b
2
2
a
–b
We can use these three formulas in two directions to
expand or to factorise any algebraic expression that fits with any of them.
Exercise 17 Calculate using a formula.
1.
(x + 2)2
2.
(x + 4)2
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3.
(2x + 1)2
4.
(x − 3)2
5.
(3x + 1)2
6.
(2x − 5)2
7.
(a + 2b)2
8.
(c
9.
(x − 2)(x + 2)
10.
(2x + 1)(2x − 1)
11.
(3 + x )(3 − x )
12.
(3y − 2x )(3y + 2x )
2
− 2a
Mathematics
)
2
Exercise 18 Factorise:
1.
x2 + 6x + 9
2.
x 2 + 2x + 1
3.
x 2 − 2x + 1
4.
x2 − 9
5.
4x 2 + 4x + 1
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6.
x 2 + 10 x + 25
7.
x 2 − 25
8.
4x2 − 9
9.
1− x 2
10.
2x 4 + 4 2 x 2 + 4
Mathematics
Exercise 19 Write down expressions for the area of these rectangles:
Exercise 20
2
This is a drawing to visualize the formula (a + b ) = a 2 + 2ab + b 2 .
2
Can you think of a similar way to visualize (a − b ) = a 2 − 2ab + b 2 ?
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Mathematics
Solutions
Exercise 1
( x − 5)·2
, 4. 4 x − t , 5. ( x + y )·2 , 6. 2a + b , 7. n2 − n
3
x
8. ( x + 2)2 , 9. 6 x , nx , 10. .
n
Exercise 2 a) x ( x + 1 ) = x² + x identity; b) 7y + 10 expression; c) V = I · R
formula; d) x² - 3x + 10 expression; e) 7x + 11 = x – 9 equation;
f) (x + 1)² = x² + 2x + 1 identity; g) A = π · r² formula; h) x² - 7x = 0 equation
1. 2x − 6 , 2. ( x + 4)2 , 3.
Exercise 3 a. 3 x + 4 = 13 , b. x + x + 1 = 81 , c. 2x − 5 = 4 , d. ( x − 11)·3 = 20 , e.
(3 x − 6)·2 = 18 , f. 5 x − 4 = 2( x + 3) , g. x + x + 1 + x + 2 + x + 3 = 90 ,
h. 2x + 1 + 2x + 3 + 2x + 5 = 177 , i. 2x + 13 = 38.
Exercise 4
Monomial
− 3x 2 y 3
7x 3 y z
4
− x3
5
32 x 3
5
Exercise 5
Monomial
5 xy
− 2x
5 x −2 y 3
xy 2
2
− x yz
− 23 x
5
7x5 y8
3
x
3 4 2 4
−
x y z t
7
Variables
x and y
Coefficient
-3
x, y and z
x
0x y
5
4
5
9
5
x3
3
x3
none
3
0
xy 2
1
x, y, z
2
x yz
−1
degree
2
1
3
4
This is not a monomial
x, y
5
x5y8
7
13
This is not a monomial
x, y, z, t
−
8
2
x3y z
Coefficient
Literal part
xy
5
−2
x
This is not a monomial
x, y
degree
5
7
−
x
none
Variables
x, y
x
Literal part
x2y3
3
7
8
0
x, y
x 4 y 2z 4t
11
None
x2y
0
No degree
Exercise 6 a. 4 x 2 + 2x , b. 7 y + 7 x , c. − 4 x + 11, d. −
f. 2x 2 + 8 x , g. 7 x 2 − 9 y , h. 8 x 2 +
7
x.
2
6-27
5 2 3
x +
x + 2 , e. 5 ,
6
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Mathematics
Exercise 7
a) 15 x 2 , b) 6 x 3 , c) − 2x 4 , d) 8 x 3 y 2 , e) − 7 x 3 y 3 , f) 5 x 4 y 3 , g) 2x 4 y , h) 30 x 6
i) − 6 x 6 , j) 6a3b , k) 2a3b3 , l) 6 x 3 z 2 .
2 3
x , c) It is not any monomial, d) 4ab , e) 15a2b ,
3
10 2 2
3
x y , h) xy 2 , i) 6 x 5 , j) It is not any monomial, k) 14a5b 2 , l) It is
f) 3 x 3 , g)
3
2
not any monomial.
Exercise 8 a) 5 x , b)
Exercise 9
Polynomial
a) 5 x 3 + 2x
b) x 2 − 4
c) 3 x 5 − 2x 2 − 3 x − 1
7
d) 3 + 2x + x 3
3
e) 5
23 2
x +x+2
3
g) 7 22 x 5 y 8 + 2x 3 y + 2xy − y − 3
Variables Degree Principal term Constant term
3
0
x
x3
2
x
2
-4
x
5
x
5
-1
3x
7 3
x
x
3
3
3
None
f) −
h) 0 x 5 − 2x 3 − 4 x − x
0
5
5
This is not a polynomial
x, y
13
x
3
7
22 x 5 y 8
-3
− 2x 3
0
Exercise 10
Polynomial
a) x = 0 b) x = 1 c) x = 2 d) x = −1
3
2
− 10
−4
−2
2
x − 2x + 3 x − 4
3
0
7
44
−7
5 x + 2x
5
2
−3
−3
81
−1
3 x − 2x − 3 x − 1
22
77
4
7
−
3 + 2x + x 3
3
3
3
3
3
Exercise 11 1. 24 x + 10 , 2. 16 x + 32 , 3. 26 x 2 + 22x + 25 , 4. 32x 2 + 31x + 28 ,
5. − 27 x 2 − 52x , 6. 6 x 2 − 32x − 15 , 7. − 15 x 2 − 21x − 11 , 8. − 27 x 2 − 25 x − 6 ,
9. 3 x 5 − 26 x 4 + 9 x , 10. − 9 x 2 − 14 .
Exercise 12 1. − 3 x + 9 , 2. − 8 x + 14 , 3. 7 x 2 − 7 x + 8 , 4. 14 x 2 − 3 x + 9 ,
5. 25 x 2 − 54 x − 2 , 6. 7 x − 16 , 7. 24 x 2 + 17 x + 21 , 8. 2x 2 − 10 x − 2 ,
9. 4 x 2 + 3 x + 34 , 10. − 9 x 7 + 17 x 5 + 19 x 2 + 11x + 32 .
Exercise 13 1. 2x + 10 , 2. 15 x − 10 , 3. 8 x − 12 , 4. − 8 x − 16 , 5. − 4 x + 20 ,
6. 27 x + 18 , 7. − 30 x + 24 , 8. x 2 + 7 x , 9. 6 x 2 + 4 x , 10. − 3 x 3 − 6 x 2 ,
11. − 10 x 3 − 6 xy 2 , 12. 6 x 3 y + 9 x 2 y 2 , 13. − 14 x 7 − 6 x 5 ,
14. 28 x 4 + 14 x 3 − 28 x 2 .
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Mathematics
Exercise 14 1. 121x 2 + 231x + 110 , 2. 27 x 2 + 54 x + 27 , 3. 40 x 2 + 143 x + 121,
4. 54 x 2 + 78 x + 28 , 5. − 42x 2 + 32x − 6 , 6. 18 x 2 − 8 x − 10 ,
7. − 3 x 3 − 7 x 2 + 52x − 32 , 8. − 4 x 3 + 11x + 3 , 9. 6 x 4 + x 3 − 4 x 2 + 15 x − 28 ,
10. − 5 x 3 − 6 x 2 + 7 x − 2 , 11. − 6 x 4 − 20 x 3 + 17 x 2 + 30 x − 12 ,
12. − 121x 3 − 11x 2 + 13 x − 1 .
Exercise 15 1. 3 x( 4 x + 1) , 2. x(2y − 3) , 3. 2x( x + 9) , 4. 2x 2 (3 x − 1) ,
5. x(3 xy + 2) , 6. 5( x + 3) .
Exercise 16 1. 48 x(6 − 7 x 2 ) , 2. 3 x( x − 1) , 3. − 3 x( x 2 + 11) , 4. 3 x(6 − 5 x ) ,
5. 4 x( x − 1)( x + 1) , 6. 20 x(8 x 2 + 5 x − 9) , 7. 19 x( x − 1)( x + 1) ,
8. 2x( 4 − 3 x 2 ) , 9. 4 x(9 x 2 − 6 x + 2) , 10. 2x(8 − 7 x ) .
Exercise 17 1. x 2 + 4 x + 4 , 2. x 2 + 8 x + 16 , 3. 4 x 2 + 4 x + 1 , 4. x 2 − 6 x + 9 ,
5. 9 x 2 + 6 x + 1, 6. 4 x 2 − 20 x + 25 , 7. a 2 + 4ab + 4b2 , 8. c 4 − 4c 2a + 4a2 ,
9. x 2 − 4 , 10. 4 x 2 − 1, 11. 9 − x 2 , 12. 9 y 2 − 4 x 2 .
2
2
2
2
Exercise 18 1. (x + 3 ) , 2. (x + 1) , 3. (x − 1) , 4. (x + 3 )(x − 3 ) , 5. (2x + 1) ,
2
6. (x + 5 ) , 7. (x + 5 )(x − 5 ) , 8. (2x + 3 )(2x − 3 ) , 9. (1 − x )(1 + x ) , 10.
Exercise 19 2x 2 − 3 x − 2 and 15 x 2 − 10 x
Exercise 20
6-29
(
)
2
2x 2 + 2 .
7 Equations
Keywords
Equation
linear equation
Variable
unknown
quadratic equation
term
1 Definition
An equation is a statement that says that two algebraic expressions are
equal.
In an equation the equality is true only for particular values of the variables.
In an identity the equality holds for all the values of the variables.
Example 1 the equality 3 x − 7 = 2 is only true when x = 3 this is the solution
of the equation.
If the degree of the polynomials is 1 the equations are called linear equations
or first degree equations.
If the degree of the polynomials is 2 the equations are called quadratic
equations or second degree equations.
Example 2
5 x − 7 = 2x + 3 Is a linear equation
5 x 2 − 7 = 5(2x + 3 ) Is a quadratic equation
2 Linear equations language in equations
Let’s work with an example
5x – 3 = 3x + 2
-
In an equation there is always an equal sign.
-
There are two sides with an algebraic expression on each side, left
side and right side.
-
Each monomial is a term, so 5 x , − 3 , 3 x and 2 are terms of this
equation. In linear equations we will have x-terms and numerical
terms.
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-
Mathematics
x is the variable or unknown.
3 Solving easy equations, basic rules
These rules must be followed, usually in this order, to solve linear equations.
Rule 1 If there are brackets we must remove them (expand the expression).
Rule 2 Any x-term or number that is adding (positive sign) moves to the other
side subtracting (negative sign) and vice versa.
Rule 3 If there are x-terms on both sides, collect them on one side and do the
same with the numbers.
Rule 4 A number that is multiplying (the whole expression) on one side
moves to the other side dividing and vice versa.
Examples 3
a) Solving 3 x − 2 = 12 , there are no brackets.
3 x − 2 = 12 ⇒ 3 x = 12 + 2 (Rule 2)
3 x = 14 (Rule 3)
x=
14
(Rule 4), this is the solution.
3
b) 5 x − 3 = 2x + 1, there are no brackets.
5 x − 2x = 1 + 3 (Rule 2)
3 x = 4 (Rule 3)
x=
4
(Rule 4), this is the solution.
3
c) 13 x + 2 = 5(2x + 1)
13 x + 2 = 10 x + 5 (Rule 1)
13 x − 10 x = 5 − 2 (Rule 2)
3 x = 3 (Rule 3)
x=
d)
3
⇒ x = 1 (Rule 4), this is the solution.
3
x
x
x
− 1 = 4 ; = 4 + 1; = 5 ; x = 5 ⋅ 3 ; x = 15 ( sol.)
3
3
3
Of course you don’t need to write the rule you are using during the process.
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Mathematics
Exercises 1
Solve:
1) x + 12 = 3
2) x + 5 = 7
3) 5 = x + 2
4) 3 = 2 − x
5) 2x = 10
6) 5 x = 4
7)
x
= 10
2
8) − 3 = 2x
9) 3 x + 6 = 12
10) 1 = 2x − 4
11) 2 = 3 x − 1
12) 3 = 2 − 4 x
13) 2x + 1 = x + 2
14) 3 x + 2 = 5 x − 3
15) 2x − 3 = x
16) 6 x − 5 = 3 x − 1
17) 6 x + 2 = 2 − 3 x
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Mathematics
18) − 2x + 1 = −3 x + 2
19) 5 x − 3 = 2 − 4 x
20) 6 x − 4 = 15 x − 3
21) 3 x − 2 + 4 x = 5 x − 2
22) x + 1 − 3 = 2x + 4 − x
23) 5 x − 3 = 2(x + 1)
24) 6(x − 2) = 7 x − 3
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Mathematics
25) 2(5 x − 3 ) = x + 1
26) 3(5 x − 1) = 2(2 − 3 x )
27) 3(x + 2) + x = 2x − 3
28)
x
+2 =1
4
29)
2x
=3
5
30)
3x
+ 5 = −2
2
31)
5x
= −2
3
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Mathematics
x
 3
32) 8 − 2  =
4
 2
 x 2 3
33) 2 −  =
2 3 5
4 Equations with denominators
When there are denominators in several terms of the equation, the first thing
we need to do is to simplify multiplying every term by the lowest common
multiple of the denominators, if there is no denominator in a term you should
consider it as 1; then we continue with the same rules that we have seen
before.
Example 4 solving x −
2 5x
=
−1
3
2
The L.C.M. of 3 and 2 is 6, so we multiply both sides by 6
2

