Year 7 · Chapter 2

Fractions, decimals and percentages

By the end of this chapter you will be able to

Change a fraction with a denominator of 2, 4, 5, 8, 10, 20, 25 or 100 into a decimal, and a decimal into a fraction
Change between fractions, decimals and percentages, and say which fractions cannot be written as terminating decimals
Find a percentage of an amount, and a fraction of an amount, using the fraction as an operator
Write one quantity as a percentage of another, including amounts over 100%
Work out a percentage increase or decrease, and find an original amount from its new value
Add, subtract, multiply and divide fractions, and choose a sensible method

You should already know

Equivalent fractions, simplifying and the place value of tenths and hundredths (Year 6, Chapter 2)
Percentages as parts out of 100, and percentage of an amount (Year 6, Chapter 6)
Place value of decimals and the order of operations (Chapter 1)

Where this leads

Standard form and rounding of decimals (Chapter 3)
Percentage change in ratio and rates of change (Chapter 8)

National Curriculum for England, Key Stage 3 mathematics, Year 7. Content: N9 change freely between terminating decimals and fractions, N10 percentages as parts per hundred; percentage change as a fraction or decimal; one quantity as a percentage of another; percentages over 100%, N11 fractions and percentages as operators (of an amount), Fraction arithmetic with the four operations (practice of N4 for fractions and mixed numbers).

Exercise P Do you remember?

Non-calculator

FWork out one half of 18.

Done

FWork out 1/2 + 1/4.

Done

FWork out 25% of 60.

Done

FWrite 3/4 as a decimal.

Done

FWhich is bigger, 3/5 or 1/2?

Done

2.1

Fractions and decimals

Year 7
N9

Video lesson: Fractions and decimalsComing soon

A bottle holds 1 litre. A jug holds 3/8 of a litre, and a cup holds 0.25 of a litre. To compare them, the amounts must be written in the same form, either both as fractions or both as decimals.

A decimal is a fraction with a denominator of 10, 100, 1000 and so on. So 0.7 is 7/10, and 0.07 is 7/100. Changing a fraction to a decimal is easy when the bottom number is a factor of a power of 10.

A fraction with a denominator made only from factors of 2 and 5 always gives a terminating decimal, which stops. For example 3/8 has denominator 8 = 2 × 2 × 2, so it does stop: 3/8 = 0.375.

To change a decimal to a fraction, write the digits over 10, 100 or 1000 by place value, then simplify.

Key ideas

A decimal with 1 digit after the point is over 10, 2 digits over 100, 3 digits over 1000.
To change a fraction to a decimal, make the bottom number 10, 100 or 1000 by multiplying the top and bottom by the same number.
Fractions whose denominators have only 2s and 5s as factors give terminating decimals, such as 3/8 = 0.375 and 7/20 = 0.35.
Simplify a fraction by dividing the top and bottom by their HCF.

Worked example 2.1.1

Write 7/20 as a decimal.

Method. Make the denominator 10, 100 or 1000, because those are easy to write as decimals.

Find what the denominator must be multiplied by. 20 times what gives 10, 100 or 1000?
20 × 5 = 100
Multiply the top and the bottom by the same number. This keeps the fraction the same size.
7/20 = (7 × 5)/(20 × 5) = 35/100
Write the denominator as a decimal. 100 means the place value is hundredths.
35/100 = 0.35

Check. 0.35 × 100 = 35, so 0.35 is 35/100. ✓

Watch out. Do not just put a decimal point in the top number. Make the bottom 10, 100 or 1000 first.

Worked example 2.1.2

Write 0.36 as a fraction in its simplest form.

Method. Read the decimal by its place value, write it over 10, 100 or 1000, then simplify.

Read the last digit. The last digit is in the hundredths place, so the bottom number is 100.
0.36 = 36/100
Find the HCF of the top and bottom. The HCF of 36 and 100 is 4.
HCF(36, 100) = 4
Divide the top and bottom by the HCF. This gives the simplest form.
36/100 = 9/25

Check. 9/25 as a decimal: 0.36. ✓

Watch out. Simplify fully. Check that the top and bottom have no common factor left.

