Year 8 · Chapter 3

Fractions and percentages

By the end of this chapter you will be able to

Add and subtract fractions with different denominators, and with mixed numbers
Multiply and divide fractions, using the reciprocal for division
Change between fractions, decimals and percentages, and put them in order
Find 10% and 5% of an amount, and build up other percentages without a calculator
Use a multiplier to increase or decrease an amount, and find a percentage change

You should already know

Equivalent fractions and simplifying (Year 7)
Finding a fraction of an amount, and place value in decimals (Year 7)
Percentages as out of 100, and simple percentages such as 50% and 25% (Year 7)

Where this leads

Compound interest and repeated percentage change (Year 9 and GCSE)
Ratio and rates, linking fractions and percentages to proportion (Year 8, Chapter 4)
Reverse percentages and fractions of quantities in algebra (GCSE)

National Curriculum for England, Key Stage 3 mathematics, Year 8. Content: Calculate with fractions, mixed numbers and improper fractions, Convert between fractions, decimals and percentages, Find a percentage of an amount, and use percentage increase and decrease, Use multipliers, and calculate a percentage change.

Exercise P Do you remember?

Non-calculator

FWhat is 3/4 of 20?

Done

FWrite 0.6 as a fraction in its simplest form.

Done

FWhat is 10% of 250?

Done

FWhat is 50% of 84?

Done

FFind the highest common factor of 12 and 18.

Done

3.1

Adding and subtracting fractions

Year 8
Calculate with fractions, mixed numbers and improper fractions

Video lesson: Adding and subtracting fractionsComing soon

Tom has 1/3 of a cake and his sister has 1/4 of the same cake. They want to know how much of the cake they have altogether before they cut it up.

You can only add or subtract fractions easily when the denominators (the bottom numbers) are the same.

To add or subtract fractions with different denominators, first find a common denominator. The lowest common denominator is the lowest common multiple (LCM) of the two bottoms.

For a mixed number such as 2 1/4, you can add the whole numbers and the fractions separately, or change the mixed number to an improper fraction first.

Key ideas

Make the denominators the same before you add or subtract.
Only the numerators (tops) are added or subtracted once the denominators match.
Add the whole numbers and the fractions separately for mixed numbers, and carry if needed.
Change a mixed number to an improper fraction: (whole × bottom + top) over the bottom.

Worked example 3.1.1

Work out 1/3 + 1/4.

Method. Make the denominators the same, then add the numerators.

Find a common denominator. The lowest common multiple of 3 and 4 is 12.
LCM of 3 and 4 = 12
Write both fractions with that denominator. Multiply the top and bottom of each fraction by the number needed to make 12.
1/3 = 4/12 and 1/4 = 3/12
Add the top numbers. The bottoms are now the same, so add only the tops.
4 + 3 = 7
Write the answer.
7/12

Check. 7/12 = 0.583 as a decimal. 1/3 + 1/4 = 0.583 as a decimal too.

Watch out. Do not add the bottoms. Only the top numbers are added once the denominators match.

Worked example 3.1.2

Work out 5/6 − 1/4.

Method. Make the denominators the same, then subtract the numerators.

Find a common denominator. The lowest common multiple of 6 and 4 is 12.
LCM of 6 and 4 = 12
Write both fractions with that denominator. Multiply the top and bottom of each fraction by the number needed to make 12.
5/6 = 10/12 and 1/4 = 3/12
Subtract the top numbers. The bottoms are now the same, so subtract only the tops.
10 − 3 = 7
Write the answer.
7/12

Check. 7/12 = 0.583 as a decimal. 5/6 − 1/4 = 0.583 as a decimal too.

Watch out. Do not add the bottoms. Only the top numbers are added once the denominators match.

Worked example 3.1.3

Work out 2 1/4 + 1 2/3. Give your answer as a mixed number.

Method. Change both mixed numbers to improper fractions, then add them.

Change the first mixed number. Multiply the whole number by the bottom, add the top, and keep the same bottom.
2 1/4 = (2 × 4 + 1)/4 = 9/4
Change the second mixed number. Do the same for the second number.
1 2/3 = 5/3
Add the fractions. Use a common denominator, 12.
9/4 + 5/3 = 47/12
Change back to a mixed number. Divide the top by the bottom to find the whole part.
3 11/12

Check. 3 11/12 is about 3.92, which matches 2.25 + 1.67.

Watch out. Keep the whole number separate when you add, or change everything to improper fractions first.

Watch out

Watch out: 1/3 + 1/4 is not 2/7. You cannot add the bottoms. Make the bottoms the same first: 4/12 + 3/12 = 7/12.

Exam tip

Exam tip. Write out the common denominator before you add. Many marks are lost by adding the bottoms.

