GCSE Maths: Fractions of amounts (Foundation)
Part of the GCSE Maths fractions chapter (Foundation). Chapter overview · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Fractions of amounts
To find a fraction of an amount, divide by the denominator, then multiply by the numerator. A bar model makes this clear.
Key ideas
- Fraction of an amount: divide by the denominator, multiply by the numerator.
- One quantity as a fraction of another: write the first over the second, then simplify (same units).
- Finding the whole: if you know a fraction of the whole, divide by the numerator to find one part, then multiply by the denominator.
Method. To find a fraction of an amount: divide by the denominator to find one part, then multiply by the numerator.
- Say what 35 of 40 means. The denominator 5 says the amount is split into 5 equal parts. The numerator 3 says we take 3 of those parts. The word "of" tells us we are finding a share of an amount.Split 40 into 5 equal parts. Take 3 parts.
- Divide 40 by the denominator 5. Sharing into 5 equal parts finds the size of one part (this is 15 of the amount).40 ÷ 5 = 8
- Multiply by the numerator 3. The numerator 3 tells us how many parts we take. Here 8 × 3 gives the size of 3 parts.8 × 3 = 24
- Write the answer. Give the answer with its units.35 of 40 = 24
Check. A different route: multiply first, then divide. 40 × 3 = 120, and 120 ÷ 5 = 24. The answers match. ✓ Also the other 2 parts make 16, and 24 + 16 = 40.
Watch out. Divide by the denominator first, then multiply by the numerator. Do not multiply by the denominator.
Method. A part as a fraction of the whole: write the part over the whole, then simplify.
- Make sure both amounts are in the same units. A fraction compares a part with a whole, so both numbers must be measured in the same units.Part = 15, whole = 40
- Write the part over the whole. The part goes on top (numerator) and the whole goes on the bottom (denominator).1540
- List the factors of the top and the bottom. We want the highest common factor (HCF) so we can simplify.Factors of 15: 1, 3, 5, 15
Factors of 40: 1, 2, 4, 5, 8, 10, 20, 40
HCF = 5 - Divide the top and the bottom by 5. Dividing both by the same number keeps the value the same but gives smaller numbers.15 ÷ 5 = 3 and 40 ÷ 5 = 8
1540 = 38
Check. 15 ÷ 40 = 0.375 and 38 = 0.375. They match. ✓
Watch out. Put the part on top and the whole on the bottom, and always use the same units for both.
Method. If you know a fraction of the whole, divide by the numerator to find one part, then multiply by the denominator to find the whole.
- Write down what we know. The fraction 38 means the whole is split into 8 equal parts, and 3 of those parts together make 27.3 parts = 27
- Find one part: divide by 3. If 3 parts make 27, then one part is 27 shared into 3.27 ÷ 3 = 9
- Find the whole: multiply by 8. The whole is all 8 parts, so multiply the size of one part by 8.9 × 8 = 72
- Write the answer. This is the whole amount (the number the fraction was taken from).The whole is 72.
Check. Work forwards: 38 of 72 = 72 ÷ 8 × 3 = 9 × 3 = 27. ✓
Watch out. Go up to the whole by multiplying by the denominator. A common mistake is to find the fraction of the part instead.
Method. Work out what fraction of his original money is left at the end, then work backwards from the £12.
- Find the fraction left after the game. He spent one third, so two thirds stay in his pocket.1 − 13 = 23
- Find the fraction left after the food. He spent half of what was left. Half of 23 is 23 ÷ 2, so the same amount remains.12 × 23 = 26 = 13
- Link this to the £12. He has 13 of his starting money left, and this is £12.13 of the start = £12, so 1 part = £12
- Find the whole by multiplying by the denominator. The whole is 3 thirds, so multiply one third by 3.£12 × 3 = £36
- Write the answer. Give the starting amount.Ben started with £36.
Check. Work forwards: £36. Game: £36 ÷ 3 = £12, so £24 is left. Food: half of £24 = £12, so £12 is left. This matches. ✓
Watch out. “Half of what is left” is half of the remaining money, not half of the original.
Method. Do the steps one at a time. The second fraction is a fraction of the money that is left after the first purchase.
- Find the cost of the coat. 27 of £84: divide by the denominator 7 to get one part, then multiply by the numerator 2.£84 ÷ 7 = £12
£12 × 2 = £24 - Find the money left after the coat. Take the cost away from the starting amount.£84 − £24 = £60
- Find the cost of the bag. It is 12 of the money that is left, which is £60 (not £84).£60 ÷ 2 = £30
- Find the money left at the end. Take the cost of the bag away from what she had before buying it.£60 − £30 = £30
- Write the answer. Answer the question that was asked.Zara has £30 left.