Worked example 2.1.3

Write 1.75 as a fraction in its simplest form.

Method. Read the decimal by its place value, write it over 10, 100 or 1000, then simplify.

Read the last digit. The last digit is in the hundredths place, so the bottom number is 100.
1.75 = 175/100
Find the HCF of the top and bottom. The HCF of 175 and 100 is 25.
HCF(175, 100) = 25
Divide the top and bottom by the HCF. This gives the simplest form.
175/100 = 7/4

Check. 7/4 as a decimal: 1.75. ✓

Watch out. Simplify fully. Check that the top and bottom have no common factor left.

Watch out

Watch out: 0.5 is one half, not 1/5. Also, do not write 0.4 as 4/100: 0.4 has one digit after the point, so it is 4/10 = 2/5.

Exam tip

Grown-up tip. Ask the pupil to say the decimal out loud: "three hundred and seventy-five thousandths" helps them see 375/1000.

2.2

Percentages, fractions and decimals

Year 7
N10

Video lesson: Percentages, fractions and decimalsComing soon

A phone shop shows a sign: "Save 40% on all cases". The sign uses a percentage, but the shelf label says 2/5 off, and the till shows 0.4. These are three ways of writing the same amount.

Per cent means out of 100. So 37% means 37 out of 100, or 37/100, or 0.37. The percent sign is a short way of writing "over 100".

Look at a 100-square grid. If 37 squares are shaded, then 37% of the grid is shaded. Each square is 1% of the whole.

To change between the forms: a percentage is a fraction over 100, and a decimal is a number with the point moved two places. So 0.37 = 37%, and 37% = 37/100.

Key ideas

Per cent means out of 100: 37% = 37/100 = 0.37.
To change a decimal to a percentage, multiply by 100 (move the point two places right).
To change a percentage to a decimal, divide by 100 (move the point two places left).
Percentages can be bigger than 100%, for quantities that are more than the whole.

Worked example 2.2.1

Write 37% as a fraction.

Method. Per cent means out of 100, so write the number over 100 and simplify.

Write it as a fraction out of 100. Per cent means per hundred.
37% = 37/100
Find the HCF of the top and bottom. The HCF of 37 and 100 is 1.
HCF(37, 100) = 1
Simplify. Divide the top and bottom by the HCF.
37/100 = 37/100

Check. 37/100 = 0.37 = 37%. ✓

Watch out. Write the answer in simplest form, not as 40/100.

Worked example 2.2.2

Write 45% as a fraction in its simplest form.

Method. Per cent means out of 100, so write the number over 100 and simplify.

Write it as a fraction out of 100. Per cent means per hundred.
45% = 45/100
Find the HCF of the top and bottom. The HCF of 45 and 100 is 5.
HCF(45, 100) = 5
Simplify. Divide the top and bottom by the HCF.
45/100 = 9/20

Check. 9/20 = 0.45 = 45%. ✓

Watch out. Write the answer in simplest form, not as 40/100.

Worked example 2.2.3

Write 0.07 as a percentage.

Method. A percentage is a number out of 100, so multiply the fraction by 100.

Start with the fraction or decimal. This is the part of the whole, where the whole is 1.
7/100 of the whole
Multiply by 100. This changes it from parts of 1 to parts of 100.
7/100 × 100 = 7
Write the answer as a percentage. Add the percent sign.
7%

Check. 7% out of 100 is the same as 7/100 of the whole. ✓

Watch out. If the answer is bigger than 100, the quantity is more than the whole. That is fine.

Watch out

Watch out: 0.5% is not 0.5 out of 10. It is 0.5 out of 100, so 0.5% = 0.005. Also do not write 45% as 1/45.

Exam tip

Grown-up tip. Ask the pupil to write the three forms for one familiar amount: half is 50%, 0.5 and 1/2.

