Download Number - Crawshaw Academy

GCSE Grade G
Number
Millions
Hundreds of
Thousands
Tens of
Thousands
Thousands
Hundreds
Tens
Unit (ones)
Place Value
9
6
7
4
1
0
8
Pencil and Paper Methods for Addition and Subtraction
Example
2296 + 1173
Answer
Rewrite the numbers in columns making sure the units are all in a column and the tens are all in a
column etc
2296
+ 1173
3469
The number 9674108 is shown above written on a place value chart
1
Large numbers are read in groups of three
The number 9674108 is read as
Always start from the units column.
nine million, six hundred and seventy four thousand, one hundred and eight.
When the total in a column is more than nine, you have to carry a digit into the next column on the left,
as shown above. It is important to write down the carried digit or you may forget to include it in the
addition.
The number system we use is a decimal system.
The place value of each number is ten times as large as the place value of the number immediately to
the right.
Example
What is the place value of the 7 in these
a.
675
Examples
1.
874 – 215
2.
300 – 163
Answer
b. 870031
86714
- 215
659
1.
Answer
a. In 675 the 7 is in the tens place. We say that the place value of the 7 is ten
b. In 870031 the 7 is in the tens of thousands place.
We say that the place value of the 7 is tens of thousands.
2 9 1
300
- 163
137
2.
Addition - ADD, PLUS, TOGETHER, SUM
Always subtract the units column first.
Example
Add together 4 + 7
Answer 11
Subtraction - MINUS, SUBTRACT, TAKE AWAY, LESS
Example 7 – 3
Answer 4
When you have to take a bigger digit from a smaller digit in a column, you must ‘borrow’ a 10 by taking
one from the column to the left and putting it with the smaller digit, as shown in the examples above
Pencil and Paper Methods for Multiplication and Division
Multiplication
Multiplication is the same as adding a number again and again
For instance 4 x 3 means 4 lots of 3 or 3 + 3 + 3 + 3
Division
Dividing a number is the same as sharing
Example - Divide 35 by 7 is the same as sharing 35 by 7
We say one number is divisible by another if there is no remainder
Example
Multiply 543 by 6
Answer
x
543
6
3258
2 1
A number is divisible by 2 if it is an even number
Starting with 6 x 3 this gives an answer of 18 so you need to carry a digit into the next column on the left.
A number is divisible by 5 if it ends in a 0 or 5
Then 6 x 4 gives an answer of 24, then add the 1 which had been carried from the previous column
giving 25, again you need to carry a digit into the next column on the left.
Order of Operations
The 6 x 5 gives an answer of 30, then add the 2 which had been carried from the previous column giving
32.
(See BIDMAS Grade C)
Division then multiplication then addition then subtraction
Example
9÷3+4x2
Answer
First divide
9÷3=3
giving 3 + 4 x 2
Then multiply
4x2=8
giving 3 + 8
Then add
3 + 8 = 11
Example
Equivalent Fractions
Divide 508 by 4
Equivalent fractions are two or more fractions that represent the same part of a whole.
Answer
Recognise using diagrams equivalent fractions.
This is set out as
1 2 7
4 51 02 8
1 2
=
2 4
An equivalent fraction can also be found from multiplying or dividing the numerator and denominator by
the same number.
First divide 4 into 5 to get 1 and remainder 1.
You can see from the diagrams that
Then, divide 4 into 10 to get 2 and remainder 2.
Finally, divide 4 into 28 to get 7.
This gives the answer of 127.
Rounding to the Nearest 10 or 100
Example
30
36 is between 30 and 40
40
36 is closer to 40 than 30
36 rounded to the nearest ten is 40
If the number is between, we round up
35 is halfway between 30 and 40, 35 to the nearest ten is 40
Fractions
Amy bought 6 sweets. She gave 4 of these to her friends
a. What fraction of the sweets did Amy give to her friends
b. what fraction did she keep
a. Amy gave
Negative Numbers - Scales and Number Lines
Scales often have positive and negative numbers on them.
For example, on a temperature scale 10° below 0° is shown as -10°.
Positive numbers such as +4 are often written without the + sign. For instance, +4 is often written as 4.
Negative numbers, such as –4, are always written with the – sign.
Zero is neither positive nor negative.
Ordering Negative Numbers
A fraction is part of a whole. The top number is called the numerator. The bottom number is called the
denominator.
