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Rational N um bers
1. Natural Numbers
The natural numbers are the counting numbers from one to infinity. We use the letter N to refer
to the set of all natural numbers:
N = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, . . .}
The set of natural numbers is endless. As there is no largest counting number, we say that the set
of natural numbers is infinite.
The factors of a natural number are all the natural numbers which divide exactly into it, leaving no
remainder.
For example, the factors of 10 are: 1, 2, 5 and 10.
5 is a factor of 20 because 5 divides exactly into 20; we can also say that 20 is divisible by 5.
The even numbers are the natural numbers which are divisible by 2. The sequence of even numbers
is: 2, 4, 6, 8, 10, 12, 14 . . . and so on.
The odd numbers are the natural numbers which are not divisible by 2. The sequence of odd
numbers is: 1, 3, 5, 7, 9, 11, 13 . . . and so on.
A prime number is a natural number which has exactly two distinct factors, itself and 1.
For example, 17 is a prime number since it has only two factors, 1 and 17.
A composite number is a natural number which has more than two factors.
For example, 26 is a composite number since it has more than two factors: 1, 2, 13 and 26.
Notice that these definitions indicate that one (1) is neither prime nor composite.
Every composite number can be factorised as a product of prime factors in one and only one
way (apart from order).
For example,
72 = 2 · 2 · 2 · 3 · 3
or
72 = 23 · 32
(in exponent form).
A multiple of any natural number is obtained by multiplying it by another natural number.
For example, the multiples of 3 are: 3, 6, 9, 12, 15, 18,. . . and these are obtained by multiplying 3
by each of the natural numbers in turn:
3 · 1 = 3,
3 · 2 = 6,
3 · 3 = 9,
1
3 · 4 = 12 etc.
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2
Example 1
a) Find the largest multiple of 9 less than 500.
b) Find the smallest multiple of 11 greater than 1000.
a) We begin dividing 500 by 9:
500
5
9
55
We have 5 remainder, so 9 ·55 is smaller than 500. Hence,
as 9 · 55 = 495, the largest multiple of 9 less than 500 is
495.
b) We begin dividing 1000 by 11:
1000
11
10
90
We have 10 remainder, so 11 · 90 is smaller than 1000:
11 · 90 = 990. Hence, the smallest multiple of 11 greater
than 1000 is 11 · 91 =1001.
The HCF (highest common factor) of two or more natural numbers is the largest factor which is
common to both of them. For example:
The factors of 24 are: 1, 2, 3, 4, 6, 8, 12, 24.
The factors of 40 are: 1, 2, 4, 5, 8, 10, 20, 40.
Hence, the HCF of 24 and 40 is 8.
In general, to find the HCF of two or more natural numbers, we write them as a product of prime
factors in exponent form, and we multiply the common factors raised to the smallest power.
For example: 24 = 23 · 3 and 40 = 23 · 5, so HCF is 23 = 8.
The LCM (lowest common multiple) of two or more natural numbers is the smallest multiple which
is common to both of them. For example:
The multiples of 6 are: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, . . .
The multiples of 8 are: 8, 16, 24, 32, 40, 48, 56,. . .
Hence, the LCM of 6 and 8 is 24.
In general, to find the LCM of two or more natural numbers, write them as a product of prime
factors in exponent form, and multiply common and non common factors raised to the highest
power.
For example: 6 = 2 · 3 and 8 = 23 , so LCM is 23 · 3 = 24.
D pto. M atemáticas. IES Jovellanos. 2011
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3
Exercises - Set A
1. Find the largest multiple of 7 which is less than 1000.
2. Find the smallest multiple of 13 which is greater than 1000.
3. Find the largest multiple of 17 which is less than 2000.
4. Find the smallest multiple of 15 which is greater than 10 000.
5. Find the HCF of 24, 72, 120.
6. Find the LCM of 12, 18, 27.
7. Three bells chime at intervals of 4, 5 and 6 seconds respectively. If they all chime at the same
instant, how long before they all chime together again?