Check. A different route: after the coat she has 57 of her money, and half of that is 514. 514 of £84 = 84 ÷ 14 × 5 = 6 × 5 = £30. ✓
Watch out. “Of what is left” means use the new, smaller amount.
Method. To find a fraction of an amount: divide by the denominator to find one part, then multiply by the numerator.
- Change the hours into minutes. To find a fraction of an amount it helps to use a unit that divides easily. There are 60 minutes in 1 hour.3 × 60 = 180 minutes
- Say what 25 of 180 minutes means. The denominator 5 says the amount is split into 5 equal parts. The numerator 2 says we take 2 of those parts. The word "of" tells us we are finding a share of an amount.Split 180 minutes into 5 equal parts. Take 2 parts.
- Divide 180 minutes by the denominator 5. Sharing into 5 equal parts finds the size of one part (this is 15 of the amount).180 minutes ÷ 5 = 36 minutes
- Multiply by the numerator 2. The numerator 2 tells us how many parts we take. Here 36 minutes × 2 gives the size of 2 parts.36 minutes × 2 = 72 minutes
- Write the answer. Give the answer with its units.25 of 180 minutes = 72 minutes
- Change to hours and minutes (to see if it is sensible). 72 minutes is more than 60, so it is 1 hour and some minutes.72 minutes = 1 hour 12 minutes
Check. A different route: multiply first, then divide. 180 × 2 = 360, and 360 ÷ 5 = 72. The answers match. ✓ Also the other 3 parts make 108 minutes, and 72 + 108 = 180.
Watch out. Change to a smaller unit first. Do not try to work with “two fifths of 3” when the answer is needed in minutes.
Exercise 3H Fractions of amounts
Non-calculatorMethod. To find a fraction of an amount: divide by the denominator to find one part, then multiply by the numerator.
- Say what 12 of 18 means. The denominator 2 says the amount is split into 2 equal parts. The numerator 1 says we take 1 of those parts. The word "of" tells us we are finding a share of an amount.Split 18 into 2 equal parts. Take 1 part.
- Divide 18 by the denominator 2. Sharing into 2 equal parts finds the size of one part (this is 12 of the amount).18 ÷ 2 = 9
- Take the number of parts the numerator asks for. The numerator is 1, so we take just one part. One part is already our answer.12 of 18 = 9
- Write the answer. Give the answer with its units.12 of 18 = 9
Check. A different route: multiply first, then divide. 18 × 1 = 18, and 18 ÷ 2 = 9. The answers match. ✓ Also the other 1 part make 9, and 9 + 9 = 18.
Watch out. Divide by the denominator first, then multiply by the numerator. Do not multiply by the denominator.
Method. To find a fraction of an amount: divide by the denominator to find one part, then multiply by the numerator.
- Say what 15 of 35 means. The denominator 5 says the amount is split into 5 equal parts. The numerator 1 says we take 1 of those parts. The word "of" tells us we are finding a share of an amount.Split 35 into 5 equal parts. Take 1 part.
- Divide 35 by the denominator 5. Sharing into 5 equal parts finds the size of one part (this is 15 of the amount).35 ÷ 5 = 7
- Take the number of parts the numerator asks for. The numerator is 1, so we take just one part. One part is already our answer.15 of 35 = 7
- Write the answer. Give the answer with its units.15 of 35 = 7
Check. A different route: multiply first, then divide. 35 × 1 = 35, and 35 ÷ 5 = 7. The answers match. ✓ Also the other 4 parts make 28, and 7 + 28 = 35.
Watch out. Divide by the denominator first, then multiply by the numerator. Do not multiply by the denominator.
Method. To find a fraction of an amount: divide by the denominator to find one part, then multiply by the numerator.
- Say what 18 of 64 means. The denominator 8 says the amount is split into 8 equal parts. The numerator 1 says we take 1 of those parts. The word "of" tells us we are finding a share of an amount.Split 64 into 8 equal parts. Take 1 part.