2
is read as two fifths. 2 is the numerator and 5 is the denominator.
5
2
means you divide a whole into 5 portions and take 2 of them.
5
2
means 2 parts out of every 5
5
2
For instance of these dots are red
5
Example
Answer
Example
3
× 4 12
→
=
4
× 4 16
4
to her friends.
6
2
b. Amy kept 2 for herself. That is, she kept .
6
-8
-7
-6
-5
-4
-3
-2
The further to the right a number is, the larger it is.
The further to the left a number is, the smaller it is.
< means less than
> means greater than
Example
Insert < or > to make these statements true.
a.
6
-6
b.
–5
-6
Answer
a. 6 > -6 (Since 6 is greater than –6)
b. –5 > -6 (Since -5 is greater than –6)
-1
0
1
2
3
4
Equivalent Fractions, Decimals and Percentages
Place Value
Convert from percentage to:
The ordinary counting system uses place value, which means that the value of a digit depends upon its
place in the number.
Decimal
Fraction
We read 432 as four hundred and thirty two.
Divide the percentage by 100
Make the percentage into a fraction with
a denominator of 100 and simplify by
cancelling down if possible
The number 52 has 5 tens and 2 ones.
Example
52% = 52 ÷ 100
= 0.52
Convert from decimal to :
Percentage
Multiply the decimal by 100%
Example
0.65 = 0.65 x 100
= 65%
Convert from fraction to :
Example
52 13
52% =
=
100 25
Putting whole Numbers in Order
Example
Rearrange this list of numbers into order of size starting with the largest number.
86, 104, 79, 88, 114, 200,
Answer
Fraction
200, 114, 104, 88, 86, 79,
If the decimal has one decimal place put
it over the denominator 10, if it has 2
decimal places put it over the
denominator 100 etc. Then simplify by
cancelling down if possible
Fractions
Example
65
13
0.65 =
=
100 20
Examples
Percentage
Decimal
If the denominator is a factor of 100
multiply numerator and denominator to
make the denominator 100, then the
numerator is the percent
Divide the numerator by the denominator
Example
3
15
=
= 15%
20 100
Or convert to a decimal and change the
decimal to a percentage
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, are called digits.
Example
9
= 9 ÷ 40 = 0.225
40
Example
7
= 7 ÷ 8 = 0.875 = 87.5%
8
Adding and Subtracting Simple Fractions
Fractions with the same denominator (bottom number) can easily be added or subtracted.
For a Grade F you would also need to simplify the answer.
1.
2.
3
4
+
10 10
Add the numerators (top number). The bottom number stays the same.
4
7
3
+
=
10 10 10
7 2
Work out 8 8
Subtract the numerators (top number). The bottom number stays the same.
Work out
Multiples (Times Tables)
Example
Write out the first five multiples of 3
Answer
3, 6, 9, 12, 15
Factors
A factor of a whole number is any whole number that divides into it exactly.
Example
List the factors of 20
Answer
Checking Calculations
1, 2, 4, 5, 10, 20
38 + 57 = 85, Round to check 40 + 60 = 100
Square Numbers
Always check your answer even when using a calculator
When you multiply any number by itself, the answer is called the square of the number or the number
squared.
This is because the answer is a square number.
1 x 1= 1
2x2=4
3x3=9
GCSE Grade F
Fractions of Quantities
Division
1
1
The word “of” can be replaced by x. For instance
of 200 means
x 200
10
10
3
Example Find of £270
5
1
Answer of £270 = 270 ÷ 5 = 54
5
3
of £270 = 54 x 3 = 162 (Divide by 5, then multiply by 3)
5
Either by using a calculator of by dividing by the denominator and then multiplying by the numerator
There are difference methods for dividing one of these methods is the chunking method.
Using Negative Numbers
Example
The temperature at 6a.m. was -5°C. By 9a.m. the temperature has risen by 7°. What was the temperature at
9a.m.
Answer
The temperature must rise 5° to 0°. By the time the temperature has risen a further 2°, it would then be 2°C.
Adding and Subtracting Negative Numbers
We can use a number line to add or subtract
To add a positive number, move to the right. To add a negative number, move to the left.
Addition and subtraction are inverse operations. To subtract, we must move in the opposite direction to that
in which we move to add.
To subtract a negative number, move to the right.
Writing down some of the multiples of 35 might be useful.