3.2

Multiplying and dividing fractions

Year 8
Calculate with fractions, mixed numbers and improper fractions

Video lesson: Multiplying and dividing fractionsComing soon

A bottle holds 3/4 of a litre of water. Each cup in the classroom holds 1/8 of a litre. Mia wants to know how many cups she can fill from one bottle.

To multiply fractions, multiply the tops and multiply the bottoms: 2/3 × 3/5 = 6/15 = 2/5.

To divide by a fraction, keep the first fraction, change the sign to multiply, and flip the second fraction. The flipped fraction is the reciprocal.

For a mixed number, change it to an improper fraction before you multiply or divide.

Key ideas

Multiply: top × top over bottom × bottom.
Divide: keep, change to multiply, flip the second fraction.
A reciprocal is the fraction turned upside down: the reciprocal of 3/4 is 4/3.
Change mixed numbers to improper fractions before you multiply or divide.

Worked example 3.2.1

Work out 2/3 × 3/5.

Method. Multiply the tops together, then multiply the bottoms together. Simplify at the end.

Multiply the top numbers. The new top is 2 × 3.
2 × 3 = 6
Multiply the bottom numbers. The new bottom is 3 × 5.
3 × 5 = 15
Simplify the fraction. Divide the top and bottom by their highest common factor.
6/15 = 2/5
Write the answer.
2/5

Check. 2/3 is 0.667, and 3/5 is 0.6. Multiplying gives about 0.400.

Watch out. Multiply across: top with top, bottom with bottom. Do not add the denominators.

Worked example 3.2.2

Work out 3/4 ÷ 1/2.

Method. Dividing by a fraction is the same as multiplying by its reciprocal (flip the fraction over).

Keep the first fraction. Keep 3/4 the same.
3/4
Change divide to multiply, and flip the second fraction. The reciprocal of 1/2 is 2/1.
3/4 × 2/1
Multiply the fractions. Multiply the tops and multiply the bottoms.
6/4
Simplify the answer.
3/2

Check. 3/2 × 1/2 = 3/4, which checks the division.

Watch out. Flip only the second fraction. Do not flip the first one.

Worked example 3.2.3

Work out 2 1/4 × 2/3. Give your answer as a mixed number.

Method. Change the mixed number to an improper fraction, then multiply.

Change 2 1/4 to an improper fraction. Multiply the whole number by the bottom and add the top: 2 × 4 + 1 = 9.
2 1/4 = 9/4
Multiply the fractions. Multiply the tops and multiply the bottoms.
9/4 × 2/3 = 18/12
Simplify. Divide the top and bottom by 6.
18/12 = 3/2
Write the answer as a mixed number. 3 divided by 2 is 1 remainder 1.
1 1/2

Check. 1 1/2 is about 1.5, and 2 1/4 × 2/3 is about 2.25 × 0.67 = 1.5.

Watch out

Watch out: to divide, flip only the second fraction. 3/4 ÷ 1/2 is 3/4 × 2/1, not 2/1 × 3/4 flipped at the start.

Exam tip

Exam tip. Check your division answer by multiplying it back. (3/2) × (1/2) = 3/4, which is the start.

3.3

Fractions, decimals and percentages

Year 8
Convert between fractions, decimals and percentages

Video lesson: Fractions, decimals and percentagesComing soon

In a school survey, 3/4 of pupils said they liked the new canteen. The headteacher wants to show this in a leaflet, so she writes it as a decimal and a percentage as well.

A fraction, a decimal and a percentage can all describe the same amount. Every one means "parts out of":

a fraction is top over bottom;
a decimal is parts out of 10, 100 or 1000;
a percentage is parts out of 100.

To change a fraction to a decimal, divide the top by the bottom. To change a decimal to a fraction, write it over 10, 100 or 1000 and simplify. To change a fraction to a percentage, change it to a decimal, then multiply by 100.

Key ideas

Fraction to decimal: divide the top by the bottom.
Decimal to fraction: write it over 10, 100 or 1000, then simplify.
Percentage to fraction: write it over 100, then simplify.
Fraction to percentage: change to a decimal, then multiply by 100.

Worked example 3.3.1

Write 3/8 as a decimal.

Method. Divide the top number by the bottom number.

Write the division. The fraction bar means divide.
3 ÷ 8
Do the division. Use short division or an equivalent fraction with a denominator of 10, 100 or 1000.
3 ÷ 8 = 0.375
Write the answer.
0.375

Check. 0.375 × 8 = 3.

Watch out. Put the decimal point in the right place. Do not stop after the first digit.

Worked example 3.3.2

Write 0.35 as a fraction in its simplest form.