2.3

Finding a percentage or a fraction of an amount

Year 7
N10, N11

Video lesson: Finding a percentage or a fraction of an amountComing soon

A bike costs £240 in a sale that offers 15% off. A school has 96 pupils, and 3/8 of them take the bus. Both problems ask for part of an amount.

To find a percentage of an amount, write the percentage as a fraction or decimal and multiply, or find 1% first. To find a fraction of an amount, divide by the bottom number and multiply by the top number. A fraction used like this is an operator, because it operates on the amount.

To write one quantity as a percentage of another, make a fraction with the part on the top and the whole on the bottom, then multiply by 100.

Key ideas

15% of £240: 1% is £2.40, so 15% is 15 × £2.40 = £36.
a/b of an amount: divide by b to find one part, then multiply by a.
Percentage = (part ÷ whole) × 100.
Look for the whole amount before you start. It is the number after the word "of".

Worked example 2.3.1

Work out 15% of £240.

Method. Find 1% first, then multiply by the number of per cents.

Find 1%. One per cent is one hundredth of the amount.
1% of 240 = 240 ÷ 100 = 12/5
Multiply by the number of per cents. We want 15 per cent, so we multiply by 15.
15 × 12/5 = 36
Write the answer. Check the units with the question.
36

Check. Check the other way: 15% = 3/20 and 3/20 × 240 = 36. ✓

Watch out. Do not add the percentage to the amount. Multiply by the percentage as a fraction or decimal.

Worked example 2.3.2

Work out 3/8 of 96.

Method. Divide by the bottom number to find one part, then multiply by the top number.

Divide by the bottom number. The bottom number tells us how many equal parts there are.
96 ÷ 8 = 12
Multiply by the top number. The top number tells us how many parts to take.
3 × 12 = 36
Write the answer. This is the value of the fraction of the amount.
36

Check. 3 of the 8 equal parts of 96: 8 parts make 96, so 3 parts are 36. ✓

Watch out. Divide first, then multiply. Multiplying first gives a number that is too big to find one part.

Worked example 2.3.3

18 out of 72 pupils walk to school. What percentage is this?

Method. Write the first number as a fraction of the second, then change it to a percentage.

Write the fraction. The number we want is the part, and the whole is the amount we compare with.
18 ÷ 72 = 1/4
Multiply by 100. A percentage is a number out of 100.
1/4 × 100 = 25
Write the answer as a percentage. Add the percent sign.
25%

Check. 25% of 72 is 18, which is 18. ✓

Watch out. The whole goes on the bottom of the fraction. Dividing the other way round gives the wrong answer.

Watch out

Watch out: "15% of 240" is not "240 + 15%". Find the part first, then add or subtract it if the question asks for a new price.

Exam tip

Grown-up tip. Ask the pupil to find 10% first (move the point one place left), then build up: 5% is half of 10%.

2.4

Percentage change and reverse percentages

Year 7
N10

Video lesson: Percentage change and reverse percentagesComing soon

A bus fare rises from £3.00 to £3.60. A town's population falls from 80,000 to 68,000 after a factory closes. In both cases we want the change as a percentage of the amount it started with.

Percentage change = change ÷ original × 100. The original is always the amount we started with, not the new amount. If the number goes up, it is an increase. If it goes down, it is a decrease.

A reverse percentage starts from the new amount and finds the original. For a 20% increase, the new amount is 120% of the original. So find 1% of the original by dividing the new amount by 120, then multiply by 100.

Key ideas

Percentage change = (new − original) ÷ original × 100.
Use the original amount as the divisor, every time.
A value of 130% means 130 out of 100: it is 30% more than 100%.
For a reverse percentage, use 100% + the increase, or 100% − the decrease, as the total percentage.

Worked example 2.4.1

A bus fare rises from 30p to 36p. Find the percentage increase.

Method. Find the change first, then compare it with the original amount. The original is the one we divide by.