- Divide 64 by the denominator 8. Sharing into 8 equal parts finds the size of one part (this is 18 of the amount).64 ÷ 8 = 8
- Take the number of parts the numerator asks for. The numerator is 1, so we take just one part. One part is already our answer.18 of 64 = 8
- Write the answer. Give the answer with its units.18 of 64 = 8
Check. A different route: multiply first, then divide. 64 × 1 = 64, and 64 ÷ 8 = 8. The answers match. ✓ Also the other 7 parts make 56, and 8 + 56 = 64.
Watch out. Divide by the denominator first, then multiply by the numerator. Do not multiply by the denominator.
Method. To find a fraction of an amount: divide by the denominator to find one part, then multiply by the numerator.
- Say what 110 of 250 means. The denominator 10 says the amount is split into 10 equal parts. The numerator 1 says we take 1 of those parts. The word "of" tells us we are finding a share of an amount.Split 250 into 10 equal parts. Take 1 part.
- Divide 250 by the denominator 10. Sharing into 10 equal parts finds the size of one part (this is 110 of the amount).250 ÷ 10 = 25
- Take the number of parts the numerator asks for. The numerator is 1, so we take just one part. One part is already our answer.110 of 250 = 25
- Write the answer. Give the answer with its units.110 of 250 = 25
Check. A different route: multiply first, then divide. 250 × 1 = 250, and 250 ÷ 10 = 25. The answers match. ✓ Also the other 9 parts make 225, and 25 + 225 = 250.
Watch out. Divide by the denominator first, then multiply by the numerator. Do not multiply by the denominator.
Method. To find a fraction of an amount: divide by the denominator to find one part, then multiply by the numerator.
- Say what 34 of 20 means. The denominator 4 says the amount is split into 4 equal parts. The numerator 3 says we take 3 of those parts. The word "of" tells us we are finding a share of an amount.Split 20 into 4 equal parts. Take 3 parts.
- Divide 20 by the denominator 4. Sharing into 4 equal parts finds the size of one part (this is 14 of the amount).20 ÷ 4 = 5
- Multiply by the numerator 3. The numerator 3 tells us how many parts we take. Here 5 × 3 gives the size of 3 parts.5 × 3 = 15
- Write the answer. Give the answer with its units.34 of 20 = 15
Check. A different route: multiply first, then divide. 20 × 3 = 60, and 60 ÷ 4 = 15. The answers match. ✓ Also the other 1 part make 5, and 15 + 5 = 20.
Watch out. Divide by the denominator first, then multiply by the numerator. Do not multiply by the denominator.
Method. To find a fraction of an amount: divide by the denominator to find one part, then multiply by the numerator.
- Say what 25 of 45 means. The denominator 5 says the amount is split into 5 equal parts. The numerator 2 says we take 2 of those parts. The word "of" tells us we are finding a share of an amount.Split 45 into 5 equal parts. Take 2 parts.
- Divide 45 by the denominator 5. Sharing into 5 equal parts finds the size of one part (this is 15 of the amount).45 ÷ 5 = 9
- Multiply by the numerator 2. The numerator 2 tells us how many parts we take. Here 9 × 2 gives the size of 2 parts.9 × 2 = 18
- Write the answer. Give the answer with its units.25 of 45 = 18
Check. A different route: multiply first, then divide. 45 × 2 = 90, and 90 ÷ 5 = 18. The answers match. ✓ Also the other 3 parts make 27, and 18 + 27 = 45.
Watch out. Divide by the denominator first, then multiply by the numerator. Do not multiply by the denominator.
Method. To find a fraction of an amount: divide by the denominator to find one part, then multiply by the numerator.
- Say what 56 of 42 means. The denominator 6 says the amount is split into 6 equal parts. The numerator 5 says we take 5 of those parts. The word "of" tells us we are finding a share of an amount.Split 42 into 6 equal parts. Take 5 parts.
- Divide 42 by the denominator 6. Sharing into 6 equal parts finds the size of one part (this is 16 of the amount).42 ÷ 6 = 7
- Multiply by the numerator 5. The numerator 5 tells us how many parts we take. Here 7 × 5 gives the size of 5 parts.7 × 5 = 35
- Write the answer. Give the answer with its units.56 of 42 = 35
Check. A different route: multiply first, then divide. 42 × 5 = 210, and 210 ÷ 6 = 35. The answers match. ✓ Also the other 1 part make 7, and 35 + 7 = 42.
Watch out. Divide by the denominator first, then multiply by the numerator. Do not multiply by the denominator.
Method. To find a fraction of an amount: divide by the denominator to find one part, then multiply by the numerator.