1 x 35 = 35
2 x 35 = 70
10 x 35 = 350
20 x 35 = 700
5 x 35 = 175
1655
- 700
955
- 700
255
-175
80
- 70
10
20 x 35
20 x 35
5 x 35
2 x 35
47
Once the remainder is found, you cannot subtract any more multiples of 35.
Add up the multiples (in red) to see how many times 35 has been subtracted.
Decimal Place Value
Examples
-7
Work out 1655 ÷ 35
So 1655 ÷ 35 = 47 remainder 10
To subtract a positive number, move to the left.
-8
Example
Remember
-6
-5
-4
-3
-2
4–1
Begin at 4, move 1 to the left
1–3
Begin at 1, move 3 to the left
1 – (-3) Begin at 1, move 3 to the right
-1 – (-3) Begin at -1, move 3 to the right
Pencil and Paper Methods
-1
0
1
2
4–1=3
4–1=3
4–1=3
4–1=3
3
4
1 can be written as 0.1 and 3 can be written as 0.3
10
10
1 can be written as 0.01 and 41 can be written as 0.41
100
100
Place Value is given by this chart.
100000
10000
1000
100
10
Multiplication
There are different methods for multiplication one of these methods is the grid method.
Example
Work out 243 x 68 without using a calculator.
Answer using the grid method. The two numbers are split into hundreds, tens and units.
Multiply all the pairs of numbers.
X
200
20
3
60
12000
2400
180
8
1600
320
24
Add all the numbers in the boxes which are not shaded.
12000 + 2400 + 180 + 1600 + 320 + 24 = 16524
So 243 x 68 = 16524
For instance, in the number 365.724
the 3 means three hundreds
the 6 means six tens
the 5 means five ones (or units)
the 7 means seven tenths
the 2 means two hundredths
the 4 means four thousandths
Reading and Writing Decimals
The number 84.73 is read as eighty four point seven three.
The number 84.73 is written with two decimal places
1
1
10
1
100
1
1000
Rounding
Pencil and Paper Methods for Adding and Subtracting Decimals
To give sensible answers to calculations we often need to approximate.
Rounded to 1 d.p.
Rounded to 2 d.p.
3.6438 is 3.6
3.6438 is 3.64
Rounded to 1 d.p.
Rounded to 2 d.p.
To add 3.42 + 15.8 we should set out our work as shown
236.847 is 236.8
236.847 is 236.85
+
To round to a given number of decimal places.
1.
Keep the number of figures asked for after the decimal point. For instance if asked to round to 3
decimal places, keep 3 figures after the decimal point.
2.
Omit all the following figures. If the first figure omitted is 5 or greater, increase the last figure kept by 1.
3.42
15 . 8
19.22
The 3 and the 5 must be lined up and added since these were both units
To do the subtraction 7.3 – 1.5 we could set out our work as shown
6
7 .1 3
- 1 .5
5 .8
Rounding to a given number of decimal places is also called approximating to that number of decimal
places.
The words “decimal places” are often abbreviated to d.p.
Example
Multiplying and Dividing Decimals by Single-digit Numbers
Examples
Round 3.6472 to
a. 2 d.p.
1.
b. 1 d.p.
4.5 x 3
Answer
a
We round to 2 d.p. which means we want 2 figures after the decimal point.
With just 2 figures after the decimal point, 3.6472 is 3.64.
The first figure omitted is 7 so we must increase the last figure kept by 1.
We then get 2.6472 = 3.65 to 2 d.p.
1
So 4.5 x 3 = 13.5
2.
We need 1 figure after the decimal point.
With just 1 figure after the decimal point, 3.6472 is 3.6.
The first figure omitted is 4 which is not large enough to alter the last figure kept.
We then get 3.6472 = 3.6 to 1 d.p._
Decimals
On a number line, the smaller a number is, the further to the left it is.
To put a list of decimal numbers in order it is a good idea to first write them all with the same number of
figures after the decimal point.
Example
2.63
2.6
3.421
4.32
3.4
4.320
3.400
3.399
4.098
Now choose the numbers with the smallest number before the decimal point
These are 2.630 2.600
Put these in order. Since 600 < 630 then 2.600 < 2.630.
The first two numbers are 2.6
2.63
Now choose the numbers with a 3 before the decimal point.