2. Integers and order of operations
The negative whole numbers, zero and the positive whole numbers (not fractions or decimals) form
the set of all integers . We use the letter Z to refer to the set of integers:
Z = {· · · − 5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5 . . .}
We can show these numbers on a number line . Zero is neither positive nor negative.
b
−5
b
−4
b
−3
b
b
−2
−1
b
b
b
b
b
b
0
1
2
3
4
5
Remember the rules for handling addition and subtraction of integers:
+
−
+
−
(positive) = (positive)
(positive) = (negative)
(negative) = (negative)
(negative) = (positive)
The following rules apply to multiplication and division with integers:
(positive) · (positive) = (positive)
(positive) · (negative) = (negative)
(negative) · (positive) = (negative)
(negative) · (negative) = (positive)
(positive) : (positive) = (positive)
(positive) : (negative) = (negative)
(negative) : (positive) = (negative)
(negative) : (negative) = (positive)
Order of operations with integers:
• Perform the operations within brackets first.
• Calculate any part involving exponents (powers, roots).
• Starting from the left, perform all divisions and multiplications as you come to them.
• Finally, restart from the left and perform all additions and subtractions as you come to them.
The word BEDMAS may help you remember this order:
Brackets
Exponents
Division
Multiplication
Addition
Subtraction
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Rules for brackets:
• If an expression contains one set of grouping symbols, i.e., brackets, work that part first.
• If an expression contains two or more sets of grouping symbols one inside the other, work the
innermost first.
• The division line of fractions also behaves as a grouping symbol. This means that the numerator
and the denominator must be found separately before doing the division.
Exercises - Set B
1. Simplify:
a) −5 + 4 · (−2 + 1)3 − (−9 + 6)2
b) 12 − 2 · [25 : (−4 − 1) + (−2) − (−6 − 10)]
c) −7 − (−3) + (−8) · (−1) − (−12) : (−4)
d) −5 − 4 · [−8 : 2 − 2 · (−3)]
e) 10 − 10 · [−6 + 5 · (−4 + 7 − 3)]
f) 6 − 5 · [−4 − 1 + (−2)2 − 32 ]
2. Simplify:
a)
12 + (5 − 7)
18 : (6 + 3)
b)
57
7 − (2 · 3)
c)
3·8+6
6
d)
(3 + 8) − 5
3 + (8 − 5)
3. Insert grouping symbols if necessary, to make the following true:
a) 120 : 4 · 2 = 15
d) 5 · 7 − 3 − 1 = 19
b) 120 : 4 · 2 = 60
e) 5 · 7 − 3 − 1 = 33
c) 5 · 7 − 3 − 1 = 15
f) 3 + 2 · 8 − 4 = 36
g) 3 + 2 · 8 − 4 = 11
h) 3 + 2 · 8 − 4 = 15
i) 8 − 6 · 3 = 6
Using the calculator:
Most modern calculators have the order of operations rules built into them. They also have grouping
symbols keys (left hand bracket and right hand bracket).
12
For example, if we consider
and key in 12 : 4 + 2 = the calculator gives an answer
4+2
of 5, which is incorrect. However, if we type 12 : (4 + 2) = we obtain the answer 2, which is
correct.
Remember to use the sign change key before a negative number.
Exercises - Set C
1. Evaluate each of the following using your calculator:
a) 17 + 23 · 15
b) (17 + 23) · 15
d) 128 : (8 + 8)
e)
89 + (−5)
−7 · 3
D pto. M atemáticas. IES Jovellanos. 2011
c) 128 : 8 + 8
f)
−15 − 5
6−8:4
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5
3. Rational numbers (fractions)
A rational number is a number which can be written as a fraction, that is, in the form
and b are both integers, and b 6= 0.
4
−1
2
0
Examples:
,
,
,
,
7
10
3
1
a
, where a
b
−2
...
1
A fraction indicates a part of a unit or a part of a quantity; it consists of two whole numbers, a
numerator and a denominator , separated by a bar symbol:
• the denominator, b, shows how many equal parts the whole has been split into.