- Say what 710 of 120 means. The denominator 10 says the amount is split into 10 equal parts. The numerator 7 says we take 7 of those parts. The word "of" tells us we are finding a share of an amount.Split 120 into 10 equal parts. Take 7 parts.
- Divide 120 by the denominator 10. Sharing into 10 equal parts finds the size of one part (this is 110 of the amount).120 ÷ 10 = 12
- Multiply by the numerator 7. The numerator 7 tells us how many parts we take. Here 12 × 7 gives the size of 7 parts.12 × 7 = 84
- Write the answer. Give the answer with its units.710 of 120 = 84
Check. A different route: multiply first, then divide. 120 × 7 = 840, and 840 ÷ 10 = 84. The answers match. ✓ Also the other 3 parts make 36, and 84 + 36 = 120.
Watch out. Divide by the denominator first, then multiply by the numerator. Do not multiply by the denominator.
Method. To find a fraction of an amount: divide by the denominator to find one part, then multiply by the numerator.
- Say what 38 of 96 means. The denominator 8 says the amount is split into 8 equal parts. The numerator 3 says we take 3 of those parts. The word "of" tells us we are finding a share of an amount.Split 96 into 8 equal parts. Take 3 parts.
- Divide 96 by the denominator 8. Sharing into 8 equal parts finds the size of one part (this is 18 of the amount).96 ÷ 8 = 12
- Multiply by the numerator 3. The numerator 3 tells us how many parts we take. Here 12 × 3 gives the size of 3 parts.12 × 3 = 36
- Write the answer. Give the answer with its units.38 of 96 = 36
Check. A different route: multiply first, then divide. 96 × 3 = 288, and 288 ÷ 8 = 36. The answers match. ✓ Also the other 5 parts make 60, and 36 + 60 = 96.
Watch out. Divide by the denominator first, then multiply by the numerator. Do not multiply by the denominator.
Method. To find a fraction of an amount: divide by the denominator to find one part, then multiply by the numerator.
- Say what 49 of 72 means. The denominator 9 says the amount is split into 9 equal parts. The numerator 4 says we take 4 of those parts. The word "of" tells us we are finding a share of an amount.Split 72 into 9 equal parts. Take 4 parts.
- Divide 72 by the denominator 9. Sharing into 9 equal parts finds the size of one part (this is 19 of the amount).72 ÷ 9 = 8
- Multiply by the numerator 4. The numerator 4 tells us how many parts we take. Here 8 × 4 gives the size of 4 parts.8 × 4 = 32
- Write the answer. Give the answer with its units.49 of 72 = 32
Check. A different route: multiply first, then divide. 72 × 4 = 288, and 288 ÷ 9 = 32. The answers match. ✓ Also the other 5 parts make 40, and 32 + 40 = 72.
Watch out. Divide by the denominator first, then multiply by the numerator. Do not multiply by the denominator.
Method. To find a fraction of an amount: divide by the denominator to find one part, then multiply by the numerator.
- Say what 23 of 45 means. The denominator 3 says the amount is split into 3 equal parts. The numerator 2 says we take 2 of those parts. The word "of" tells us we are finding a share of an amount.Split 45 into 3 equal parts. Take 2 parts.
- Divide 45 by the denominator 3. Sharing into 3 equal parts finds the size of one part (this is 13 of the amount).45 ÷ 3 = 15
- Multiply by the numerator 2. The numerator 2 tells us how many parts we take. Here 15 × 2 gives the size of 2 parts.15 × 2 = 30
- Write the answer. Give the answer with its units.23 of 45 = 30
- Add the units. The amount is money, so write the pound sign.£30
Check. A different route: multiply first, then divide. 45 × 2 = 90, and 90 ÷ 3 = 30. The answers match. ✓ Also the other 1 part make 15, and 30 + 15 = 45.
Watch out. Divide by the denominator first, then multiply by the numerator. Do not multiply by the denominator.
Method. To find a fraction of an amount: divide by the denominator to find one part, then multiply by the numerator.
- Change the kilograms into grams. Fractions of an amount are easier with a smaller unit that divides by 5. There are 1000 g in 1 kg.2 kg = 2 × 1000 = 2000 g
- Say what 35 of 2000 g means. The denominator 5 says the amount is split into 5 equal parts. The numerator 3 says we take 3 of those parts. The word "of" tells us we are finding a share of an amount.Split 2000 g into 5 equal parts. Take 3 parts.