 5x

6 x −  = 6
− 1 ; 6 x − 4 = 15 x − 6 , and then:
3

 2

6 x − 15 x = −6 + 4 ⇒ −9 x = −2 ⇒ x =
2
−2
⇒ x =
−9
9
Solving a complex equation requires organising the calculations in these
steps
1. If there are brackets remove them (expand)
2. If there are denominators remove them (multiplying both sides by the
LCM of the denominators)
3. Transpose the like terms (move to one side the x-terms and the
numbers to the other side)
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Mathematics
4. Combine like terms
5. Isolate the unknown (move the coefficient of the x to the other side)
Exercises 2
Solve:
1) −
3 x
1 x
+
=− −
5 10
5 5
2) 2(3x − 1) = 3( x − 1)
3)
x +1 x −1
=
3
4
4) 4( x − 1) − 2 = 3x
5)
2x − 1 x
=
3
2
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Mathematics
6) 4(1 − 2x ) = 3(2 − x )
7) 3(2x + 1) + 2( x − 1) = 23
8) 5(1 − 2x ) − 3( 4 + 4x ) = 0
9)
12
=4
2x − 3
10)
5
15
=
x+5 x+7
11)
x x
+ =1
3 4
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Mathematics
Exercises 3 Find the number in each question by forming an equation
and solving it.
1) If you multiply the number by 2 and then subtract 5, the answer is 4
2) If you multiply the number by 10 and then add 19, the answer is 16
3) If you add 3 to the number and then multiply the result by 4, the
answer is 10
4) If you subtract 11 from the number and then treble the result, the
answer is 20
5) If you double the number, add 4 and then multiply the result by 3, the
answer is 13
6) If you treble the number, take away 6 and then multiply the result by
2, the answer is 18
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Mathematics
7) If you double the number and subtract 7 you get the same answer as
when you add 5 to the number.
8) If you multiply the number by 5 and subtract 4, you get the same
answer as when you add 3 to the number and then double the result.
9) If you multiply the number by 6 and add 1, you get the same answer
as when you add 5 to the number and then treble the result.
10) If you add 5 to the number and then multiply the result by 4, you get
the same answer as when you add 1 to the number and then multiply
the result by 2.
5 Solving problems using linear equations
So far you have concentrated on solving given equations. Making up your
own equations helps you to solve difficult problems.
There are four steps.
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Mathematics
I Call the unknown number x (or any other suitable letter) and state the units
where appropriate. Write down what we call “x” is.
II Write the problem in the form of an equation. Read the problem carefully
III Solve the equation and give the answer in words.
IV Check your solution using the problem and not your equation.
Exercises 4 Solve each problem by forming an equation. The first
questions are easy but should still be solved using an equation, in
order to practise the method.
1 The length of a rectangle is twice the width. If the perimeter is 20 cm,
find the width.
x
2 The width of a rectangle is one-third of the length. If the perimeter is
96 cm, find the width.
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3 The sum of three consecutive numbers is 276. Find the numbers.
Let the first number be x.
4 The sum of four consecutive numbers is 90. Find the numbers.
5 The sum of three consecutive odd numbers is 177. Find the numbers.
6 Find three consecutive even numbers which add up to 1524.
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7 When a number is doubled and then added to 13, the result is 38. Find
the number.
8 If AB is a straight line, find x.
9 The difference between two numbers is 9. Find the numbers, if their
sum is 46.
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Mathematics
10 The three angles in a
triangle are in the ratio 1:3:5.
Find the angles.
11 The sum of three numbers is 28. The second number is three times
the first and the third is 7 less than the second. What are the numbers?
12 If the perimeter of this triangle is
22 cm, ,
find the length of the
shortest side.
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Mathematics
13 David weighs 5 kg less than John, who in turn is 8 kg lighter than
Paul. If their total weight is 197 kg, how heavy is each person?
14 The perimeter of the rectangle is 34 cm. Find x.
15 The diagram shows a rectangular lawn surrounded by a footpath x m
wide.
a) Show that the area of the
path is 4x 2 + 14x
b) Find an expression, in
terms of x, for the distance
around the outside edge of
the path.
c) Find the value of x when
this perimeter is 20 m.
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Mathematics
16 A man is 32 years older than his son. Ten years ago he was three
times as old as his son was then. Find the present age of each one.
17 A man runs to a telephone and back in 900 seconds. His speed on
the way to the telephone is 5 m/s and his speed on the way back is 4
m/s. Find the distance to the telephone.
18 A car completes a journey in 10 minutes. For the first half of the
distance the speed was 60 km/h and for the second half the speed was
40 km/h. How far is the journey?
19 A bus is travelling with 48 passengers. When it arrives at a stop, x
passengers get off and 3 get on. At the next stop half the passengers
get off and 7 get on. There are now 22 passengers. Find x.
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20 A bus is travelling with 52 passengers. When it arrives at a stop, y
passengers get off and 4 get on. At the next stop one-third of the
passengers get off and 3 get on. There are now 25 passengers. Find y.
21 Mr Lee left his fortune to his 3 sons, 4 nieces and his wife. Each son
received twice as much as each niece and his wife received £6000,
which was a quarter of the money. How much did each son receive?
6. Quadratic equations
Quadratic equations always have a x 2 term; in general they have also a xterm and a number. Generally they have two different solutions.
First we are going to solve quadratic equations with only two terms.
6.1 Equations in the form ax 2 + c = 0
Examples 5
a) Solve 2x 2 − 8 = 0 ; we transpose 8 to the right side
2x 2 = 8 ; We isolate x 2
x2 =
8
→ x 2 = 4 From here we can take the square root and the solutions
2
are x = 2 and x = −2
b) Solve 3x 2 + 5 = 0 ; we transpose 5 to the right side
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5
3 x 2 = −5 → x 2 = − , there is not any solution because the square of all
3
number gives a positive number.
6.2 Equations in the form ax 2 + bx = 0
Examples 6
a) Solve 3x 2 + 6x = 0
We factorise x (3x + 6 ) = 0
The product of two numbers can only be zero if one or both are zero, so
x = 0