These are 3.421 3.400 3.399
Since 399 < 400 < 421, then 3.399 < 3.400 < 3.421
The first five numbers are 2.6
2.63
3.399
3.4
3.421
Left are 4.098 and 4.320
Since 98 < 320, then 4.098 < 4.320
b
The numbers in order smallest to largest
2.6 2.63 3.399 3.4 3.421 4.098
8.25 ÷ 5
1. 6 5
5 8.1215
So 8.25 ÷ 5 = 1.65
Reading Calculator Displays
There is a maximum numbers of digits that can be displayed on any calculator screen.
On a calculator with an 8 digit screen display the answer to the division 7 ÷ 3 is displayed as 2.3333333.
Other calculators may display more than or fewer than 8 digits.
We often need to give the answer to a calculation to the nearest whole number
3.399 4.098
To put these numbers in order from the smallest to largest
Rewrite the numbers as
2.630 2.600 3.421
4.5
x 3
13.5
Examples
1. A calculator displayed the answer to 7 ÷ 3 as 2.3333333
To the nearest whole number the answer to 7 ÷ 3 is 2
2. A calculator displayed the answer to 53 ÷ 7 as 7.5714286
To the nearest whole number the answer to 53 ÷ 7 is 8
Sometimes we need to give the answer to a calculation to the closest but smaller whole number.
Example
Oranges are 52p each. How many can be bought with £5.00
Answer
On the calculator the answer to 500 ÷ 52 is 9.615384615
Only 9 oranges can be bought
In the previous example the answer was given as the closest but smaller whole number.
Using the Calculator for Calculations
4.32
The calculator is often used for calculations. It is very useful when the numbers are large or have many
digits
Always have a rough idea of the size of the answer you expect to get. It is very easy to press a wrong key
or to forget to press the = key at the end of a calculation
Improper Fractions and Mixed Numbers
Square Roots
A fraction with the numerator (top number) smaller than the denominator (bottom number) is called a proper
fraction.
4
An example of a proper fraction is .
5
An improper fraction has a bigger numerator (top number) than the denominator (bottom number).
9
An example of an improper fraction is . It is sometimes called a top heavy fraction.
5
A mixed number is made up of a whole number and a proper fraction.
The square root of a given number is a number that, when multiplied by itself, produces the given number.
3
The answer to “what number squared gives 4” is 2, 2 is called the “square root of 4”
The sign for a square root is 4 .
For instance
4 means “the square root of 4”
Squaring and finding the square root are inverse operations. One “undoes” the other
For instance 22 = 4 and
4 =2
Square roots are found using the calculator
An example of a mixed number is 1 .
The answer to “what number cubed gives 64” is 4, 4 is called the “cube root of 64”
Examples
14
Convert
into a mixed number.
5
Answer
14
means 14 ÷ 5.
5
The sign for a cube root is 3 64 .
4
Dividing 14 by 4 gives 2, with a remainder of 4. (5 fits into 14 two-times with
14
4
=2
5
5
Example
1
Convert 3 into an improper fraction.
4
Answer
So
Each whole is four – quarters.
So 3 whole ones are 12 quarters.
1
So 3 is 12 quarters + 1 quarter = 13 quarters.
4
1 13
3 =
4
4
For instance 3 64 means “the cube root of 64”
Cubing and finding the cube root are inverse operations. One “undoes” the other
For instance 43 = 64 and 3 64 = 4
Cube roots are found using the calculator
4
left over)
5
GCSE Grade E
Percentages of Quantities
Equivalent Ratios
The word “of” can be replaced by x. For instance 23% of 200 means 23% x 200
Equivalent ratios are equal. For instance 8 : 2 and 4 : 1 are equivalent ratios.
23
Since 23% means 23 out of 100, 23% may be written as 100
We can divide both sides of a ratio by the same number.
We can also multiply both sides of a ratio by the same number.
Example
A ratio is in its simplest form if the numbers in the ratio are the smallest possible whole numbers
Find 23% of 200cm
Example
Answer
Write these ratios in their simplest form.
23% of 200cm = 23% x 200
a. 1mm : 5m
23
= 100
x 200cm
b. 45p : £2
c. £3.50 : £2.
Answer
= 46cm
Writing Ratios
a. 1mm : 5m
= 1mm : 5000mm
= 1 : 5000
b.
= 45p : 200p
= 45 : 200
= 9 : 40
45p : £2
The ratio of two quantities x and y is written as x : y.
x : y is read as “the ratio of x to y”.