- Divide 2000 g by the denominator 5. Sharing into 5 equal parts finds the size of one part (this is 15 of the amount).2000 g ÷ 5 = 400 g
- Multiply by the numerator 3. The numerator 3 tells us how many parts we take. Here 400 g × 3 gives the size of 3 parts.400 g × 3 = 1200 g
- Write the answer. Give the answer with its units.35 of 2000 g = 1200 g
- Change back to kilograms (if you like). The question does not say which unit to use, so both answers are correct.1200 g = 1.2 kg
Check. A different route: multiply first, then divide. 2000 × 3 = 6000, and 6000 ÷ 5 = 1200. The answers match. ✓ Also the other 2 parts make 800 g, and 1200 + 800 = 2000.
Watch out. Divide by the denominator first, then multiply by the numerator. Do not multiply by the denominator.
Method. To find a fraction of an amount: divide by the denominator to find one part, then multiply by the numerator.
- Say what 58 of 4 m means. The denominator 8 says the amount is split into 8 equal parts. The numerator 5 says we take 5 of those parts. The word "of" tells us we are finding a share of an amount.Split 4 m into 8 equal parts. Take 5 parts.
- Divide 4 m by the denominator 8. Sharing into 8 equal parts finds the size of one part (this is 18 of the amount).4 m ÷ 8 = 0.5 m
- Multiply by the numerator 5. The numerator 5 tells us how many parts we take. Here 0.5 m × 5 gives the size of 5 parts.0.5 m × 5 = 2.5 m
- Write the answer. Give the answer with its units.58 of 4 m = 2.5 m
Check. A different route: multiply first, then divide. 4 × 5 = 20, and 20 ÷ 8 = 2.5. The answers match. ✓ Also the other 3 parts make 1.5 m, and 2.5 + 1.5 = 4.
Watch out. Divide by the denominator first, then multiply by the numerator. Do not multiply by the denominator.
Method. To find a fraction of an amount: divide by the denominator to find one part, then multiply by the numerator.
- Change the hour into minutes. There are 60 minutes in 1 hour. 60 divides by 12 exactly, which makes the working easy.1 hour = 60 minutes
- Say what 712 of 60 minutes means. The denominator 12 says the amount is split into 12 equal parts. The numerator 7 says we take 7 of those parts. The word "of" tells us we are finding a share of an amount.Split 60 minutes into 12 equal parts. Take 7 parts.
- Divide 60 minutes by the denominator 12. Sharing into 12 equal parts finds the size of one part (this is 112 of the amount).60 minutes ÷ 12 = 5 minutes
- Multiply by the numerator 7. The numerator 7 tells us how many parts we take. Here 5 minutes × 7 gives the size of 7 parts.5 minutes × 7 = 35 minutes
- Write the answer. Give the answer with its units.712 of 60 minutes = 35 minutes
Check. A different route: multiply first, then divide. 60 × 7 = 420, and 420 ÷ 12 = 35. The answers match. ✓ Also the other 5 parts make 25 minutes, and 35 + 25 = 60.
Watch out. Divide by the denominator first, then multiply by the numerator. Do not multiply by the denominator.
Method. To find a fraction of an amount: divide by the denominator to find one part, then multiply by the numerator.
- Say what 34 of 2 litres means. The denominator 4 says the amount is split into 4 equal parts. The numerator 3 says we take 3 of those parts. The word "of" tells us we are finding a share of an amount.Split 2 litres into 4 equal parts. Take 3 parts.
- Divide 2 litres by the denominator 4. Sharing into 4 equal parts finds the size of one part (this is 14 of the amount).2 litres ÷ 4 = 0.5 litres
- Multiply by the numerator 3. The numerator 3 tells us how many parts we take. Here 0.5 litres × 3 gives the size of 3 parts.0.5 litres × 3 = 1.5 litres
- Write the answer. Give the answer with its units.34 of 2 litres = 1.5 litres
Check. A different route: multiply first, then divide. 2 × 3 = 6, and 6 ÷ 4 = 1.5. The answers match. ✓ Also the other 1 part make 0.5 litres, and 1.5 + 0.5 = 2.
Watch out. Divide by the denominator first, then multiply by the numerator. Do not multiply by the denominator.
Method. To find a fraction of an amount: divide by the denominator to find one part, then multiply by the numerator.
- Change the pounds into pence. It is easier to divide 350 by 5 than to divide £3.50. There are 100p in £1.£3.50 = 350p
- Say what 25 of 350p means. The denominator 5 says the amount is split into 5 equal parts. The numerator 2 says we take 2 of those parts. The word "of" tells us we are finding a share of an amount.Split 350p into 5 equal parts. Take 2 parts.