Method. Write the decimal as a fraction over 10, 100 or 1000, then simplify.

Count the decimal places. There are 2 decimal places, so the denominator is 100.
0.35 has 2 decimal places
Write it as a fraction. The digits go on the top, and the denominator is 10, 100 or 1000.
35/100
Simplify the fraction. Divide the top and bottom by the highest common factor.
35/100 = 7/20
Write the answer.
7/20

Check. 7/20 = 0.35 as a decimal, so the answer is right.

Worked example 3.3.3

Write 3/5 as a percentage.

Method. Change the fraction to a decimal, then multiply by 100 to get a percentage.

Change to a decimal. Divide the top by the bottom.
3 ÷ 5 = 0.6
Multiply by 100. A percentage is out of 100, so multiply the decimal by 100.
0.6 × 100 = 60
Write the answer.
60%

Check. 60% of 5 is 3, which is the top number 3.

Watch out

Watch out: 3/4 is 75%, not 34%. Make sure the top of the fraction is the number of parts, not the bottom.

Exam tip

Exam tip. When you put fractions, decimals and percentages in order, change them all to decimals first.

3.4

Finding a percentage of an amount

Year 8
Find a percentage of an amount without a calculator

Video lesson: Finding a percentage of an amountComing soon

A school has 360 pupils. The head teacher says that 45% of them walk to school. She wants to know how many children that is, and she works it out without a calculator.

Percent means out of 100. So 10% is 10 out of 100, which is one tenth. To find a percentage of an amount:

find 10% by dividing the amount by 10;
find 5% by halving 10%;
multiply or add to build up the percentage you want.

Other easy ones: 50% is half, 25% is a quarter, and 1% is one hundredth.

Key ideas

10% of an amount is the amount divided by 10.
5% is half of 10%.
50% is half, 25% is a quarter, and 75% is three quarters.
Build any percentage from 10% and 5% (for example 35% = 3 × 10% + 5%).

Worked example 3.4.1

Work out 35% of 60.

Method. Find 10% first, then build up to 35%.

Find 10% of the amount. To find 10%, divide by 10.
10% of 60 = 60 ÷ 10 = 6
Multiply 10% by 3. 35% is 3 times 10%, plus 5% if the percentage ends in 5.
3 × 6 = 18
Find 5% and add it. 5% is half of 10%.
5% of 60 = 3, and 18 + 3 = 21
Write the answer. So 35% of 60 is 21.
21

Check. Check another way: 35% = 35 ÷ 100, and 35 ÷ 100 × 60 = 21.

Watch out. Find 10% first. Do not divide by the percentage: 35% is not 35 divided by x.

Worked example 3.4.2

Work out 45% of 360.

Method. Find 10% first, then build up to 45%.

Find 10% of the amount. To find 10%, divide by 10.
10% of 360 = 360 ÷ 10 = 36
Multiply 10% by 4. 45% is 4 times 10%, plus 5% if the percentage ends in 5.
4 × 36 = 144
Find 5% and add it. 5% is half of 10%.
5% of 360 = 18, and 144 + 18 = 162
Write the answer. So 45% of 360 is 162.
162

Check. Check another way: 45% = 45 ÷ 100, and 45 ÷ 100 × 360 = 162.

Watch out. Find 10% first. Do not divide by the percentage: 35% is not 35 divided by x.

Worked example 3.4.3

Work out 15% of 240.

Method. Find 10% first, then build up to 15%.

Find 10% of the amount. To find 10%, divide by 10.
10% of 240 = 240 ÷ 10 = 24
Multiply 10% by 1. 15% is 1 times 10%, plus 5% if the percentage ends in 5.
1 × 24 = 24
Find 5% and add it. 5% is half of 10%.
5% of 240 = 12, and 24 + 12 = 36
Write the answer. So 15% of 240 is 36.
36

Check. Check another way: 15% = 15 ÷ 100, and 15 ÷ 100 × 240 = 36.

Watch out. Find 10% first. Do not divide by the percentage: 35% is not 35 divided by x.

Watch out

Watch out: finding 10% by multiplying by 10 is wrong. 10% of an amount is the amount divided by 10.

Exam tip

Exam tip. Write down the 10% value first. Then build the percentage you need. This is quicker than a calculator and shows your working.

3.5

Percentage increase, decrease and multipliers

Year 8
Use percentage increase and decrease, and multipliers, Calculate a percentage change

Video lesson: Percentage increase, decrease and multipliersComing soon

A bike costs £200. The shop puts its prices up by 10% in January, and then has a sale in July with 25% off. Tom wants to know the price at each stage.

To increase an amount by a percentage, find the percentage and add it. To decrease it, find the percentage and subtract it.