The order is important x : y is different from y : x.
c. £3.50 : £2
= 3.5 : 2
=7:4
A ratio compares quantities of the same kind.
Multiplying and Dividing Decimals
Example
Example
On a bus there are 17 adults and 5 children
Find the ratio of
a. adults to children
b. children to adults.
Answer
a.
In our ratio we must have the number of adults first, then the number of children.
The ratio of adults to children is 17 : 5.
b.
In our ratio we must have the number of children first, then the number of adults. The ratio of
children to adults is 5 : 17
Using Ratios
Ratios are used in the drawing of plans.
The plan will be the same shape as the original only smaller.
The ratio used may be 1 : 2 which means that the original will be twice as large as the plan. In other
words the plan will be ½ the size of the original.
Calculate
16.7 x 0.8
Answer
Ignore the decimal point and calculate
167 x 8
X
100
60
7
8
800
480
56
800 + 480 + 56 = 1336
The ratio used may be 1 : 5 which means that the original will be five times as large as the plan. The plan
Count how many digits after the decimal point in both numbers and then count from the right. In 16.7
there is 1 and in 0.8 there is 1 so count 2 from the right.
So 16.7 x 0.8 = 13.36
would be 1 the size of the original.
Negative Numbers
5
Many other ratios could be used.
Multiplying and Dividing
The rules for multiplying and dividing with negative numbers are: When the signs are the same the answer is positive
When the signs are different the answer is negative
Examples
1.
-2 x -2 = +4
2.
-4 x 6 = -24
3.
7 x -2 = -21
4.
24 ÷ -3 = -8
5.
-16 ÷ 8 = -2
6.
-14 ÷ -7 = +2
Approximation, Estimation
Multiplying and Dividing
Example
To multiply fractions, you multiply the numerators together and you multiply the denominators together.
Write 0.0504715 to 4 s.f.
Example
Answer
Counting from the first non zero figure and keeping 4 figures after this 0.0504715 is 0.05047
The first figure omitted is 1 which is not large enough to alter the last figure kept
We then get 0.0504715 = 0.05047 to 4 s.f.
When we approximate large number to s.f. we may have to insert zeros so the size of the number is
unchanged
For instance 34592 to 1 s.f. is written as 30000 not as 3.
Because the second digit is less than five then we leave the first digit as it is. If the value of the second
digit is equal to or greater than five then add one to the first digit.
Example 45281 would round to 50000 to 1 significant figure.
Work out
1
2
of .
4
5
Answer
1 2 1 ×2
2
1
x =
=
=
4 5 4 ×5
20 10
To divide a fraction, you turn the fraction upside )finding its reciprocal), and then multiply
Example
3
4 3 ×4 12
3÷ =3x =
=
=4
4
3
3
3
For Grade C you would have to multiply and divide with mixed numbers and improper fractions.
Prime Numbers
Approximations with Decimals
A prime number only has two factors: 1 and the number itself.
Rounded to 1 significant figure 3.6482 is 4
The prime numbers up to 50 are
Rounded to 2 significant figures 3.6482 is 3.6
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47
Rounded to 3 significant figures 3.6482 is 3.65
Powers
Estimating Answers using Approximation
Powers are a convenient way of writing repetitive multiplications. They are also called indices.
We can check that the answer to a calculation such as 423 x 76 is about the right size by approximating.
7 x 7 = 72
We can approximate 423 as 400. We can approximate 76 as 80
7 x 7 x 7 = 73
which is 7 to the power 3 or 7 cubed.
So we estimate 423 x 76 to be about 32000
7 x 7 x 7 x 7 = 74
which is 7 to the power 4.
which is 7 to the power 2 or 7 squared.
Always estimate answers when using the calculator.
Using Place Value of Multiply and Divide
Calculator Errors
Using Place value 23 x 10 = 230, 23 x 100 = 2300
When using the calculator, it is a good idea to have a rough idea of the answer.
We can often use multiplication by 10 or 100 or 1000 etc. to help us when we are multiplying by 80 or
800 or 8000 etc.
It is easy to key a calculation into the calculator incorrectly.
If the answer given by the calculator is very different from the rough estimate, the calculation should be
keyed in again
Fractions
We can often use division by 10 or 100 or 1000 etc. to help us when we are dividing by 40 or 400 or
4000 etc.