- Divide 350p by the denominator 5. Sharing into 5 equal parts finds the size of one part (this is 15 of the amount).350p ÷ 5 = 70p
- Multiply by the numerator 2. The numerator 2 tells us how many parts we take. Here 70p × 2 gives the size of 2 parts.70p × 2 = 140p
- Write the answer. Give the answer with its units.25 of 350p = 140p
- Change back to pounds. 100p = £1, so move the decimal point two places.140p = £1.40
Check. A different route: multiply first, then divide. 350 × 2 = 700, and 700 ÷ 5 = 140. The answers match. ✓ Also the other 3 parts make 210p, and 140 + 210 = 350.
Watch out. Divide by the denominator first, then multiply by the numerator. Do not multiply by the denominator.
Method. If you know a fraction of the whole, divide by the numerator to find one part, then multiply by the denominator to find the whole.
- Write down what we know. The fraction 14 means the whole is split into 4 equal parts, and 1 of those parts together make 12.1 part = 12
- Find one part: divide by 1. If 1 parts make 12, then one part is 12 shared into 1.12 ÷ 1 = 12
- Find the whole: multiply by 4. The whole is all 4 parts, so multiply the size of one part by 4.12 × 4 = 48
- Write the answer. This is the whole amount (the number the fraction was taken from).The whole is 48.
Check. Work forwards: 14 of 48 = 48 ÷ 4 × 1 = 12 × 1 = 12. ✓
Watch out. Go up to the whole by multiplying by the denominator. A common mistake is to find the fraction of the part instead.
Method. If you know a fraction of the whole, divide by the numerator to find one part, then multiply by the denominator to find the whole.
- Write down what we know. The fraction 25 means the whole is split into 5 equal parts, and 2 of those parts together make 18.2 parts = 18
- Find one part: divide by 2. If 2 parts make 18, then one part is 18 shared into 2.18 ÷ 2 = 9
- Find the whole: multiply by 5. The whole is all 5 parts, so multiply the size of one part by 5.9 × 5 = 45
- Write the answer. This is the whole amount (the number the fraction was taken from).The whole is 45.
Check. Work forwards: 25 of 45 = 45 ÷ 5 × 2 = 9 × 2 = 18. ✓
Watch out. Go up to the whole by multiplying by the denominator. A common mistake is to find the fraction of the part instead.
Method. If you know a fraction of the whole, divide by the numerator to find one part, then multiply by the denominator to find the whole.
- Write down what we know. The fraction 37 means the whole is split into 7 equal parts, and 3 of those parts together make 21.3 parts = 21
- Find one part: divide by 3. If 3 parts make 21, then one part is 21 shared into 3.21 ÷ 3 = 7
- Find the whole: multiply by 7. The whole is all 7 parts, so multiply the size of one part by 7.7 × 7 = 49
- Write the answer. This is the whole amount (the number the fraction was taken from).The whole is 49.
Check. Work forwards: 37 of 49 = 49 ÷ 7 × 3 = 7 × 3 = 21. ✓
Watch out. Go up to the whole by multiplying by the denominator. A common mistake is to find the fraction of the part instead.
Method. If you know a fraction of the whole, divide by the numerator to find one part, then multiply by the denominator to find the whole.
- Write down what we know. The fraction 56 means the whole is split into 6 equal parts, and 5 of those parts together make 35.5 parts = 35
- Find one part: divide by 5. If 5 parts make 35, then one part is 35 shared into 5.35 ÷ 5 = 7
- Find the whole: multiply by 6. The whole is all 6 parts, so multiply the size of one part by 6.7 × 6 = 42
- Write the answer. This is the whole amount (the number the fraction was taken from).The whole is 42.
Check. Work forwards: 56 of 42 = 42 ÷ 6 × 5 = 7 × 5 = 35. ✓
Watch out. Go up to the whole by multiplying by the denominator. A common mistake is to find the fraction of the part instead.
Method. A part as a fraction of the whole: write the part over the whole, then simplify.