Find the change. Subtract the old amount from the new amount, and ignore the sign for now.
|36 − 30| = 6
Divide by the original amount. Percentage change is always compared with the original, not the new amount.
6 ÷ 30 = 1/5
Multiply by 100 and say whether it is an increase or a decrease. The number went increase, so the answer is a increase.
1/5 × 100 = 20% increase

Check. 20% of 30 is 6, and 30 + 6 = 36. ✓

Watch out. Divide by the original amount. Dividing by the new amount gives the wrong percentage.

Worked example 2.4.2

The number of pupils in a club falls from 80 to 68. Find the percentage decrease.

Method. Find the change first, then compare it with the original amount. The original is the one we divide by.

Find the change. Subtract the old amount from the new amount, and ignore the sign for now.
|68 − 80| = 12
Divide by the original amount. Percentage change is always compared with the original, not the new amount.
12 ÷ 80 = 3/20
Multiply by 100 and say whether it is an increase or a decrease. The number went decrease, so the answer is a decrease.
3/20 × 100 = 15% decrease

Check. 15% of 80 is 12, and 80 − 12 = 68. ✓

Watch out. Divide by the original amount. Dividing by the new amount gives the wrong percentage.

Worked example 2.4.3

After a 20% increase, the price of a jacket is £36. What was the original price?

Method. Work out what 100% of the original is first. After the increase, it is (100 + the increase) per cent.

Write what the new amount is as a percentage. The original plus 20% is 100% + 20% = 120%.
120% = 36
Find 1%. Divide the amount by the number of per cents.
36 ÷ 120 = 3/10
Find 100%. Multiply by 100 to get the original.
3/10 × 100 = 30

Check. 30 increased by 20% is 30 × 6/5 = 36. ✓

Watch out. Do not take 20% of the new amount and subtract it. That gives the wrong original.

Watch out

Watch out: a 50% increase followed by a 50% decrease does not come back to the start. Check with real numbers, such as 100 and 150.

Exam tip

Grown-up tip. Ask the pupil to write the original and the new amount on two lines, then subtract, so the original is always clear.

2.5

Fractions: adding, subtracting, multiplying and dividing

Year 7
N4, N11

Video lesson: Fractions: adding, subtracting, multiplying and dividingComing soon

A recipe needs 2/3 of a cup of flour and 3/4 of a cup of oats. Lena wants to know how much dry mix she needs altogether. To add the two amounts, the fractions must be written with the same bottom number.

To add or subtract fractions, write both with the same denominator (the lowest common multiple of the two bottom numbers is a good choice). Then add or subtract the top numbers and keep the bottom number.

To multiply, multiply the tops and multiply the bottoms, and cancel any common factor first if you can. To divide, turn the second fraction upside down and multiply.

Key ideas

Add or subtract: use a common denominator, then add or subtract the tops.
Multiply: multiply the tops, then multiply the bottoms. Simplify at the end.
Divide: multiply by the reciprocal (turn the second fraction upside down).
Whole numbers are fractions with a bottom number of 1, so 8 = 8/1.

Worked example 2.5.1

Work out 2/3 + 3/4.

Method. Use the rule for the operation, then simplify.

Write both fractions with the same bottom number. The lowest common multiple of 3 and 4 is 12.
2/3 = 2/3 and 3/4 = 3/4
Add the top numbers. Keep the bottom number. The parts are now the same size, so we add the parts.
8 + 9 = 17, so the total is 17/12
Simplify. Divide the top and bottom by any common factor.
17/12

Check. 17/12 is about 1.42, and 2/3 + 3/4 is about 1.42. ✓

Watch out. Find a common bottom number before you add or subtract. Multiplying the bottoms is not right for addition.

Worked example 2.5.2

Work out 5/6 − 1/4.

Method. Use the rule for the operation, then simplify.