• the numerator, a, tells us how many of those equal parts are being described.
The set of rational numbers is represented by the letter Q:
o
na
, a, b ∈ Z, b =
6 0
Q=
b
Remember that in general:
−a
a
a
=
=−
b
−b
b
However:
−a
a
=
−b
b
How to say a fraction:
Here are some examples:
1
a half or one half
2
2
two fifths or two over five
5
8
eight over thirty-five
35
1
a third or one third
3
4
four sevenths or four over seven
7
54
fifty-four over seventy-two
72
1
4
a quarter
Proper and improper fractions:
A fraction which has numerator less than its denominator is called a proper fraction .
3
For example, is a proper fraction.
4
A fraction which has numerator greater than or equal to its denominator is called an improper fraction .
7
For example, is an improper fraction.
4
When an improper fraction is written as a whole number and a fraction, it is called a mixed number .
7
3
For example, can be written as the mixed number 1 43 , as it is really 1 + .
4
4
Example 2
Write the improper fraction
29
as a mixed number.
3
First, we divide 29 by 3:
29
2
3
9
Hence:
29
2
= 9 + = 9 32
3
3
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Simplifying fractions:
We can simplify a fraction by cancelling common factors in the numerator and denominator.
When a fraction is written as a rational number with the smallest possible denominator, we say
it is in lowest terms .
We can simplify a fraction to its lowest terms dividing both numerator and denominator by their
HCF.
Example 3
Simplify:
16
24
As HCF of 16 and 24 is 8, dividing both numerator and denominator by 8 we get:
16
2
= ,
24
3
which is the fraction in its lowest terms
Two fractions are equal or equivalent if they can be written in the same lowest terms.
We can convert a fraction to an equivalent fraction by multiplying or dividing both the numerator
3
6
9
12
and denominator by the same non-zero number:
= =
=
= ...
4
8
12
16
We can also test if two fractions are equivalent by cross-multiplying their numerators and denomi12
24
nators. For example,
and
are equivalent because 12 · 40 = 20 · 24 = 480.
20
40
Comparing fractions:
To compare two or more fractions:
• Find the common denominator , which is the LCM of the original denominators.
• Work out the equivalent fractions.
• Write the fractions in ascending or descending order. Remember that if two fractions have the
same denominator, the greater one is the fraction with the highest numerator.
D pto. M atemáticas. IES Jovellanos. 2011
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Example 4
Which fraction is bigger,
4
7
or ?
5
9
The common denominator of
4
36
=
5
45
7
35
=
9
45
4
7
and is 45, the LCM of 5 and 9.
5
9
(multiply numerator and denominator by 9).
(multiply numerator and denominator by 5).
The order is
36
35
4
7
>
, so > .
45
45
5
9
Exercises - Set D
1. Express with denominator 12:
a)
2
3
b)
3
4
5
6
c)
d)
6
18
e)
15
45
d)
24
28
e)
18
42
e)
33
77
2. Express with numerator 12:
a)
3
7
b)
6
5
4
9
c)
3. Express in lowest terms:
a)
6
10
6
18
b)
c)
25
10
d)
14
35
f)
48
72
g)
78
117
h)
125
1000
4. Simplify:
a)
8−2·5
4·3
b)
−1 + 8 : 2
8−5
c)
32 − 2 · 5
8 − 32
d)
−5 · (−6)
−11 − 4
e)
6−3:3
2 + 10 : 2
f)
12 + 8 : 2
12 − 8 · 2
g)
6 + 22
6 − 23
h)
11 − 3
16 : 4
5. Place these fractions in ascending order:
a)
2 3 3 17
, , ,
5 8 4 40
b)
4 7 5
3
6
, , ,− ,−
3 5 7
4
11
c)
1
2 3
1
3
,− ,
,− ,−
8
3 11
6
4
6. Place these fractions in descending order:
a)
5 11 7 5
,
,
,
6 24 12 8
b)
2 3 4 5 6
, , ,
,
5 7 9 13 10
5
1
4
7
6
c) − , − , − , − , −
8
2
7
11
13
7. Write as a mixed number:
a)
45
4
b)
11
2
a)
37
5
D pto. M atemáticas. IES Jovellanos. 2011
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4. Operations with fractions
Addition and subtraction:
To add or subtract fractions:
• If necessary, convert the fractions so they have the lowest common denominator.