- Make sure both amounts are in the same units. Both amounts are numbers of items, so the units already match.Part = 15, whole = 40
- Write the part over the whole. The part goes on top (numerator) and the whole goes on the bottom (denominator).1540
- List the factors of the top and the bottom. We want the highest common factor (HCF) so we can simplify.Factors of 15: 1, 3, 5, 15
Factors of 40: 1, 2, 4, 5, 8, 10, 20, 40
HCF = 5 - Divide the top and the bottom by 5. Dividing both by the same number keeps the value the same but gives smaller numbers.15 ÷ 5 = 3 and 40 ÷ 5 = 8
1540 = 38
Check. 15 ÷ 40 = 0.375 and 38 = 0.375. They match. ✓
Watch out. Put the part on top and the whole on the bottom, and always use the same units for both.
Method. A part as a fraction of the whole: write the part over the whole, then simplify.
- Make sure both amounts are in the same units. Both amounts are numbers of items, so the units already match.Part = 12, whole = 18
- Write the part over the whole. The part goes on top (numerator) and the whole goes on the bottom (denominator).1218
- List the factors of the top and the bottom. We want the highest common factor (HCF) so we can simplify.Factors of 12: 1, 2, 3, 4, 6, 12
Factors of 18: 1, 2, 3, 6, 9, 18
HCF = 6 - Divide the top and the bottom by 6. Dividing both by the same number keeps the value the same but gives smaller numbers.12 ÷ 6 = 2 and 18 ÷ 6 = 3
1218 = 23
Check. 12 ÷ 18 = 0.6667 and 23 = 0.6667. They match. ✓
Watch out. Put the part on top and the whole on the bottom, and always use the same units for both.
Method. A part as a fraction of the whole: write the part over the whole, then simplify.
- Make sure both amounts are in the same units. Both amounts are in pounds, so the units match.Part = 4, whole = 20
- Write the part over the whole. The part goes on top (numerator) and the whole goes on the bottom (denominator).420
- List the factors of the top and the bottom. We want the highest common factor (HCF) so we can simplify.Factors of 4: 1, 2, 4
Factors of 20: 1, 2, 4, 5, 10, 20
HCF = 4 - Divide the top and the bottom by 4. Dividing both by the same number keeps the value the same but gives smaller numbers.4 ÷ 4 = 1 and 20 ÷ 4 = 5
420 = 15
Check. 4 ÷ 20 = 0.2 and 15 = 0.2. They match. ✓
Watch out. Put the part on top and the whole on the bottom, and always use the same units for both.
Method. A part as a fraction of the whole: write the part over the whole, then simplify.
- Make sure both amounts are in the same units. The amounts are in different units: 30 minutes and 2 hours. Change the hours into minutes: 2 × 60 = 120 minutes.Part = 30 min, whole = 120 min
- Write the part over the whole. The part goes on top (numerator) and the whole goes on the bottom (denominator).30120
- List the factors of the top and the bottom. We want the highest common factor (HCF) so we can simplify.Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30
Factors of 120: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120
HCF = 30 - Divide the top and the bottom by 30. Dividing both by the same number keeps the value the same but gives smaller numbers.30 ÷ 30 = 1 and 120 ÷ 30 = 4
30120 = 14
Check. 30 ÷ 120 = 0.25 and 14 = 0.25. They match. ✓
Watch out. Put the part on top and the whole on the bottom, and always use the same units for both.
Method. To find a fraction of an amount: divide by the denominator to find one part, then multiply by the numerator.
- Say what 38 of 840 means. The denominator 8 says the amount is split into 8 equal parts. The numerator 3 says we take 3 of those parts. The word "of" tells us we are finding a share of an amount.Split 840 into 8 equal parts. Take 3 parts.
- Divide 840 by the denominator 8. Sharing into 8 equal parts finds the size of one part (this is 18 of the amount).840 ÷ 8 = 105
- Multiply by the numerator 3. The numerator 3 tells us how many parts we take. Here 105 × 3 gives the size of 3 parts.105 × 3 = 315
- Write the answer. Give the answer with its units.38 of 840 = 315
- Answer in words. The amount we found is the number of pupils who walk.315 pupils walk to school.
Check. A different route: multiply first, then divide. 840 × 3 = 2520, and 2520 ÷ 8 = 315. The answers match. ✓ Also the other 5 parts make 525, and 315 + 525 = 840.
Watch out. Divide by the denominator first, then multiply by the numerator. Do not multiply by the denominator.
Method. The pupils who do not walk are the rest of the school. Either subtract the walkers from the total, or find the fraction who do not walk.
- Use the answer to part (a). 315 pupils walk and there are 840 pupils altogether.840 pupils − 315 walkers
- Subtract. The pupils who do not walk are what is left.840 − 315 = 525
- Write the answer. Say what the number means.525 pupils do not walk.