x (3x + 6 ) = 0 ⇒ 
The solutions are x = 0 and x = −2

3x + 6 = 0 → x = −2
b) Solve 2x 2 − 3x = 0 ; x (2x − 3 ) = 0 ; x = 0 and 2x − 3 = 0 → x =
Note that in these cases one of the solutions is always x = 0 .
Exercises 5
Solve:
1) x 2 − 3 x = 0
2) 2x 2 − 6 = 0
3) 3 x 2 − 5 x = 0
4) 2x 2 + 7 = 0
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5) x 2 − 9 = 0
6) 2x 2 − 16 x = 0
6.3 Equations in the form ax 2 + bx + c = 0 (complete form)
If we fail to factorise using any simple method we use this formula that has to
be memorised.
− b ± b 2 − 4ac
x=
2a
Example 7
Solve x 2 − x − 2 = 0
Start writing down the values of a, b and c
a =1
− (− 1) ±
b = −1 and using the formula x =
c = −2
1+ 3
=2
2
The solutions are
1− 3
x2 =
= −1
2
x1 =
Exercises 6
Solve
1) x 2 − 3 x + 2 = 0
7-19
(− 1)2 − 4 ⋅ 1⋅ (− 2) = 1 ±
2 ⋅1
2
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Mathematics
2) x 2 − 6 x + 5 = 0
3) x 2 − 3 x − 4 = 0
4) 2x 2 + 2x − 12 = 0
5) 2x 2 − x − 6 = 0
6) 6 x 2 + 5 x − 6 = 0
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7) 3 y 2 − 2y + 1 = 0
8) 2y 2 − 7 y + 2 = 0
9) x 2 − 5 x + 3 = 0
10) (2x − 3 )(x + 1) = 2x + 5
11) a 2 − 2a(a + 1) = −8
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12) z 2 − 3z + 2 = 0
13)
3z − 1
2
=
3
z+2
14)
x − 1 x( x − 2 )
+
= x2 − 1
2
3
15) x( x − 2) − 3( x − 1) = x 2 − 2
Exercises 7
1 Two numbers which differ 3, have a product of 88. Find them
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2 The product of two consecutive odd numbers is 143. Find the
numbers
3 The height of a photo exceeds the width by 7 cm. If the area is 60 cm2,
find the height of the photo.
4 The length of a rectangle exceeds the width by 2 cm and the diagonal
is 10 cm long, find the width of the rectangle.
5 The area of the
rectangle exceeds the
area of the square by
24m2. Find x.
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6 Three consecutive integers are written as x, x + 1, x + 2. The square of
the largest number is 45 less than the sum of the squares of the other
numbers, Find the three numbers
7 (x- 1), x and (x + 1) represent three positive integers. The product of
the three numbers is five times their sum.
a) Write an equation in x,
b) Show that your equation simplifies to x3 -16x = 0.
c) Factorise x3 -16x completely,
d) Finally find the three positive integers.
8 An aircraft flies a certain distance on a bearing of 45º, and then twice
the distance on a bearing of 135°, its distance from the starting point is
then 350 km. Find the length of the first part of the journey.
9 The area of rectangle A is twice
the area of B. Find x.
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10 The perimeter of a rectangle is 68 cm, if the diagonal is 26 cm, find
the dimensions of the rectangle.
11 A stone is thrown in the air. After t seconds its height, h, above sea
level is given by the formula h = 80 + 3t- 5t2, Find the value of t when the
stone falls into the sea.
12 The total surface area of a cylinder, A, is given by the formula
A = 2 π r 2 + 2 π rh . Given that A = 150π cm2 and h = 10 cm , find the
value of r.
13 A rectangular pond, 6 m x 4 m, is
surrounded by a uniform path of
width x. The area of the path is equal
to the area of the pond. Find x.
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14 The perimeters of a square and a rectangle are equal. The length of
the rectangle is 11 cm and the area of the square is 4cm2 larger than the
area of the rectangle. Find the side of the square.
15 The sequence 3, 8, 15,24, ...can be written (1 x 3), (2 x 4), (3x5),(4x6)...
a) Write an expression for the nth term of the sequence.
One term in the sequence is 255.
b) Form an equation and afterwards find what number term it is.
16 A cyclist travels 40 km at a speed x km/h. Find the time taken in
terms of x. Find the time taken when his speed is reduced by 2km/h. If
the difference between the times is 1 hour, find the original speed, x
km/h.
17 An increase of speed of 4 km/h on a journey of 32 km reduces the
time spent by 4 hours. Find the original speed.
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18 A train normally travels 60 miles at a certain speed. One day, due to
bad weather, the train's speed is reduced by l0 mph, so the journey
takes 3 hours longer. Find the normal speed.
19 A number exceeds four times its reciprocal by 3. Find the number.
20 Two numbers differ by 3. The sum of their reciprocals is
numbers.
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7
; find the
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Solutions
Exercises 1
1) x = −9 , 2) x = 2 , 3) x = 3 , 4) x = −1, 5) x = 5 , 6) x =
4
, 7) x = 20 ,
5
3
5
1
5
, 9) x = 2 , 10) x = , 11) x = 1 , 12) x = − , 13) x = 1 , 14) x = ,
2
2
4
2
4
5
1
15) x = 3 , 16) x = , 17) x = 0 , 18) x = 1 , 19) x = , 20) x = − , 21) x = 0
3
9
9
5
7
1
9
22) No solution, 23) x = , 24) x = −9 , 25) x = , 26) x = , 27) x = −
3
9
3
2
15
14
6
35
29
28) x = −4 , 29) x =
, 30) x = −
, 31) x = − , 32) x =
, 33) x =
.
2
3
5
4
15
8) x = −
Exercises 2
4
1
2
11
1) x = , 2) x = − , 3) x = −7 , 4) x = 6 , 5) x = 2 , 6) x = − , 7) x =
,
3
3
5
4
7
12
8) x = −
, 9) x = 3 , 10) x = −4 , 11) x =
.
22
7
Exercises 3
9
3
1
, 2) 10 x + 19 = 16, x = − , 3) 4(x + 3 ) = 10, x = − ,
2
10
2
53
1
, 5) 3(2x + 4 ) = 13, x = , 6) 2(3 x − 6 ) = 18, x = 5 , 7)
4) 3(x − 11) = 20, x =
3
6
10
14
2x − 7 = x + 5, x = 12 , 8) 5 x − 4 = 2( x + 3), x =
, 9) 6 x + 1 = 3( x + 5), x =
3
3
10) 4(x + 5 ) = 2( x + 1), x = −9
1) 2x − 5 = 4, x =
Exercises 4
)
1 Width x, 2x + 4 x = 20; x = 3.3 cm. 2 Width x 2x + 6 x = 96; x = 12 cm.
3 x + x + 1 + x + 2 = 276; x = 91, the numbers are 91, 92 and 93.
4 x + x + 1 + x + 2 + x + 3 = 90; x = 21, the numbers are 21, 22, 23 and 24.
5 2x + 1 + 2x + 3 + 2x + 5 = 177; x = 28 , the numbers are 57, 59 and 61
6 2x + 2x + 2 + 2x + 4 = 1523; x = 253 , the numbers are 506, 508, 510.
7 2x + 13 = 38; x = 12.5 ; 8 x + 4 x + 3 x + 20 = 180; x = 20 .
9 x + x + 9 = 46; x = 18.5 , the numbers are 18.5 and 27.5.
10 x + 3 x + 5 x = 180; x = 20º ,the angles are 20º, 60º and 100º.
11 x + 3 x + 3 x − 7 = 28; x = 5 , the numbers are 5, 15 and 8.
12 3 x + 1 + 2x − 1 + x + 4 = 22; x = 3 , the shortest side is 2x − 1 = 5 cm.
)
)
)
13 x + x + 5 + x + 13 = 197; x = 59.6 , David weighs 59.6 kg, John 64.6 kg and
)
Paul 72.6 kg. 14 x + 4 + x − 1 = 17; x = 7 cm. 15 b) 8 x + 14 , c) x = 0.75 m.
16 The man is 58 years old and his son is 26, 17 Distance x;
x x
x
x
1
+ = 900, x = 2000 m , 18 Distance x;
+
= , x = 4 Km. 19 x = 21
5 4
60 40 6
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20 y = 23 , 21 £3600.
Exercises 5
1) x = 0, x = 3 , 2) x = ± 3 , 3) x = 0, x =
5
,4) There is not any real solution,
3
5) x = ±3 , 6) x = 0, x = 8 .
Exercises 6
3 +1
3 −1
1) x1 =
= 2, x 2 =
= 1, 2) x1 = 5, x 2 = 1, 3) x1 = 4, x 2 = −1
2
2
3
2
3
4) x1 = −3, x 2 = 2 , 5) x1 = − , x 2 = 2 , 6) x1 = , x 2 = −
2
3
2
7 ± 33
y1 ≈ 3.19, y 2 ≈ 0.31
7) There is not any real solution, 8) y =
4
5 ± 13
3 ± 73
9) x =
, x1 ≈ 4.3, x2 ≈ 0.697 , 10) x =
, x1 ≈ 2.89, x2 ≈ −1.37
2
4
3
8
11) a1 = −4, a2 = 2 , 12) z1 = 2, z2 = 1, 13) z1 = 1, z 2 = − , 14) x1 = , x 2 = −1 ,
4
3
15) x = 1
Exercises 7
1 x1 = −11, x2 = −8 and x1 = 8, x2 = 11 , 2 x1 = −13, x2 = −11 and x1 = 11, x2 = 13
3 The height 12 cm and the width 5 cm, 4 6 cm, 5 x = 11 m, 6
S1 : 8, 9, 10
S2 : −6, − 5, − 4
7 a) ( x − 1)x ( x + 1) = 15 x , c) x ( x − 4)( x + 4) , d) 0, 4, − 4 , 8 70 5 ≈ 156.5 km
9 x = 4 , 10 24 x10 cm, 11 t = 4.31 s, 12 r = 5 cm , 13 x = 1 m, 14
40
40
13 cm, 9 cm , 15 a) n(n + 2) , b) n(n + 2) = 255, n = 15 , 16 t1 =
, t2 =
,
x
x−2
x = 10 km/h, 17 x = 4 km/h, 18 20 mph, 19 x = 4 , 20 x1 = 2, x 2 = 5 and
15
6
x1 = − , x 2 =
7
7
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8 Graphs and simultaneous
equations
Keywords
Coordinate/Cartesian plane/grid
x-axis
grid
y-axis
origin
axes
coordinates
increasing
decreasing
gradient
linear
substitution method
equating method
graph
function
ordered pair
quadrant
points
domain
continuous
range
discontinuous
slope
y-intercept
parallel line
elimination method
1 Coordinating the plane
1.1 The coordinate grid.
In the sixteenth century, the French mathematician René Descartes developed
a grid of numbers to describe the location of any point in the plane.
In the diagram below we have a horizontal line and a vertical one meeting at the
point 0. It is called the coordinate plane (or grid) or the Cartesian Plane in honour
of Descartes:
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The horizontal line is called the x-axis.
The vertical line is called the y-axis.
The point where the two lines meet is the origin.
1.2 Plotting points in the plane.
Let’s plot for example the point (2, 1): starting at the origin, we move 2 units to
the right along the x-axis; then move vertically up 1 unit. We describe the
location of this point as the ordered pair (2, 1). This ordered pair is also called
as the coordinates of the point. When giving the coordinates of a point, the xvalue comes first. Some points are plotted in the coordinate plane below:
1.3 The four quadrants
The diagram below shows that the x-axis and the y-axis divide the plane into four
quarters. Each of these is called a quadrant:
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Exercise 1
Draw a grid as shown. Join these points in order, using the same grid.
a) (4, 6), (5, 7), (6, 6), (4, 6).
b) (5, 8), (4, 8), (4, 7), (5, 8), (6, 8), (6, 7), (5, 8).
c) (4, 5), (5, 4), (6, 5), (5, 3), (4, 5)
d) (5, 2), (3, 4), (3, 5), (2, 5), (2, 8), (3, 8), (3, 9), (7, 9), (7, 8), (8, 8), (8, 5) (7, 5),
(7, 4), (5, 2).
10
9
8
7
6
5
4
3
2
1
1
2
3
4
5
6
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Exercise 2
The diagram below shows the face of a dice showing a 5. Write a set of
instructions that would give the face of the dice that shows a 5.
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2 Functions
2.1 Definition.
A function is a relation between two variables called x and y in which:
•
•
•
x is the independent variable
y is the dependent variable
Every x -value is related to one and only one y-value.
Functions can be represented using grids and points. This is important when
the function behaviour needs to be visualized. But be careful, because there are
graphs which are not functions. For example:
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Mathematics
In the second case, for example the x-value -1 has three different
correspondences, so, it is not a function.
The set of all real numbers variable x can take such that the expression
defining the function is real is the domain of the function, and the set of all
values that the function takes when x takes values in the domain is called the
range. For example:
The graph of a function has to be studied from the left to the right, that is to say,
how the y-coordinate varies when the x-coordinate increases.
If the graph of a function is going up from left to right, then it is an increasing
function.
If the graph is going down from left to right, then it is a decreasing function.
Exercise 3
Find the domains and ranges of the following functions:
a)
b)
7
y
6
5
4
3
2
1
x
-9
-8
-7
-6
-5
-4
-3
-2
-1
1
-1
-2
-3
-4
-5
-6
-7
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Mathematics
d)
Exercise 4
This function shows the temperature in a city during a day.
Answer these questions:
a) What are the x-variable and the y-variable?
b) What is the domain? And the range?
c) When is the function increasing and decreasing?
A continuous function is a function whose graph can be drawn without lifting
the chalk from the blackboard, (or the pen from the notebook). Otherwise, the
function is discontinuous. This is only an intuitive definition.
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3.5
y
Continuous Function
3
2.5
2
1.5
1
0.5
x
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0.5
1
1.5
2
2.5
3
3.5
-0.5
-1
-1.5
-2
-2.5
y
7
Discontinuous function
6
5
4
3
2
1
x
-9
-8
-7
-6
-5
-4
-3
-2
-1
1
2
-1
-2
-3
-4
-5
-6
-7
Exercise 5
This is the graph of a discontinuous function:
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Invent a situation that could be described in the graph.
Exercise 6
This is the price list of a car park:
Open from 9 h. to 22 h.
First and second hours.................................Free
Third hour and consecutive or fraction.........1 €
each.
Daily maximum.............................................12 €
Draw the graph of the function which relates park timing to its price. Is it a
continuous function?
2.2 Another way of seeing the concept of function.
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In mathematics we use the word FUNCTION for any rule, where for any input
number; there is only one output number. The input and the output numbers
may be written as an ordered pair (or a couple)
A set of couples will be a function if no two couples have the same first
component.
Thus, {(1, 4), (2, 5), (3, 6), (4, 7)} is a function, but {(2, 7), (3, 8), (3, 9), (4, 12)}
is not a function as (3, 8) and (3,9) have the same first component.
In a set of couples, for example:
{(1, 1), (2, 4), (3, 9), (4, 16), (5, 25)}
The set of first components is the domain:
Domain: {1, 2, 3, 4, 5}
The set of second components is the range:
Range: {1, 4, 9, 16, 25}
From this, you can see that a function may be thought of as a machine which
processes numbers:
0
1
2
3
2x + 3
3
5
7
Input
9
Output
Function
machine
2.3 Notation for a function
Notice that all the functions that we have seen until now have been studied with
a graph which allows us to know some characteristics of the function. But there
is a great amount of functions given by an analytical expression which
connects x and y variables algebraically.
We generally use the letter f to represent a function. If a function tell us to
double a number and add 4, i.e. 2x + 4, the function may be written in any of
these ways:
i) f (x) = 2x + 4
ii) y = 2x + 4
Example
Complete this table relating the base and height of rectangles whose
area is 12 m²:
Base x (m)
Height y (m)
1
2
3
4
6
12
x
a. Represent the function in the form of a graph.
b. Find which of these expressions corresponds to this function
x
12
y=
y=
y = 12 x
12
x
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Solution:
We complete the table and draw the graph:
Base x (m)
Height y (m)
1
12
2
6
3
4
4
3
6
2
12
1
x
12/x
y
12
11
10
9
8
7
6
5
4
3
2
1
x
-4
-3
-2
-1
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
-1
So, the right expression for this function is y =
12
x
2.4 Drawing the graph of a function given by a formula
To draw the graph of a function:
• Write a table of values
• Calculate the value of y for each value of x
• Draw a suitable grid
• Plot the pairs (x,y) and join them with a line
For example: draw a graph of the function y = 3x +1
First, make a table of easy values for the x-coordinates. Then calculate the
corresponding values of y, using y = 3x +1. These y-coordinates are also put in
the table:
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x
y
-1
-2
Mathematics
0
1
1
4
2
7
3
10
Then, construct a grid, plot the coordinates and join them to give a straight line
graph.
Note that this graph passes through countless other coordinates, all of which
obey the same rule of the function. You can choose any point on the line which
has not been plotted to show that this is true.
y
11
10
9
8
7
6
5
4
3
2
1
x
-5
-4
-3
-2
-1
1
2
3
4
5
6
7
8
9
10
11
12
13
-1
-2
-3
-4
Example: Make a table of values of the function y = x2. Then draw the graphs in
a suitable grid.
x
-2
-1
0
1
2
3
y
4
1
0
1
4
9
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y
10
9
8
7
6
5
4
3
2
1
x
-9
-8
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
7
8
9
-1
-2
-3
-4
Exercise 7
Copy and complete the table below for the functions:
i) y = x + 2
ii) y = x2
x
y=x+2
y = x2 – 4
-2
-1
0
Then draw both graphs on a suitable grid
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3 Linear graphs
3.1 Functions with the expression y = mx
Let’s work with some examples.
Draw the graph of y = 2 x and on the same grid the graph of y = 3 x and
compare them.
First we make a table with some points of y = 2 x
x-value
y-value
Point
0
0
O(0,0)
1
2
A(1,2)
2
4
B(2,4)
-1
-2
C(-1,-2)
-2
-4
D(-2,-4)
And then we represent all the points
y
9
8
7
6
5
B
4
3
A
2
1
x
-9
-8
-7
-6
-5
-4
-3
-2
-1
O
1
2
3
4
5
6
7
8
9
-1
C
-2
-3
D
-4
-5
-6
-7
-8
-9
If we make a table with some points of y = 3 x
x-value
y-value
Point
0
0
O(0,0)
1
3
A(1,3)
2
6
B(2,6)
And then we represent both graphs
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y
9
8
7
6
5
B
4
3
A
2
1
x
-9
-8
-7
-6
-5
-4
-3
-2
-1
1
O
2
3
4
5
6
7
8
9
-1
C
-2
-3
D
-4
-5
-6
-7
-8
-9
As it can be seen, the line y = 3 x is steeper than y = 2 x , the x-coefficient is
the gradient and indicates how steep the line is. All the functions with an
expression in the form y = mx go through the origin.
If we represent the lines y = −3 x and y = −2 x , we get:
y
f(x)=-3x
f(x)=-2x
9
y = -3x
y = -2x
8
7
6
5
4
3
2
1
x
-9
-8
-7
-6
-5
-4
-3
-2
-1
1
2
3
-1
-2
-3
-4
-5
-6
-7
-8
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We get decreasing functions and they go through the origin as well.
If we represent the functions y = 2 x + 1 and y = 2 x − 2 we get:
y
9
8
7
y = 2x+1
6
y=2x-2
5
4
3
2
1
x
-9
-8
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
7
8
9
-1
-2
-3
-4
-5
-6
-7
-8
-9
Comparing their graphs we can see that they are straight lines, their gradient is
2, so they are parallel lines, and they cut the y-axis in the points (0,1) and (0,-2)
1 and -2 are called y-intercept.
Summarising all this we make the following definitions:
3.2 Definitions:
All straight lines have a similar expression:
y = mx + n
where:
m is the gradient or slope, and indicates how steep it is.
n is the y-intercept, the point where the line cuts the y-axis.
Parallel lines have the same gradient.
If the equation of a straight line has no y-intercept, the line goes through the
origin and can be related to ratios.
Example:
When a car is filled with petrol, both the amount and the cost of the petrol are
displayed on the pump. One litre of petrol costs 1.20 €. So, 2 litres cost 2.40 €
and 5 litres cost 6 €.
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The table below shows the different quantities of petrol as displayed on the
pump:
Amount of
petrol (litres)
5
6
Cost (€)
10
12
15
18
20
24
25
30
30
36
The information can be graphed, as shown below. Notice that for every 5 litres
6
across the graph, the graph rises by 6 €. The ratio is . This is the reason why
5
the graph is a straight line. This idea can be used to solve a number of different
types of problem.
y
48
42
Cost (€)
36
30
24
18
12
6
x
5
10
15
20
25
30
35
40
45
Amount of petrol (litres)
Exercise 8
Mr Smith wants to convert all students’ scores in an English test to
percentages. He uses the facts given below to help him to draw a
conversion graph:
English score
Percentage
0
0
He uses the above two points to draw a linear graph.
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Mathematics
y
What percentage does Stephanie
get if she scores 30? What is
Joe’s score if his percentage is
63?
90
80
70
60
Percentage
50
40
30
20
10
-10
10
20
30
40
Student score
50
60
x
70
You can calculate the equation of a straight line from its graph. You just need to
find the gradient and the y-intercept:
Change in y − value ∆y
Gradient =
=
Change in x − value ∆x
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∆y 7 − 1
=
= 2 it means that for every change in de x∆x 3 − 0
value the correspondent change in the y-value is always the double.
As the gradient is 2 and the y-intercept 1 the quation of the straight line is
y = 2x + 1
In this case Gradient =
There are some graphs which are very useful when we are working with
functions. These are the horizontal graphs.
On this horizontal graph, the points are (-3,4), (-2, 4), (-1, 4), (0, 4), (1, 4), (2, 4).
The y-coordinate is always 4, so the equation of the graph is y = 4.
y
12
11
10
9
8
7
6
5
4
3
2
1
x
-9
-8
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
7
8
9
10
-1
-2
-3
The equation of any horizontal line is always y = a, where a is a real number
Exercise 9
Match the equation cards with the graph cards, explaining how you made your
choice.
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y
7
y
y
7
7
6
6
5
5
4
4
3
3
6
5
4
3
2
2
2
1
1
1
x
-9
-8
-7
-6
-5
-4
-3
-2
-1
-1
-1/2
1
2
3
4
5
6
7
8
x
x
9
-9
-8
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
7
8
9
-9
-8
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
7
8
9
-1
-1
-1
-2
-2
-3
-3
-4
-4
-5
-5
-6
-6
-6
-7
-7
-7
-2
-3
-4
-5
y = 3x + 2
y = 2x – 1
y = 5x + 1
Exercise 10
Answer ‘true’ or ‘false’ to the following:
a) y = 5x is steeper than y = 3x
b) y = 2x is parallel to y = 2x + 4
c) y = 4x is steeper than y = x – 5
d) y = 2x is parallel to y = x + 2.
EXERCISES:
11 Which is the graph for each of the following situations? Indicate what
the independent and dependent variable represents in each case:
a) Route made by an urban bus.
b) Bike ride round a park, stopping once to drink some water.
c) Distance made by a racing car in a circuit section.
d) A postman delivering the post.
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12 Match every statement with the corresponding graph and indicate what
the independent and dependent variable represent in each case:
a) Height of a ball that bounces constantly.
b) Cost of a phone call according to its time.
c) Distance to get home during a 30-minute walk.
d) Level of the water in an empty swimming pool when filling it.
13 Four friends are talking about their walk to school this morning.
- Jessica: I came by motorbike but I forgot an essay I have to hand
in and I had to go back home. Then I ran as fast I could to go to the
school.
- Brian: My mother brought me by car; but we found a traffic jam at
the traffic lights situated half way and we were too late.
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- Elena: I met a friend who attends a different school at the front
door of my house. We walked together part of the way, and when we
parted I had to hurry up. I was too late because of our walk.
- Andy: I left home very quickly because I had arranged to meet
Maria and it was late. Then we walked together slowly.
All of them go to the same secondary school, and each graph shows, in a
different order, the path followed since they left home until the entrance to
the school.
The same scale has been used in every graph.
Match each boy/girl with the graph that represents their walk to school.
14 Write the equations of the following straight-line graphs:
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15 Write the equations of the following straight-line graphs:
a) Gradient is 5 and cuts the y-axis at (0, 1).
b) Gradient is
1
and cuts the y-axis at (0, -1)
2
c) Parallel to y = 3x + 5 and cuts the y-axis at (0, 7)
d) Twice as steep as y = x – 1 and cuts the y-axis as the same place as y =
x+3
e) Goes through the points (1, 5) and (2, 9)
16 Select the equations that produce horizontal graphs and then make a
sketch of them indicating where they cut the y-axis:
a) y = 3x2
b) y = - 4
c) y = 2x – 6
d) y = 2
e) x = - 1
4 Simultaneous equations
Simultaneous equations are a set of two or more equations containing
some variables or unknowns, simultaneous equations can also be called a
system of equations.
We will work with sets of two equations containing two variables.
A solution is a pair of values which replace the unknowns and makes both
equalities true.
4 x + 3 y = 11 
 is a set of linear equations or a system of equations
3 x − 2y = −13
and you can check that x = −1 and y = 5 is a solution.
Example:
5 Solving simultaneous equations
5.1 Graphical solution.
Each one of the equations in the system is in fact a function when we isolate y,
so, we can draw both functions on the same set of axes; each point that we plot
is a solution of this particular equation, because it makes the equality true. The
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common point of the two straight lines gives us the solution of the simultaneous
equations.
Example: Solve using graphical method
3x − y = 1