• Add or subtract the new numerators. The denominator stays the same.
For example:
3 2 1
9
8
6
9−8+6
7
− + =
−
+
=
=
4 3 2
12 12 12
12
12
When adding or subtracting mixed numbers, you can first convert them to improper fractions and
then perform the operation. However you can also add the whole numbers and fractions separately,
then combine the result.
Example 5
Find: 2 13 − 3 12 + 1 41
First, we convert the mixed numbers into improper fractions:
1
6 1
7
1
6 1
7
1
4 1
5
2 13 = 2 + = + = ; 3 12 = 3 + = + = ; 1 14 = 1 + = + =
3
3 3
3
2
2 2
2
4
4 4
4
Hence: 2 31 − 3 21 + 1 41 =
7 7 5
28 42 15
1
− + =
−
+
=
3 2 4
12 12 12
12
Multiplication:
To multiply two fractions, we multiply the two numerators to get the new numerator, and multiply
the two denominators to get the new denominator:
a c
a·c
· =
b d
b·d
To help make multiplication easier, we can cancel any common factors in the numerator and denominator before we multiply.
For example:
4 3
4·3
4·1
4
· =
=
=
9 5
9·5
3·5
15
Two numbers are reciprocals of each other if their product is one.
For any fraction
a
a b
a
b
, we notice that · = 1. So, the reciprocal of is .
b
b a
b
a
Division:
To divide two fractions, multiply the first by the reciprocal of the second:
a c
a d
a·d
: = · =
b d
b c
b·c
For example: 1 31 : 3 21 =
4 7
4·2
8
: =
=
3 2
3·7
21
D pto. M atemáticas. IES Jovellanos. 2011
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9
Exercises - Set E
1. Find:
a) 1 32 − 2
b) 3 43 − 1 21
c)
3
− 2 21
4
d) 1 23 + 3 14
e) 4 31 + 2 61
f) 2 32 − 5 56
g) −2 41 + 3 18
h) 4 15 − 2 16
2. Find:
a)
2 1 3
· ·
3 4 5
3 5 4
− ·
5 2 3
2
3
2
− · 1 32
g)
3
4
d)
b)
3
4
2
· −
· −
8
3
5
c)
2 3 2
+ ·
3 4 3
e)
3 1 2 1
· + ·
5 3 3 4
f)
4 1 1 2
· − ·
3 2 6 3
d)
4
:3
5
h) 4 · 1 13 − 5 ·
2
7
3. Find:
2 1
a) :
3 6
e)
1
−
2
3
g) 1 21 : −
4
5 1
b) :
7 3
1
: 1 23
4
f) 2 34 :
3
c) :
4
2
3
h) 3 15 : 1 31
4. Find:
2 3 4
a) − :
3 2 5
5 1 4
b) : +
3 2 3
1 2 3 6
c) · − :
2 5 4 5
2
d) :
5
1
3 2
−
+ ·
2
4 5
5. Find and give your answers in their simplest form:
a)
3 7 7
+ −
4 6 8
:
25
12
3 1
(−3) ·
−
5 3
d)
4 6
(−2) ·
−
3 5
b)
7
9
13
13
−
·
+ −
15 25
22
33
1
3
2
3− ·
−
4
5 15
e)
4
1 3
6+
·
−
25
2 4
c)
1
−
2
3
−1
4
3
+1
4
2 5
3 5
−
·
−
3 9
4 6
f) 7
5
4
−
· +1
12 6
3
5. Problem solving
To find a fraction of a quantity we multiply the quantity by the numerator, and divide the result by
the denominator.
3
3
3 · 85
For example:
of 85 e are:
· 85 =
= 51 e.