Write both fractions with the same bottom number. The lowest common multiple of 6 and 4 is 12.
5/6 = 5/6 and 1/4 = 1/4
Subtract the top numbers. Keep the bottom number. Subtract the second top number from the first.
10 − 3 = 7
Simplify. Divide the top and bottom by any common factor.
7/12

Check. 7/12 + 1/4 = 5/6, so the subtraction is right. ✓

Watch out. Find a common bottom number before you add or subtract. Multiplying the bottoms is not right for addition.

Worked example 2.5.3

Work out 4/5 ÷ 2/3.

Method. Use the rule for the operation, then simplify.

Turn the second fraction upside down. Dividing by a fraction is the same as multiplying by its reciprocal.
2/3 becomes 3/2
Multiply the fractions. Multiply the tops and the bottoms.
4/5 × 3/2 = 6/5
Simplify. Divide the top and bottom by any common factor.
6/5

Check. 6/5 × 2/3 = 4/5, which is the original. ✓

Watch out. Find a common bottom number before you add or subtract. Multiplying the bottoms is not right for addition.

Watch out

Watch out: when adding fractions, do not add the bottom numbers. 1/2 + 1/3 is not 2/5. Find a common bottom number first.

Exam tip

Grown-up tip. Use a real amount, such as a pizza cut into 12 slices, to show why 1/2 + 1/3 needs twelfths.

Chapter summary

Skill	Method
Fraction to decimal	Make the bottom 10, 100 or 1000 (multiply top and bottom); terminating only if the bottom has only 2s and 5s
Decimal to fraction	Write over 10, 100 or 1000 by place value, then simplify
Percentage	Out of 100: 37% = 37/100 = 0.37
Percentage of an amount	Find 1% (divide by 100), then multiply by the percentage; or write it as a fraction or decimal
Fraction of an amount	Divide by the bottom number, then multiply by the top number
One quantity as a percentage	(part ÷ whole) × 100
Percentage change	(new − original) ÷ original × 100, with the original as the divisor
Reverse percentage	Find 1% of the original from the new amount, using (100 + increase)% or (100 − decrease)%
Fraction arithmetic	Add/subtract with a common denominator; multiply tops and bottoms; divide by multiplying by the reciprocal

Can you…?

I can change a fraction to a decimal, and a decimal to a fraction, in its simplest form
I can say when a fraction gives a terminating decimal and when it does not
I can change between fractions, decimals and percentages
I can find a percentage of an amount, and a fraction of an amount
I can write one quantity as a percentage of another
I can work out a percentage increase or decrease, including over 100%
I can find an original amount from its new value (reverse percentage)
I can add, subtract, multiply and divide fractions

Key terms

terminating decimal: a decimal that stops, such as 0.375
recurring decimal: a decimal with a pattern that repeats forever, such as 0.333...
per cent: out of 100; the sign % means per cent
operator: a fraction or percentage that acts on an amount to find a part of it
common denominator: a shared bottom number for two or more fractions
reciprocal: the fraction turned upside down; the reciprocal of 2/3 is 3/2
percentage change: the change in an amount, as a percentage of the original amount
reverse percentage: finding the original amount from a new amount after a percentage change

Total 21 marks. Allow about 21 minutes.

← Chapter 1: Integers, factors and powers Chapter 3: Standard form, rounding and estimation →

Fractions, decimals and percentages

By the end of this chapter you will be able to

You should already know

Where this leads

Exercise P Do you remember?

Fractions and decimals

Key ideas

Watch out

Exam tip

Exercise 2A Fractions and decimals

Percentages, fractions and decimals

Key ideas

Watch out

Exam tip

Exercise 2B Percentages, fractions and decimals

Finding a percentage or a fraction of an amount

Key ideas

Watch out

Exam tip

Exercise 2C Percentages and fractions of amounts

Percentage change and reverse percentages

Key ideas

Watch out

Exam tip

Exercise 2D Percentage change and reverse percentages

Fractions: adding, subtracting, multiplying and dividing

Key ideas

Watch out

Exam tip

Exercise 2E Fraction calculations

Chapter summary

Can you…?

Key terms

Exercise 2Z Check-up

Exercise 2X Practice test questions