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Fractions/Decimals/Percentages
Pupil Notes and worked examples
Key Skill 1: Create equivalent fractions and simplify fractions
We use the term numerator to describe the term on the top of a fraction.
We use the term denominator to describe the term on the bottom of a fraction.
Examples
numerator
1
2
1
2
3
17
denominator
2
3
4
5
10
100
Equivalent fractions
To create equivalent fractions you multiply, or divide, the numerator and denominator of a
fraction by the same number.
2
2x2
=
4
=
3
3x2
2
2 x 100
(two-thirds is equivalent to four-sixths)
6
=
3
200
=
(two-thirds is equivalent to two hundred three-hundredths)
3 x 100
300
Simplifying fractions
To simplify fractions, divide the numerator and denominator by the same number.
20  10
20
=
30
2
=
30  10
3
81  9
81
=
90
90  9
9
=
10
It may take more than one division to fully simplify a fraction.
24  4
24
=
32
32  4
6  2
6
=
=
8
8  2
3
=
4
Written by John Donnelly
Key Skill 2: Determine a fraction of a quantity
To calculate a fraction of a quantity we divide the quantity by the denominator and multiply the
result by the numerator.
You may have heard you teacher say: “divide by the bottom, times by the top”
Examples
Find
(a)
2
(b)
4
of 642
3
3
7
of 9860
5
21 4
6 4 12
x
(c)
of 7850
5
214
2
428
10
1 5 70
7 28 35 0
10
0 7 8 5
7 78 85 50
1570
x
4
6280
785
x 7
5495
2
53
Key Skill 3: Correctly change a mixed number to an improper fraction and vice-versa
A mixed number contains a whole number with a fraction. Examples of mixed numbers are:
2
4
3
2
6
3
7
10
4
9
3
10
An improper fraction is when a fraction has a larger numerator than its denominator.
Examples of improper fractions are:
22
13
92
17
3
4
3
10
Written by John Donnelly
Key Skill 3 (continued): Correctly change a mixed number to an improper fraction and vice-versa
Converting a mixed number into an improper fraction
Multiply the whole number by the denominator and add on the numerator.
This gives the new numerator. The denominator remains the same.
Examples
2
3
4
2
6
10
3
4
4x3+2
=
3
6x4+3
=
10 x 3 + 2
=
3
4
14
3
27
=
32
=
=
3
4
3
Converting an improper fraction into a mixed number
Find how many times the denominator divides into the numerator, this gives the whole number.
The remainder gives the new numerator, the denominator remains the same.
Examples
22
13
17
3
4
10
1
=
7
3
=
3
3
7
=
4
1
10
3 divides into 22
4 divides into 13
10 divides into 17
7 times with a
3 times with a
1 time with a
remainder of 1
remainder of 3
remainder of 7
Written by John Donnelly
Key Skill 4: Put in order a range of fractions
To compare fractions you should convert them into equivalent fractions with a common
denominator. (When the denominators are the same).
Example
Order the list of fractions from lowest to highest
1
2
1
2
3
2
3
4
5
10
All 5 fractions need to converted into sixtieths
1 x 30
2 x 20
1 x 15
2 x 12
3x6
2 x 30
3 x 20
4 x 15
5 x 12
10 x 6
30
40
15
24
18
60
60
60
60
60
Order the original list after comparing the numerators:
1
3
2
1
2
4
10
5
2
3
Key Skill 5: Identify and use a range of commonly used fraction with their decimal and
percentage equivalents
Fraction
1
Decimal
Percentage
0.5
50%
2
1
Decimal
Percentage
0.75
75%
0.4
40%
0.6
60%
0.7
70%
0.375
37.5%
4
2
0.25
25%
4
1
5
3
0.2
20%
5
1
5
7
0.1
10%
10
1
10
3
0.125
8
Fraction
3
12.5%
8
Written by John Donnelly
Key Skill 6: Put in order a range of decimal fractions, to at least 3 decimal places
Decimal fractions are just decimals!!!
Order the following decimals from lowest to highest:
0.2323,
0.2,
0.3,
0.3232,
0.23,
0.32,
0.235
0.23,
0.32,
0.235
Consider those with the smallest tenths digit
0.2323,
0.2,
0.3,
0.3232,
Those numbers with 2 tenths are
0.2323,
0.2,
0.23,
0.235
0.23,
0.235
0.2 is the smallest number
0.235
0.23 is the 2nd smallest number
0.2323 is the 3rd smallest number
0.235 is the 4th smallest number
Compare their hundredths digit
0.2323,
0.20,
Compare their thousandths digit
0.2323,
0.230,
The beginning of our list goes: 0.2, 0.23, 0.2323, 0.235
Those numbers with 3 tenths are
0.3,
0.3232,
0.32,
Compare their hundredths digit
0.30,
0.3232,
0.32,
0.3 is the 3rd largest number
0.320,
0.32 is the 2nd largest number
0.3232 is the largest number
Compare their thousandths digit
0.3232,
The end of our list goes: 0.3, 0.32, 0.3232
Our ordered list is as follows: 0.2, 0.23, 0.2323, 0.235, 0.3, 0.32, 0.3232
Written by John Donnelly
Key Skill 7: Add, subtract, multiply and divide decimals
Adding decimals
Set out your calculations by putting tenths, hundredths, thousandths etc… below each other.