Check. A different route: the fraction who do not walk is 1 − 38 = 58. 840 ÷ 8 = 105, and 105 × 5 = 525. ✓
Watch out. Make sure you answer the question that is asked: this part asks for the pupils who do not walk.
Method. Find the number of red and blue counters, add them, then subtract from the total to find the green ones.
- Find the red counters. 13 of 60: divide by the denominator 3. The numerator is 1, so one part is the answer.60 ÷ 3 = 20 red
- Find the blue counters. 14 of 60: divide by 4.60 ÷ 4 = 15 blue
- Add the red and blue counters. These are the counters that are not green.20 + 15 = 35
- Subtract from the total. The rest are green.60 − 35 = 25 green
Check. A different route: the fraction that is red or blue is 13 + 14 = 412 + 312 = 712. So green is 512, and 60 ÷ 12 × 5 = 25. ✓
Watch out. The green counters are what is left over. Do not find a fraction of the wrong total.
Method. The reduction is a fraction of the price. Find the reduction and take it away from the original price.
- Find the reduction. The price is reduced by 14 of £80. Divide by the denominator 4.£80 ÷ 4 = £20
- Subtract the reduction from the original price. The sale price is the original price take away the reduction.£80 − £20 = £60
- Write the answer. Say what the number means.The sale price is £60.
Check. A different route: if 14 is taken off, 34 is left. 34 of £80 = 80 ÷ 4 × 3 = 20 × 3 = £60. ✓
Watch out. Do not stop at the reduction (£20). The question asks for the sale price.
Method. Do the two steps one at a time, and use the amount left after the first step.
- Find the cost of the book. 13 of £60: divide by 3.60 ÷ 3 = £20
- Find the money left after the book. Take the cost away from the starting amount.60 − 20 = £40
- Find the cost of lunch. It is 14 of what is left, which is £40 (not £60).40 ÷ 4 = £10
- Find the money left at the end. Take the cost of lunch away from what she had before lunch.40 − 10 = £30
Check. A different route: after the book she has 23 of her money. After lunch she has 34 of that: 34 × 23 = 12. Half of £60 = £30. ✓
Watch out. “Of what is left” means use the remaining amount, not the starting amount.
Method. Work out both amounts and compare.
- Work out 23 of 30. Divide by the denominator 3, then multiply by the numerator 2.30 ÷ 3 = 10
10 × 2 = 20 - Work out 34 of 24. Divide by the denominator 4, then multiply by the numerator 3.24 ÷ 4 = 6
6 × 3 = 18 - Compare the two answers. Both are numbers, so the bigger number is the bigger amount.20 > 18
- Write the conclusion. Answer the question that was asked.23 of 30 is bigger.
Check. A different route: 23 × 30 = 603 = 20 and 34 × 24 = 724 = 18. The answers match. ✓
Watch out. Do not compare the fractions 23 and 34 alone. The amounts they are taken from are different (30 and 24).
Method. Track the fraction of the money that is left after each step, then work backwards from the £12.
- Find the fraction left after the game. Ben spent 13, so 23 is left.1 − 13 = 23
- Find the fraction left after the food. He spent half of what was left, so half of 23 remains.12 × 23 = 26 = 13
- Use the £12. He has 13 of his money left, and this is £12.13 of the start = £12
- Find the whole. The whole is 3 thirds, so multiply by 3.3 × £12 = £36
Check. Work forwards from £36. Game: £36 ÷ 3 = £12 spent, so £24 left. Food: half of £24 = £12 spent, so £12 left. This matches. ✓
Watch out. “Half of the rest” means half of what is left after the game.
Method. Find the fraction left after both gifts, then work backwards from the 21 marbles.
- Add the fractions given away. Use a common denominator of 15 (the LCM of 5 and 3).15 + 13 = 315 + 515 = 815
- Find the fraction left. Take the fraction given away from 1 whole (15/15).1515 − 815 = 715
- Use the 21 marbles. 715 of the marbles is 21, so one fifteenth is 21 ÷ 7.1 part = 21 ÷ 7 = 3
- Find the whole. The whole is 15 parts.15 × 3 = 45 marbles
Check. Work forwards from 45. Bo gets 45 ÷ 5 = 9 and Cy gets 45 ÷ 3 = 15, so 45 − 9 − 15 = 21 are left. ✓
Watch out. Add the fractions given away first. Do not take the fractions from 21.