x + 2y = 5
1. First of all we isolate y in the first equation: y = 3 x − 1 , this is a function.
2. We make a table and draw the graph
x 0 1 2
y -1 2 5
3. We do the same with the second equation y = −
x -1 0
y 3 5
2
2
3
2
4. We draw this function on the same
set of axes
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As the common point is (1,2) that is x = 1 and y = 2 , it satisfies both equations
simultaneously, this is the solution.
Usually we can get only an approximation of the solution when using this
method.
5.2 Algebraic solution.
There are three algebraic methods: The substitution method, the equating
method and the elimination method. As an example we will solve the same
3x − y = 1
system
 using all three methods, but in each case we must choose
x + 2y = 5
the most suitable one.
5.2.1 Substitution method.
On this method we isolate one of the variables in one equation and substitute its
expression in the other and after we must solve this equation.
3x − y = 1 (I)
Lets solve using this method:
 .
x + 2y = 5 (II)
1. Isolate y in (I) y = 3 x − 1 (*)
2. Substitute this expression in (II) x + 2(3 x − 1) = 5
3. Solve the equation x + 6 x − 2 = 5 → 7 x = 7 → x = 1
4. If we substitute this value in (*) y = 3 ⋅ 1 − 1 → y = 2
The solution is x = 1 and y = 2 , we must check this solution in the system.
This method should be used when at least one equation contains a single y or a
single x, because then we can avoid using fractions when they are isolated.
5.2.2 Equating method.
In this method we isolate the same variable in both equations and then equal
the expressions that we have already got.
As an example we will work with the same system, even though we already
know the solution.
3x − y = 1 (I)

x + 2y = 5 (II)
1. Isolate y in (I) y = 3 x − 1
5−x
2. Isolate y in (II) y =
2
5−x
2
4. Solve the equation 6 x − 2 = 5 − x → 7 x = 7 → x = 1
And by substituting as we have done before, we get our solution: x = 1 , y = 2 .
3. Equal both expressions for y 3 x − 1 =
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This method is suitable when at least one of the variables, x or y, is contained in
both equations with coefficient 1.
5.2.3 Elimination method.
This method must be used when the other two are unsuitable. This is the most
general method and some people prefer to use it all the time.
This method consists of multiplying, when needed, one or both of the equations
by a suitable number so as we can get the opposite coefficient of one of the
variables and when we add the equations this variable cancels out.
We use the same system again.
3x − y = 1 (I)

x + 2y = 5 (II)
1. Multiply (I) by 2:
6 x − 2y = 2 (I)

x + 2y = 5  (II)
6x − 2y = 2 (I)

2. Add (I) and (II) x + 2y = 5  (II) so x = 1 and working as we have done in the
(I) + (II) → 7x = 7
previous methods: y = 2 .
Exercises
17 Use the graphical method to solve: (Draw them in your notebook)
a)
x + y = 2

x − y = 3
b)
x − 2y = 1 

2 x + y = −3 
c)
2x − 3 y = 12

3 x + 2y = 5 
18 Solve using the substitution method and draw a graph of a)
x − 3 y = −8 
a)

− 3 x + 2y = 17
b)
2x − 3 y = 9 

5 x + y = −3 
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c)
Mathematics
− 3 x + y = −5

x + 5 y = 23 
19 Solve using the equating method and draw the graph of b)
3 x + y = 4
a)

2 x − y = 1
b)
x − 5 y = −12

x+y=0 
c)
2x + y = 4 

− 8 x + y = −1
20 Solve using the elimination method and draw a graph of c)
a)
2 x + 3 y = 2

3 x + 2y = 1
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b)
− x + 4 y = −1

3 x + 2y = 17 
c)
2x + y = 3 

3 x + 2 y = 5
Mathematics
21 Solve using the most suitable method
a)
5 x + y = 10