5
5
5
D pto. M atemáticas. IES Jovellanos. 2011
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Example 6
If
3
of a shipping container holds 2100 identical cartons, how many cartons will fit into:
8
a)
5
of the container
8
b) the whole container?
3
1
of the container holds 2100 cartons; hence of the container holds
8
8
2100 : 3 = 700 cartons.
5
holds 700 · 5 = 3500 cartons
8
8
b) The whole is which is 700 · 8 = 5600 cartons.
8
a) So,
Example 7
Rob eats
remains?
1
3
of a watermelon one day and of it the next day. What fraction of the watermelon
3
8
We let the whole watermelon be represented by 1. The fraction remaining will be:
24
8
9
7
1 3
1− − =
−
−
=
3 8
24 24 24
24
Exercises - Set F
1. Millie calculated that her bicycle cost
$, what did her bicycle cost?
1
of the cost of her father’s car. If the car cost 38 014
83
1 1
1
, and of the brickwork of his new garage. What
3 5
4
fraction must he complete on the fourth and final day?
2. Over three successive days Colin builds
3. 200 kg of sugar must be poured into packets so there is
packets will be filled?
2
kg of sugar per packet. How many
5
4. 2400 kg of icecream is put into plastic containers which hold
containers are needed?
3
kg each. How many plastic
4
5. John says that his income is now 3 21 times what it was 20 years ago. If his current annual
income is 63 000 e , what was his income 20 years ago?
1
1
1
1
of its weekly budget on rent, on food, on clothes,
on entertainment
3
4
8
12
and the remainder is banked. How much is banked if the weekly income is 864.72 e?
6. A family spends
D pto. M atemáticas. IES Jovellanos. 2011
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11
Exercises - Set G
1. Renee used
pipe is left?
2.
1
2
of a length of pipe and later used of what remained. What fraction of the
3
3
2
1
of an amount of money is 519 $. Find: a) of the money
3
3
b) the whole amount.
3
of a field was searched for truffles and 39 were found. How many truffles would we expect
13
to find in the remainder of the field?
3.
4. The seven tenths of a quantity are 210. What is the quantity?
5. Alfredo sent
2
2
of his potato crop to market last week. This week he sent of the remainder.
5
3
a) What fraction of his crop has now gone to market?
b) If he has 860 kg remaining, what was the original weight of the crop?
2
9
of the weight of a loaf of bread comes from the flour used in making the bread. If of
10
9
the weight of the flour is protein, what fraction of the weight of a loaf of bread is protein?
6.
7. A tree is losing its leaves. Two thirds fall off in the first week, two thirds of those remaining
fall off in the second week and two thirds of those remaining fall off in the third week. Now
there are 37 leaves. How many leaves did the tree have originally?
8. Out of a group of 40 people in a shop, 32 were aged thirty or over. What fraction of the
people were under thirty?
4
1
of the cars on the road are saloons. Of these saloons are red. What fraction of the cars
5
8
on the road are red saloons?
9.
13
of the distance between college and home. She wants to split
28
the remaining distance into 5 equal parts. What fraction of the whole journey is each part?
10. Sadie has already driven
1
of his time when awake playing on his
8
computer. What fraction of his day does Marc spend playing on the computer?
11. Marc spends a third of the day asleep. He spends
12. A 9 tonne load of top soil is divided into
can be filled?
3
tonne plant containers. How many containers
25
2
4
of a bar of chocolate. Linda eats of what remains. What fraction of the bar
5
9
of chocolate have they eaten between them?
13. John eats
14. In his will a man leaves half his estate to his wife and the rest is shared equally among his
five sons. What fraction of his estate does each son inherit?
15. Out of a deposit of oil you empty one half. Out of what remains, you empty one half again,
11
and then you empty
of what remains. Finally, there are 36 litres left. How many litres were
15
there at the beginning?
✁✃✁✃✁✃✁✃✁✃✁✃✁✃
D pto. M atemáticas. IES Jovellanos. 2011
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