A multiplier does the whole change in one step. For an increase of 15%, multiply by 1.15 (100% + 15% = 115%). For a decrease of 20%, multiply by 0.8 (100% − 20% = 80%).

To find a percentage change, use: (change ÷ original amount) × 100. Divide by the original amount.

Key ideas

Increase: original + the percentage of the original.
Decrease: original − the percentage of the original.
Multiplier for an increase of p%: 1 + p/100. For a decrease: 1 − p/100.
Percentage change = (change ÷ original) × 100. Always divide by the original.

Worked example 3.5.1

Increase 240 by 15%.

Method. Work out 15% of 240, then increase the amount by it.

Find 10% of the amount. Divide by 10.
10% of 240 = 24
Find 15%. 15% is 1 times 10%, plus 5%.
36 = 15% of 240
Increase by that amount. The new amount is the old amount + the increase.
240 + 36 = 276
Write the answer.
276

Check. Multiplier check: 23/20 × 240 = 276.

Watch out. Use the original amount for the percentage. The increase is measured from the original amount, not from 100%.

Worked example 3.5.2

Find the multiplier for a 30% increase.

Method. A multiplier is one number that does the change in one step.

Write the percentage as a decimal. 30% = 30 ÷ 100.
30% = 0.3
Add or subtract from 1. For a increase, the original is 100%, which is 1 as a multiplier.
1 + 0.3 = 1.3
Write the multiplier. Multiply the original amount by this to get the new amount.
1.3

Check. If the original is 100, the new amount is 100 × 1.3 = 130.

Worked example 3.5.3

A shop price goes from 40 to 50. Find the percentage change.

Method. Percentage change = (change ÷ original) × 100.

Find the change. The change is the new amount minus the original amount.
50 − 40 = 10
Divide by the original. Always divide by the original amount, not the new amount.
10 ÷ 40 = 1/4
Multiply by 100. Change the fraction to a percentage.
1/4 × 100 = 25
Write the answer. The amount has gone up or down by this percentage.
25% increase

Check. 25% of 40 is 10, and 40 + that is 50.

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

Watch out

Watch out: a 20% increase followed by a 20% decrease does not bring you back to the start. 100 × 1.2 = 120, and 120 × 0.8 = 96.

Exam tip

Exam tip. A percentage change is always from the original. Write the original first, then the new amount.

Chapter summary

Skill	Method
Add or subtract fractions	Find the lowest common denominator, make both fractions match, then add or subtract the tops
Multiply fractions	Multiply the tops and the bottoms, then simplify
Divide fractions	Keep the first, change to multiply, flip the second, then multiply
Fraction, decimal, percentage	Fraction to decimal: divide. Decimal to fraction: over 10, 100 or 1000. Percentage: over 100
Percentage of an amount	10% is divided by 10; 5% is half of 10%; build the percentage from these
Increase and decrease	Add or subtract the percentage of the original; or use a multiplier 1 ± p/100
Percentage change	(change ÷ original) × 100

Can you…?

I can add and subtract fractions with different denominators, including mixed numbers
I can multiply and divide fractions, and use the reciprocal
I can change between fractions, decimals and percentages
I can find 10%, 5% and other percentages of an amount without a calculator
I can increase or decrease an amount by a percentage, and use a multiplier
I can calculate a percentage change, dividing by the original amount

Key terms

numerator: the top number of a fraction
denominator: the bottom number of a fraction
mixed number: a whole number and a fraction, such as 2 1/4
improper fraction: a fraction whose top number is bigger than its bottom number, such as 9/4
reciprocal: the fraction turned upside down; the reciprocal of 3/4 is 4/3
multiplier: a number that multiplies an amount to change it by a percentage, such as 1.15
percentage change: the change as a percentage of the original amount

Total 28 marks. Allow about 28 minutes.

← Chapter 2: Surds, rounding and bounds Chapter 4: Ratio and proportion →

Fractions 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?

Adding and subtracting fractions

Key ideas

Watch out

Exam tip

Exercise 3A Adding and subtracting fractions

Multiplying and dividing fractions

Key ideas

Watch out

Exam tip

Exercise 3B Multiplying and dividing fractions

Fractions, decimals and percentages

Key ideas

Watch out

Exam tip

Exercise 3C Fractions, decimals and percentages

Finding a percentage of an amount

Key ideas

Watch out

Exam tip

Exercise 3D Percentages of amounts without a calculator

Percentage increase, decrease and multipliers

Key ideas

Watch out

Exam tip

Exercise 3E Percentage increase, decrease and change

Chapter summary

Can you…?

Key terms

Exercise 3Z Check-up

Exercise 3X Exam-style questions