3 x + 2y = 1
b)
− 3 x + 2y = 5

5 x + 4 y = −1 
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c)
Mathematics
x − 3y = 7 

− x + 4 y = 2
y

= 3
d)

2
x + 7 y = −1
2x +
e)
− x + 3 y = 15 

− 2 x + 6 y = −3 
1 
3 
f)
3x
7
−
+ 2y = 
2
2
7x − y =
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Solutions
1
.
2
a) (2, 2), (1, 3), (1, 8), (2, 9), (7, 9), (8, 8), (8, 3), (7, 2), (2, 2).
b) (2, 3), (2, 4), (3, 4), (3, 3), (2, 3).
c) (6, 7), (6, 8), (7, 8), (7, 7), (6, 7).
3 a) Domain: {0,1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, Range: {150, 160, 170, 180, 190,
200, 210, 220, 230, 240, 250}; b) Domain: - {0}. Range: ; c) Domain [-3,4].
Range: [0,16]; d) Domain: [0, 24]. Range [-2, 4]
4 a) x-variable: hour of the day; y-variable: temperature. b) Domain [0, 24].
Range: [12, 40]. c) Increasing: (6, 15). Decreasing (0, 6)∪(15, 24)
5 For example, the amount of money during a day inside a water dispenser
machine in a school, when the machine is on.
Exercise 6
30
y
28
26
24
22
20
18
16
14
12
10
8
6
4
2
x
2
4
6
8
10
12
14
16
18
20
22
24
26
28
30
32
34
36
(x-axis in hours, y-axis in €)
7
x
y=x+2
y = x2 – 4
-2
0
0
-1
1
-3
0
2
-4
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1
3
-3
2
4
0
3
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y
y
10
7
9
6
8
5
7
4
6
3
5
2
4
1
3
x
-9
-8
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
7
8
9
2
-1
1
x
-5
-4
-3
-2
-1
1
2
3
4
5
6
7
8
9
10
11
12
-2
13
-3
-1
-4
-2
-3
-5
-4
-6
-5
-7
8 Stephanie gets 50%. Joe´s score is 38
9
y
7
y
y
7
7
6
6
5
5
4
4
3
3
6
5
4
3
2
2
2
1
1
1
x
-9
-8
-7
-6
-5
-4
-3
-2
-1
-1
-1/2
1
2
3
4
5
6
7
8
x
x
9
-9
-8
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
7
8
-9
9
-8
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
-1
-1
-1
-2
-2
-3
-3
-4
-4
-2
-3
-4
-5
-5
-5
-6
-6
-6
-7
-7
-7
y = 3x + 2
y = 2x – 1
y = 5x + 1
10 a) True. b) True. c) True. d) False,
11 a-III, b-IV, c-I, d-II,
12 a-II, b-III, c-IV, d-I;
13 Jessica – B, Brian – D, Elena – A, Andy – C
14 a) y = 2x + 2, b) y = 4x – 1,
1
15 a) y = 5x +1, b) y = x – 1 , c) y = 3x + 7
2
d) y = 2x + 3, e) y = 4x + 1;
16 Horizontal graphs: b) y = -4 and d) y = 2
y
y
7
7
6
6
5
5
4
4
3
3
2
2
1
1
x
-9
-8
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
7
8
9
x
-9
-8
-7
-6
-1
-5
-4
-3
-2
-1
1
2
-1
It cuts the y-axis in (0, -4)
It cuts the y-axis in (0, 2)
-2
-2
-3
-3
-4
-4
-5
-5
-6
-6
-7
-7
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4
5
6
7
8
9
5
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17
a)
b)
x = 2.5, y = −0.5
x = −1, y = −1
c)
x = 3, y = −2
18 a) x = −5, y = 1; b) x = 0, y = −3 ; c) x = 3, y = 4
19 a) x = 1, y = 1; b) x = −2, y = 2 ; c) x =
1
,y = 3
2
1
4
20 a) x = − , y = ; b) x = 5, y = 1 ; c) x = 1, y = 1
5
5
18 a)
19 b)
20 c)
19
25
43
10
,y = −
; b) x = −1, y = 1 ; c) x = 34, y = 9 ; d) x =
,y = −
7
7
27
27
1
e) No solution; f) x = , y = 2 .
3
21 a) x =
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9 Statistics
Data
interval
pyramid
median
Keywords:
frequency table
bar chart
pie chart
tally
pictogram
population
temperature and Rainfall chart
mean
mode
average deviation
Previous ideas
a) Sort data into discrete, continuous and categorical.
b) Choose the most appropriate graph to display data: bar chart, pie chart,
pictogram,
1. Constructing a frequency table
Example 1 Twenty families are asked about how many children they have.
These are the answers:
3
2
3
2
4
3
1
2
2
4
3
2
2
5
Data
0
1
2
3
4
5
5
3
1
4
0
3
Frequency
1
2
6
6
3
2
Now, we can draw a bar chart or a pie chart:
Families' children
Number of
children
Families' children
Number of families
7
10%
6
5%
0
10%
15%
5
1
2
4
30%
3
2
30%
1
3
4
5
0
0
1
2
3
4
5
Number of children
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Histogram:
The histogram is used for variables whose values are numerical and measured
on an interval scale.
A histogram divides up the range of possible values in a data set into classes or
groups. For each group, a rectangle is constructed with an area proportional to
the frequency, (if the bars have equal width the height of each bar corresponds
to the frequency).
Notes:
A vertical bar graph and a histogram differ in these ways:
In a histogram, frequency is measured by the area of the column.
In a vertical bar graph, frequency is measured by the height of the bar.
Example 2 Mario decided to collect data about the height of his partner in the
school. These are the data of 40 children (in cm):
163
167
165
159
164
168
161
164
163
164
165
163
167
165
164
164
168
161
164
165
163
164
170
160
157
167
165
172
165
167
164
164
168
161
164
163
164
155
158
162
It is useful to make a grouped frequency table
for this case:
Interval
[155-160)
[160-165)
[165-170)
[170-175)
Tally
IIII
IIII IIII IIII IIII I
IIII IIII III
II
Frequency
4
21
13
2
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Mathematics
Example 3 These are the data about employment in a town.
Men
Women
Total
Unemployment Employment
50
150
100
100
150
250
Total
200
200
400
We draw a composite bar chart to display the data.
Employment
300
250
people
200
Women
Men
150
100
50
0
Unemployment
Employment
In a composite bar chart you can compare totals easily, but is hard to see the
separate data.
2. Interpreting diagrams
Pie Charts
A pie chart is a circle graph divided into sectors, each displaying the size of
some related piece of information. Pie charts are used to display the sizes of
parts that make up some whole.
Example 4 Favorite movie genres in Mrs. Lucia's Film class
In the pie chart above, the legend is made properly and the percentages are
included for each of the pie sectors. However, there are too many items in the
pie chart to quickly give a clear picture of the distribution of movie genres. If
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there are more than five or six categories, consider using another graph to
display the information.
a) Suppose 100 students were in all. Then there were 27 students that
prefer comedy movies, 18 that prefer action movies, 14 romances, 14
drama too, 11 horror, 8 foreign and 8 science fiction.
b) Suppose 300 students. Then there were 71 students that prefer comedy
movies, …
Exercise 1 a) In my high school there are
200 students. They were asked about their
favourite pet. See the figure and answer the
questions:
Favourite pet
10%
How many people answer “dogs”? And
“cats”?
30%
15%
Dogs
Cats
Birds
Fish
20%
25%
Pictogram
b) These are the number of photographic
cameras sold the last year
10 photographic cameras
Cameras photographic
January
February
March
April
May
June
July
August
September
October
November
December
How many cameras had been sold every month? Which one was the best
month for the salesman? Why do you thing was this month?
Population pyramid
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Example 1
This population pyramid has a
very wide base, showing that
this country has a high birth
rate.
This is likely to be the
population pyramid of a less
developed country and because
of a high infant death.
There is a sharp indent in the male side from 15 years upwards. This could be
because of a conflict or migrated to other countries to get jobs.
There is a low life expectancy.
Example 2
The largest grouping is
between 30 and 45.
The problem in the future will
be when there few people
paying taxes to support the
pensions.
There are large numbers of
people living into their 80s.
Exercise 2
See these three population pyramids:
a)
b)
Could you say which one is from Mexican or UK?
Climate Data: Temperature and Rainfall Charts
Example
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The graphic shows average rainfall and temperature in Manaus, Brazil, in the
Amazon rainforest.
Total rainfall is 2104mm per year, most of it during the rainy season from
December to May.
Notice how much the rainfall varies over the year: the highest monthly rainfall is
in March with over 300mm, while the lowest is in August with less than 50mm.
Meanwhile the temperature only varies by 2°C.
Exercise 3
These are the data from Ronaldsway airport in Isle of Man (between
England, Scotland and Gales):
Could you explain each other?
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What about your city? Are similar?
3. Parameter statistics
Mean
The mean of a list of numbers is also called the average. It is found by adding
all the numbers in the list and dividing by the number of numbers in the list.
Example 1 Find the mean of 10, 11, 7, and 8 to the nearest hundredth.
mean ≡ x ≡
10 + 11 + 7 + 8
=9
4
Median
The median of a list of numbers is found by ordering them from least to
greatest. If the list has an odd number of numbers, the middle number in this
ordering is the median. If there is an even number of numbers, the median is
the sum of the two middle numbers, divided by 2. Note that there are always as
many numbers greater than or equal to the median in the list as there are less
than or equal to the median in the list.
Example 2 My marks in Maths were 7, 10, 8, 10, 7 last year. Find the media.
Placed in order, the age’s marks were 7, 7, 8, 10, 10. The number 8 is the
median.
Imagine my marks were 7, 10, 8, 10, 7 and 7. So the middle numbers are 7 and
8, which are the 3th and 4th.
The median is the average of these two numbers:
7 + 8 15
=
= 7 .5
2
2
Mode
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The mode in a list of numbers is the number (or numbers) that occurs most
often.
Example
The students in Bjorn's class have the following ages: 5, 9, 1, 3, 4, 6, 6, 6, 7, 3.