GCSE Grade D
Fractions
Multiplying Fractions
A fraction is written in its lowest terms or as the simplest fraction, if the numbers in the fraction are the
smallest possible whole numbers.
Multiply the numerators to obtain the numerator of the answer and multiply the denominators to obtain the
denominator of the answer.
Example
24
Write
in its lowest terms
32
Answer
When multiplying a mixed number, change the mixed number to an improper fraction before you start
multiplying.
The largest number that divides into both 24 and 32 is 8
24 ÷ 8 3
=
=
32 ÷ 8 4
An amount or quantity can be given as a fraction of another.
Example
Write £5 as a fraction of £20
Answer
5
As a fraction this is written as .
20
1
This cancels to .
4
1
So £5 is of £20.
4
Adding and Subtracting Fractions
To add (or subtract) fractions, the fractions must be the same type
Fifths may be added to fifths, quarters may be added to quarters but to add fifths to quarters we must
rewrite as the same type.
Examples
Work out
a.
Answer
4
9
4
9
×
3
10
b.
2
2
5
×1
7
8
3
2 is a factor of 4 and 10. 3 is a factor of 3 and 9.
10
2
2
1
Simplifying the fractions, cancelling by 2 and 3 gives
× =
3
15
5
12
2
7
b.
2 × 1 Converting the mixed numbers into improper fractions gives
5
5
8
3
9
3
1
Simplifying the fractions by cancelling by 4 and 5 gives × =
=4
1
2
2
2
Ratios as Fractions
a.
×
×
15
.
8
A ratio in its simplest form can be expressed as portions of a quantity.
Example
A garden is divided into lawn and shrubs in the ratio 3 : 2.
What fraction of the garden is covered by
a.
b.
Lawn
Shrubs
We need to find equivalent fractions which have the same denominator.
Answer
Examples
2 1 3
+ =
5 5 5
5 3 2 1
– = =
8 8 8 4
4 3
4
3
To find + first find fractions equivalent to and which have the same denominator
5 4
5
4
4 8
12 16
=
=
=
5 16 15 20
3 6
9
12 15
= =
=
=
4 8 12 26 20
4 3 16 15 31
11
Then + =
+
=
=1
5 4 20 20 20
20
The denominator (bottom number) of the fraction comes from adding the numbers in the ratio 2 + 3 = 5
3
a. The lawn covers of the garden
5
2
b. And the shrubs covers of the garden
5
Speed, Distance and Time
The relationship between speed, distance and time can be expressed in three ways
distance
speed =
time
distance = speed x time
distance
time =
speed
In problems relating to speed, you usually mean average speed.
Example
Paula drove a distance of 270 miles in 5 hours.
What was her average speed.
Answer
Paula’s average speed = distance she drove = 270 = 54 miles per hour (mph)
time she took
5
Using Place Value to Multiply and Divide
Percentage Increase and Decrease
Examples
To increase a quantity by a percentage, work out the increase and add it on to the original amount.
Use place value to find the answers.
a. 1.34 x 10
c. 2.7 x 1000
Answers
a. 13.4
Example
b. 203.6 x 100
d. 1.34 ÷ 10
(move each digit one place to the left)
e. 58 ÷ 100
Increase £6 by 5%.
Answer
Work out 5% of £6.
b. 20360 (move each digit two places to the left)
1% = £6 ÷ 100 = 6 pence
c. 2700
5% = 6 pence x 5 = 30 pence or £0.30
(move each digit 3 places to the left)
d. 0.134 (move each digit one place to the right)
Add £0.30 to the original amount
e. 0.58
To decrease a quantity by a percentage, work out the decrease and subtract it from the original amount.
(move each digit two places to the right)
£6 + £0.30 = £6.30
Example
Decrease £8 by 4%
Answer
1% = £8 ÷ 100 = 8 pence
4% = 8 pence x 4 = 32 pence or £0.32
Subtract £0.32 from the original amount
£8 - £0.32 = £7.68
GCSE Grade C
Order of Operations
Dividing in a given Ratio
There is an order of operations which you must follow when working out calculations.
To answer questions like this, you must follow the BIDMAS rule. This tells you the order in which you
must do the operations
Example
Brackets
Answer
Indices (powers)
1.
2+5=7
Add together the ratios
Division
2.
280 ÷ 7 = 40
Divide the total amount by (1)
Multiplication
3.