Add trailing zeros to ensure all numbers have the same number of digits after the decimal point.
(a)
6.456 + 17.78
(b)
6.456
+ 17.780
24.236
1.8 + 3.785 + 55.243
1.800
3.785
+ 55.243
60.828
11 1
1 1 1
Subtracting decimals
Set out your calculations by putting tenths, hundredths, thousandths etc… below each other.
Add trailing zeros to ensure all numbers have the same number of digits after the decimal point.
(a)
9.475 – 5.6
(b)
9 – 3.654
8 9.1475
9 9
8 9 .10 10 10
- 5. 600
3. 875
-3.6 5 4
5.3 4 6
Written by John Donnelly
Key Skill 7 (continued): Add, subtract, multiply and divide decimals
Multiplying decimals by a single digit
Multiply each digit of the decimal by the single digit, starting with the digit furthest right.
(a)
4.563 x 7
(b)
4.563
x
7
31.941
236.4 x 8
236.4
x 8
1691.2
3 42
2 53
Dividing decimals by a single digit
Divide each digit of the decimal by the single digit, starting with the digit furthest left.
Any remainders carry over to the next digit.
7.41  3
(a)
3
2. 4 7
7 . 14 21
(b)
78.55  5
5
1 5. 71
7 28 . 35 5
Written by John Donnelly
Key Skill 8: Calculate a percentage change
100% is everything!!!
A percentage increase is when you find a percentage of an amount and add it to the original amount
A percentage decrease is when you find a percentage of an amount and subtract it from the original amount
Percentage to find
10%
Method
Divide by ten as 10% is
1
of 100%
10
Divide by five as 20% is
1
of 100%
5
20%
or
Divide by ten (10%) then multiply by two (20% = 10% x 2)
30%
Divide by ten (10%) then multiply by three (30% = 10% x 3)
Divide by ten (10%) then multiply by nine(90% = 10% x 9)
90%
or
5%
Divide by ten (10%) and subtract it from 100%
1
Divide by twenty as 5% is
of 100%
20
or
15%
25%
75%
1%
3%
9%
11%
23%
99%
0.5%
1.5%
110%
125%
Divide by ten (10%) then divide by 2 (half)
15% = 10% + 5%
1
Divide by four as 25% is of 100%
4
or
Halve 50%
or
75% = 50% + 25%
1
Divide by one hundred as 1% is
of 100%
100
or
75% = 25% x 3
Divide by ten (10%) and then divide by ten
Divide by one hundred (1%) then multiply by three
Divide by one hundred (1%) then multiply by nine
11% = 10% + 1%
23% = 10% + 10% + 3%
or
23% = 20% + 3%
99% = 100% - 1%
or
99% = (10% x 9) + (10% x 9)
Divide by one hundred (1%) then divide by two (halve)
1.5% = 1% + 0.5%
110% = 100% + 10%
125% = 100% + 25%
Written by John Donnelly
Key Skill 9: Compare and order fractions, decimals and percentages to make choices
Comparing fractions, decimals and percentages
Now, you can find a fraction of an amount and a percentage of an amount before making a
comparison.
Example
Which is larger
3
of £460 or 80% of £400?
4
3
of £460 = £345
4
4
80% of £400 = £320
1 1 5
4 6 20
10% of £400 = £40
115
x 3
345
40
x 8
320
1
3
of £450 (£345) is larger than 80% of £400 (£320) by £25 (£345 - £320)
4
Ordering fractions, decimals and percentages
To order fractions, decimals and percentages convert them all into percentage before comparing them.
Example
Order the following from lowest to highest
0.35 =
0.35,
35
= 35%
100
43%,
3
,
5
39.5%,
3
,
10
0.4
43%
3 3  20
60
=
=
= 60%
5 5  20
100
39.5%
3
3  10
30
=
=
= 30%
10 10  10 100
0.4 =
4
4  10
40
=
=
= 40%
10 10  10 100
So the correct order in terms of percentages is 30%, 35%, 39.5%, 40%, 43%, 60%
So the correct order is
3
,
10
0.35, 39.5%,
0.4,
43%,
3
5
Written by John Donnelly