Find the mode of their ages. The most common number to appear on the list is
6, which appears three times. No other number appears that many times. The
mode of their ages is 6.
Average deviation
Deviation is a measure of difference for interval and ratio variables between the
observed value and the mean.
Average deviation is calculated using the absolute value of deviation (it is the
sum of absolute values of the deviations) divided by the number of data.
The ages of my cousins are 10, 15, 12, 13, 10, 12
We calculate the mean:
Mean ≡ x =
10 + 15 + 12 + 13 + 10 + 12
= 12
6
Data
Absolute value of
deviation
Average deviation ≡
10
15
12
13
10
12
2
3
0
1
2
0
2 + 3 + 0 + 1+ 2 + 0
= 1.33
6
Exercises 4
4.1 The tallest 4 trees in a park have heights in meters of 40, 52, 50, 55.
Find the median of their heights.
4.2 Find the mean, median, mode and average deviation for the following
data:
a) 10, 12, 13, 12, 13, 10, 14 and 13.
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4.3 Twenty families are asked about how many children they have. These
are the answers:
3
2
3
2
4
3
1
2
2
4
3
2
2
5
5
3
1
4
0
3
a) Find the mean, median, mode and average deviation.
b) Make the frequently table. Can you use it for a)
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Mathematics
Solutions
1 a) Dogs 60 people, cats 50 people.
b)
Month
January
February
March
April
May
June
July
August
September
October
November
December
Cameras
60
30
20
30
40
30
50
50
30
20
30
50
The best month for the salesman was January, it
is Christmas time.
2 option b)
3 Temperatures and sunshine present the highest quantities during summer,
while the rainfall is in their lowest values during that season. Yes
Exercises 4
4.1 51 m 4.2 mean 12.12, median 12.5, mode 13, average deviation 1.125.
4.3
Children (x) f
0
1
2
3
4
5
Total
x*f
1
2
6
6
3
2
20
mean =
dm =
abs(d)
0
2
12
18
12
10
54
2,7
1,03
abs(d)*f
2,7
1,7
0,7
0,3
1,3
2,3
median =
modes
9-10
2,7
3,4
4,2
1,8
3,9
4,6
20,6
3
3 and 2
10 Similarity
Keywords:
Similarity
Scale Thales position
similarity criterion
Previous ideas
1 Triangle
A triangle is a three-sided polygon. We
use capital letters to name the vertices
and lower case letters to the opposite
sides like this
2 Angles in two intersecting lines crossed by two parallels
Alternate Interior Angles c = b and e = h
Alternate Exterior Angles f = g and a = d
Corresponding Angles e = f and a = b
f = h or b = d, for example, are Vertical Angles.
1 Similar shapes
Two geometrical objects are similar when one is the result of enlarging or
shrinking the other, corresponding angles are equal and the length of
corresponding segments are in the same ratio
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Example These two shapes are similar
Ex
er
cise 1 Use a protractor to
check that corresponding
angles are equal and use a
ruler to find out which is the
ratio between the
corresponding sides of the
shapes.
Exercise 2 Find out if these two pictures are similar
The height of the door on the left is 2 m. Which is the height of the door
on the right?
Calculate the real height of the house on the left.
2 Maps
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Maps are representations of an area. There is a scale, the ratio between the
real objet and the map, and then we can know the real distance between two
points.
Example in this map the scale is 1:100 it means that one cm of the map is 1
m in the house
Exercise 2 Find the dimensions of each room of this house. Calculate
the size of the flat in m2
2 Thales Theorem (Intercept theorem)
If two non parallel lines intersect with parallel lines, the ratios of any two
segments on the first line are equal to the ratios of the corresponding
segments on the second line
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That is:
AB A ' B'
AB
BC
=
, you can also say that
=
BC B' C'
A ' B' B' C'
Exercise 3 Calculate the ratio (use a ruler)
AB
BC
or
A ' B'
B' C'
of the picture
above.
Exercise 4 Consider that lines a, b and c are parallel lines and
AB = 2 cm ,
EF = 2.8 cm and
BC = 3.5 cm .
Calculate DE
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3 Triangles put in the Thales position. Similar triangles
If we use the Thales theorem in the drawing above in which the non parallel
lines intercept at A, the triangle ABC is inside and fitted on the triangle ADE,
we say that these two triangles are in the Thales position and they are similar
because corresponding angles are equal and corresponding sides are in
proportion
Exercise 5 Use the Thales theorem to find DE in the picture below
D
AD=10,50 cm
B
AB= 6,20 cm
BC= 4,86 cm
A
C
E
Use a ruler to measure the real lengths and a protractor to measure the
angles in the two triangles. Check that the triangles ABC and ADE have
the same angles and their sides are in proportion. What can you say
about these triangles?
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3 Similar triangles, similarity criteria
To be sure that two triangles are similar, we do not need to see if all the three
corresponding angles are equal and corresponding sides are in proportion.
We can assure that they are similar with fewer conditions. These sets of
conditions are called similarity criteria.
3.1 First similarity criterion
Two triangles ABC and A’B’C’ are similar if  =  ' and B̂ = B̂ ' because then
Ĉ = Ĉ' . Remember that  + B̂ + Ĉ =  '+B̂'+Ĉ' = 180º .
3.2 Second similarity criterion
Two triangles ABC and A’B’C’ are similar if all the three sides are in
a b c
proportion that is
= = , this is true because these triangles could be
a' b' c '
put in the Thales position on any vertex.
3.2 Third similarity criterion
b c
= , this is like
b' c '
that because these triangles could be put in the Thales position on the vertex
A.
Two triangles ABC and A’B’C’ are similar if  =  ' and
Exercises 6
Exercise 6.1
Measuring the length of the shadow of a stick, we can calculate the
height of a tree. Calculate the height of the tree from the picture below
considering that the length of the stick is 1.25 m its shadow is 1.52 m
and the shadow of the tree is 6.3 m
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6.2 Look at the scale of this map and calculate:
a) The distance covered by a person that goes from our school in A
to their house in Arquitecto Fernández (D) following the route
ABCD
b) The shorter distance that has to be covered if we want go from A
to E
6.3 The two polygons are similar. Find out:
a) FE
b) Ê'
d) What is the ratio between every pair of sides?
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D
6,12 cm
C
D'
74,2 ° E
3,42 cm
C'
E'
B
B'
24,2 °
A
6,62 cm
2,94 cm
24,2 °
A'
F
F'
6.4 The three sides of a triangle ABC are a = 2 cm , b = 2.7 cm and
c = 3.4 cm . The triangle A´B´C´ is similar and a′ = 2.5 cm . Which are the
values of the other two sides? Draw these triangles using a ruler and a
compass.
6.5 The triangles ABC and A`B`C` have  =  ' = 45,1º and Ĉ = Ĉ ' = 102,6º
a) Check that they are similar triangles
b) Find the value of the angle B̂
C'
102,6 °
C
102,6 °
45,1 °
B'
À'
45,1 °
B
A
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c) Calculate the similarity ratio
6.6 Prove without measuring that these triangles are similar. Check that
their corresponding angles are equal.
4,01 cm
4,22 cm
6,75 cm
7,14 cm
7,12 cm
12,04 cm
6.7 Prove without measuring these triangles are similar. Check that
corresponding angles are equal and sides a and d are in proportion
with the other corresponding sides
6.8 Prove that the triangles AA`C and AA`B are similar. Which are the
corresponding sides of these triangles?
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Solutions
1.1 Ratio 1:2 (approximately); 1.2 yes, they are similar, about 3 m;
2 Estimations:Left bedroom: 3.30x4.20 m, midle bedroom: 3.5x3 m, right
bedroom: 4x2.6 + 1.5x1.5 m + bathroom of 2.40x1.50 m, living room: 8x4.20
m, kitchen: 2.30x2.50 m, second bathroom: 2.30x1.70 m, size of the flat:
about 87 m2;
AB 2.7 A ' B'
4
=
=
3
,
(the lines are not parallel, the picture is only an
BC 2.4 B' C' 3.4
AB BC
approximation), 4 DE = 1.6 cm ; 5
=
≈ 0.6 → DE ≈ 8.1 cm , Â ≈ 37º ,
AD DE
B̂ = D̂ ≈ 93º , Ĉ = Ê ≈ 50º , they are similar triangles.
Exercises 6
6.1 The height of the tree is 5.18 m. 6.2 a) 210+600+120=930 m, b) 810 m
6.3 a) FE = 5.26 , b) Ê' = 74.2º
c) A ′F′ = 3.7cm d) the ratio is 1’79;
6.4 b′ = 3.375 cm c ′ = 4.25 cm
(Aproximation of the drawing)
6.5 b) B̂ = 32.3º , c) the similarity ratio is 1.35. 6.6
12.4 6.75 7.12
=
=
= 1.68
7.12 4.01 4.22
6.55 8.13
≈
= 1.707 and  = D̂ = 43º . 6.8 ∠C = ∠A ′AB , ∠B = ∠CAA ' ;
3.84 4.76
A ′C ⇔ AA ′ , AA ′ ⇔ A ′B and AC ⇔ AB .
6.7
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11 3-D shapes
Keywords:
Cuboid
cube
slant height
prism
trunk
pyramid
base
apex
cylinder
cone
sphere
Remember
Polyhedrons
Polyhedrons are geometric solids whose faces are formed by polygons
Components: Faces, Edges, Vertices, Dihedron angle
Regular polyhedrons:
Tetrahedron
Cube
Octahedron Dodecahedron Icosahedron
1 Cuboid
In cuboids, faces are rectangles. They have 6 faces, 12 edges and 8 vertices
1.1 Area and volume of a cuboid
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If we name the edges of the cuboid a, b and c, its area is:
A = 2ab + 2ac + 2bc = 2(ab + ac + bc ) And the volume is V = a ⋅ b ⋅ c
1.2 Area and volume of a cube
The special case of cuboids in which all the faces are squares is the cube.
The area of a cube is A = 6a 2
The volume is V = a 3
1.3 Diagonal of a cuboid