2 x 40 = 80 and 5 x 40 = 200
Multiply the ratios by (2)
Addition
One family gets 80 apples, the other gets 200 apples.
Subtraction
Prime Numbers, Highest Common Factor (HCF), Lower Common Multiple (LCM)
Example
The highest common factor (HCF) of two numbers is the greatest number that is a factor of both
of the given numbers
60 – 5 x 32 + (4 x 2)
First work out the brackets
(4 x 2) = 8
giving 60 – 5 x 32 + 8
The apples in a container are to be divided between two families in the ration 2 : 5
How many apples does each family get if there are 280 apples in the container?
Then the index (power)
3 =9
giving 60 – 5 x 9 + 8
The lowest common multiple (LCM) of two numbers is the smallest number that is a multiple of both
of the given numbers.
Then multiply
5 x 9 = 45
giving 60 – 45 + 8
Example
Then add
60 + 8
giving 68 – 45
Find
Finally subtract
68 – 45 = 23
2
a.
the (HCF) of 24 and 54
Fractions
b.
the (LCM) of 24 and 54
To add (or subtract) mixed numbers we can begin by writing each mixed number as an improper fraction.
Answer
24
Examples
2
3
11 7
44
21
65
5
3 +1 =
+ =
+
=
=5
3
4
3
4
12
12
12
12
1
1
19
3
19
9
10
4
2
3 −1 =
− =
− =
=1 = 1
2
This prime factor tree can be used to rewrite 24 as 2 x 2 x 2 x 3
12
2
6
6
2
6
2
6
6
6
6
3
Another method of adding (or subtracting) mixed numbers is to add (or subtract) the whole numbers then
add or subtract the fractions
2
If more than one operation is involved in the same calculation then follow BODMAS (GCSE Grade B)
3
54
This prime factor tree can be used to rewrite 54 as 2 x 3 x 3 x 3
Dividing Fractions
2
To divide by a fraction, multiply by the reciprocal.
27
Examples
3
4
1.
3.
3
4
3
6
4
1
3
1
= ×
4
6
1
1
= ×
4
2
1
=
8
5
3
12
= ×
12
4
5
3
3
= ×
1
5
9
=
5
4
=1
5
÷6 = ÷
÷
2.
6÷
3
4
=
=
=
6
3
÷
1
4
6
4
×
1
3
2
4
×
1
1
3
9
3
3
24 = 2 x 2 x 2 x 3 54 = 2 x 3 x 3 x 3
2 x 3 is common to both numbers
Hence the HCF is 6
=8
4.
2
1
3
÷
1
4
6
5
3
5
×
3
1
×
1
2
5
= ÷
=
=
=
25
6
6
25
2
5
The smallest number that both 24 and 54 divide into exactly is
2 x 2 x 2 x 3 x 3 x 3 Hence the LCM is 216
24
54
L
C
M
2
HCF
2
3
2
3
3
Indices
Reciprocals
The reciprocal of any number is one divided by the number.
Rules of Indices
Example
33 x 35 = (3 x 3 x 3) x (3 x 3 x 3 x 3 x 3) = 38 which is 3(3 + 5)
m
n
1
2
The reciprocal of 0.25 is 1 ÷ 0.25 = 4
The reciprocal of 2 is 1 ÷ 2 =
m+n
a xa =a
When multiplying add the powers
You can find the reciprocal by swapping the numerator and denominator
7 ÷ 7 = (7 x 7 x 7 x 7 x 7 x 7) ÷ 7 = 7 x 7 x 7 x 7 x 7 = 7 which is 7
Example
am ÷ an = am - n
The reciprocal of
6
5
When dividing subtract the powers
Examples
Use the rules of indices to write the following as single powers of 7
a.
74 x 79 = 74 + 9 = 713
b.
78 ÷ 72 = 78 – 2 = 76
(6 – 1)
2 3
is
3 2
4 7
The reciprocal of is
7 4
Percentage Change
To find a percentage increase (or decrease) we compare the actual increase with the initial value
% Increase = Actual Increase x 100%
Original amount
% Decrease = Actual Decrease x 100%
Original amount
Example
A jersey which was priced at £39 is reduced to £28 in a sale. What percentage reduction is this?
Answer
Actual decrease in price
= £39 - £28
= £11
% Decrease = Actual Decrease x 100%
Original amount
11
=
x 100%
39
= 28% (to the nearest percentage)