In this cuboid the triangle with sides
a, b and e is a right triangle, so using
the Pythagorean Theorem we can
say e 2 = a 2 + b 2 , but the triangle with
sides c, e and d is a right triangle as
well in which d is the hypotenuse
and d 2 = e 2 + c 2 , that is:
d 2 = a 2 + b 2 + c 2 And finally
d = a2 + b2 + c 2
Exercise 1 Find the area, the volume and the diagonal of each solid
shown below:
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a)
b)
Exercise 2 Find the area, the volume and the missing length of these
cuboids:
a)
b)
Exercise 3 Find the area and the volume of
each solid shown below:
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a)
b)
2 Prisms
They have a constant cross-section, side faces are parallelograms
The distance between the two bases is the height of the prism.
Depending on the polygons of the bases they can be:
Triangular prisms, square prisms, pentagonal prisms, etc.
This is an hexagonal prism
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2.1 Area and volume of prisms
The area of a prism is found by adding the areas of its faces, if we call l the
side of the base, a the apothem and h the height of the prism, the total area
is A = n ⋅ l ⋅ a + n ⋅ l ⋅ h
Since the area of each base, which is a regular polygon of n sides, is
n⋅l⋅a
Ab =
and we have n rectangles each one of width l and height h
2
The volume is the area of the cross-section (base) multiplied by the height
V = base area × height
When the prism is oblique the
formula is the same but the height
is not the length of the edge.
Exercise 4 Find the area and the volume of each prism shown below:
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a)
b)
c)
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3 Pyramids
In a pyramid one of its faces is a polygon called the base and the other faces
are triangles that join at a point that is the apex.
The height of a pyramid (h) is the distance from the base to the apex.
This is a square pyramid
3.1 Area of a pyramid
The area of a pyramid whose base is a regular polygon of n sides of length b
and apothem g, and calling a to the height of the triangular faces (slant
height) is:
n⋅b ⋅g
A = A b + A l , The base is a regular polygon with area A b =
and there
2
n⋅b ⋅a
are n triangular faces with a total area of A l =
2
n ⋅ b ⋅ g n ⋅ b ⋅ a n ⋅ b ⋅ (g + a)
A=
+
=
2
2
2
If it is a square based pyramid like the one in the picture, the area is
A = b2 + 2 ⋅ b ⋅ a
3.2 Volume of a pyramid
The volume of a pyramid is V =
area of the base × height
3
If it is a square pyramid like the one in the picture V =
3.3 Trunk of a pyramid
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1 2
b ⋅h
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A trunk of a pyramid is the part of a pyramid which is between two parallel
planes. The faces of the solid obtained by cutting it are called bases of the
trunk, and the Distance between the two cutting planes is the height of the
trunk. The side faces are trapeziums.
The area of a trunk of a pyramid with bases which are regular polygons of n
sides, is
l + l'
a , being l and l’ the
2
sides of the base polygons and a the height of the
trapeziums.
A = A base1 + A base 2 + n
In the square based pyramid on the right the area
(l + l' ) ⋅ a
of a trapezium is A t =
and the area of
2
2
the bases are A 1 = l 2 and A 2 = (l')
The area of the trunk is
2
A = 2(l + l' ) ⋅ a + l 2 + (l')
Exercise 5 Find the area and the volume of these pyramids:
a)
4,8 cm
4,5 cm
3,8 cm
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b)
Exercise 6 Find the area and the volume of the trunk of pyramid shown
below:
4 Cylinders
A cylinder is a curvilinear geometric solid formed by a curved surface and
two circles as bases.
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The curved surface unrolled is a rectangle that measures h by 2π r ; the
radius of the cylinder is the radius of any of the two bases.
4.1 Area of a cylinder
The area of a cylinder is A = 2π r h + 2π r 2 and the volume is V = π r 2 h
If the cylinder is an oblique cylinder, the formula for the volume is the same,
but the perpendicular height is not equal to the height of the curved surface.
Exercise 7 Find the area and the volume of these cylinders
a) r = 15 cm h = 12 cm
b)
c) r = 8 mm h = 34 cm
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5 Cones
A cone is a solid bounded by a curved surface that has a common point
(vertex) and a circle as the base of the cone.
Vertex or apex is the top of the cone (V).
Slant height of the cone is the straight line that joins the vertex with the
circle of the base (g).
5.1 Area and volume of a cone
The area of the curved surface of the cone is the area of a sector and it is
given by the formula A = π r g and the area of the base is A b = π r 2 so the
total area of the cone is A = π r g + π r 2
The volume of the cone is V =
1 2
πr h
3
5.2 Trunk of a cone
A trunk of a cone is the part of the cone which is between two parallel planes.
The faces of the solid obtained by cutting it are called trunk bases, and the
distance between the two cutting planes is the height of the trunk.
The area of any trunk of a cone is the area of the cone minus the area of the
cone that has been removed
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As can be seen in a vertical cut of the cone on the right the triangles VBC
and VAD are similar and some of the measures can be calculated from the
others.
Exercise 8 Find the area and the volume of these solids:
a)
b)
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Exercise 9 The height of the cone of the picture is 38 cm and the radio r
is 15 cm. It has been cut by a plane at 12 cm from the vertex. Calculate:
a) The area of the trunk of the cone
b) The volume of the trunk of the cone
6 Sphere
In a sphere all points are at the same distance
r from the centre of the sphere C.
The distance from the centre to the surface of
the sphere is called the radius of the sphere r
Area and volume of a sphere
The area of a sphere of radio r is A = 4π r 2 and
4
the volume is V = π r 3
3
Exercise 10 Find the area and the volume of the
sphere
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Exercise 11 Find the total area and the volume of this solid
Exercise 12 Find the area and the volume of an hexagonal prism where
the base edge is 5 cm, the apothem 4 cm and the perpendicular height
13 cm
Exercise 13 Calculate the volume and the area of a squared-based right
pyramid. The edge of the base is 13.2 cm and the perpendicular height
is 17 cm long.
Exercise 14 The volume of this sphere is 32 cm3. find its radius and its
area.
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Exercise 15 The volume of a cone is 785.4 cm3 and the perpendicular
height is 10 cm. Find the area.
Exercise 16 One Egyptian pyramid has a base side of 112 m and a
height of 130m. Work out which volume of
stone would be needed to build this
pyramid.
Exercise 17 A tomato tin is cylindrical. Each tin has a capacity of 1 litre
and a base radius of 5 cm. Find the height and the area of the tin.
Exercise 18 The paddling pool in the picture is an hexagonal-based
prism, the base is a regular polygon, the edge
of the base is 1,5 m and the depth of the water
is 60 cm. Calculate the volume of water that
this pool contains. Calculate the weight of this
water in Kg.
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Exercise 19 A solid consists of a cylinder with a diameter of 15 cm and
a height of 7 cm glued together to an hemisphere, on one side, and to a
cone with a height of 10 cm, on the other. Find the total volume and
area of the solid
Exercise 20 One corner of a 13 cm sided cube has been removed by
cutting through the midpoints of the edges like in the picture. Find the
volume and the area of the remaining piece.
Exercise 21 A rectangular sheet of paper 15 cm by 29 cm is rolled up to
make a tube. Find the radius and the volume of the tube
a) If the long sides are joined
b) If the short sides are joined
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Solutions
1 a) A = 294 cm 2 , d = 12.12 cm , V = 343 cm 3 ; b) A = 355.6 cm 2 ,
d = 16.72 cm , V = 372 cm 3 .
2 a) a = 13.23 cm , A = 296.18 cm 2 , V = 278.89 cm 3 , b) b = 5.29 cm ,
A = 289.28 cm 2 , V = 317.4 cm 3
3 a) Vertical front area 592, horizontal outside area 1624, vertical outside
area 504, vertical inside area 560 and horizontal inside area 616
cm2 A T = 10032 cm 2 , V = 33152 cm 3 . b) A = 6228 cm 2 , V = 20088 cm 3
4 a) Side of the triangle 3.86 cm , A = 84.48 cm 2 , V = 40.64 cm 3 ,
b) A = 61.95 cm 2 , V = 33.075 cm 3 , c) A = 108.8 cm 2 , V = 67.2 cm 3
5 a) A = 50.92 cm 2 , V = 21.66 cm 3 ; b) A = 82.12 cm 2 , V = 31.62 cm 3
6 A = 975 u.a. , V = 1911.5 u.v. ;
7 a) A = 2544.7 cm 2 , V = 8482.3 cm 3
b) A = 1009 cm 2 , V = 2461 cm 3 c) A = 174.92 cm 2 , V = 68.36 cm 3
8 a) A = 417.6 cm 2 , V = 556.95 cm 3 b) A = 119.6 units , V = 84.7 units ;
9 a) A = 2514.7 cm 2 , b) V = 8672.4 cm 3 ;
11 A = 461.81 cm 2 , V = 718.38 cm 3 ;
12 A = 510 cm 2 , V = 780 cm 3 ;
13 V = 987.36 cm 3 , A = 655.78 cm 2 ;
14 r = 1.97 cm , A = 48.77 cm 2 ;
15
16
17
18
A = 595.57 cm 2 ;
V = 543573.3 m 3 ;
h = 12.7 cm A = 557.1 cm 2
V = 3.5m 3 , wheight = 3500kg ;
19 A = 977.82 cm 2 V = 2709.6 cm 3 ;
20 A = A cub − A faces + A triang face = 1014 − 63.38 + 36.7 = 987.3 cm 2
V = Vcube − Vtetrahedro n = 2197 − 45.8 = 2151.2 cm 3 ;
21 a) r = 2.38 cm, V = 520 cm 3 b) R = 4.61 cm, V = 1005.8 cm 3
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