GCSE Maths: Ratio problems: one part known, differences and combining ratios (Foundation)
Part of the GCSE Maths ratio and proportion chapter (Foundation). Chapter overview · Higher version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Ratio problems: one part known, differences and combining ratios
In many problems you are not told the total. Instead you know one share, or the difference between two shares. The trick is always the same: find the value of one part, then use it.
One share known. Red and blue counters are in the ratio 3 : 5 and there are 12 red counters. Red is 3 parts, so 3 parts = 12 and one part = 12 ÷ 3 = 4. Then blue is 5 × 4 = 20. Divide by the ratio number of the share you were given, not by the total number of parts.
Difference known. Ali and Bea share money in the ratio 7 : 4. Ali gets £36 more than Bea. The difference in the number of parts is 7 − 4 = 3 parts, and this is worth £36. So one part = £36 ÷ 3 = £12.
Combining ratios. Suppose A : B = 2 : 3 and B : C = 4 : 5. B is in both ratios, but it is 3 in one and 4 in the other. To join the ratios, make the B numbers equal. The lowest common multiple of 3 and 4 is 12, so we multiply the first ratio by 4 and the second ratio by 3.
| A | B | C | ||
|---|---|---|---|---|
| A : B | 2 | 3 | × 4 | |
| becomes | 8 | 12 | ||
| B : C | 4 | 5 | × 3 | |
| becomes | 12 | 15 | ||
| A : B : C | 8 | 12 | 15 |
Key ideas
- One share known: divide that share by its ratio number to get the value of one part.
- Difference known: subtract the ratio numbers, divide the difference by the answer to get the value of one part.
- Then multiply the value of one part by any ratio number, or by the total number of parts.
- Combine ratios by making the number for the shared quantity the same in both ratios (use the LCM).
Key words. To share in a ratio means to split an amount into equal-sized parts, then give each person the number of parts shown in the ratio. The ratio 3 : 5 means 3 parts for the first person and 5 parts for the second.
Method. When one share is known, divide it by its number of parts to find one part. Then multiply to find any other share.
- Write down what we know. We are told the value of one of the shares. That tells us how many equal parts it is worth.Red has 3 parts, worth 12
- Find the value of one part. 3 parts are worth 12, so divide by 3.12 ÷ 3 = 4 for one part
- Multiply to find what the question asks for. Multiply the value of one part by the number of parts needed.Blue: 5 × 4 = 20
Check. Check the share we were given: 3 × 4 = 12, which matches. ✓
Watch out. The amount you are given belongs to one share, not to the whole. Do not divide by the total number of parts.
Key words. To share in a ratio means to split an amount into equal-sized parts, then give each person the number of parts shown in the ratio. The ratio 3 : 5 means 3 parts for the first person and 5 parts for the second.
Method. Subtract the ratio numbers to find the difference in parts. The known difference is worth that many parts, so divide to find one part.
- Find the difference in parts. The difference between two shares is the number of extra parts that the bigger share has.7 − 4 = 3 parts
- Find the value of one part. We are told the difference is £36. That is worth 3 parts, so divide by 3.£36 ÷ 3 = £12 for one part
- Multiply to find what the question asks for. Multiply the value of one part by the number of parts needed.Ali: 7 × £12 = £84
Bea: 4 × £12 = £48
Check. Check the difference: 7 × £12 − 4 × £12 = £84 − £48 = £36, which is the difference we were given. ✓
Watch out. Subtract the ratio numbers (e.g. 7 − 4 = 3 parts). Do not add them. The sum gives the total, the subtraction gives the difference.
Key words. A ratio compares the sizes of two or more quantities, written with a colon, for example 2 : 3 (read “2 to 3”). The numbers in a ratio are called parts. The simplest form of a ratio has no whole number (other than 1) that divides into every part.
Method. To combine ratios, make the shared quantity (B) the same number in both ratios, using the LCM, then join them.
- Spot the quantity that is in both ratios. B appears in both ratios, so it links them together. To combine, B must stand for the same number of parts in both.A : B = 2 : 3
B : C = 4 : 5 - Find the lowest common multiple (LCM) of the two B numbers. We make both B numbers equal to the LCM of 3 and 4.Multiples of 3: 3, 6, 9, 12
Multiples of 4: 4, 8, 12
LCM = 12 - Multiply each ratio so that B becomes 12. Multiplying every part of a ratio by the same number gives an equivalent ratio.A : B = 2 : 3 (multiply by 4) = 8 : 12
B : C = 4 : 5 (multiply by 3) = 12 : 15 - Write the combined ratio A : B : C. B is now the same in both, so the three numbers can be joined.A : B : C = 8 : 12 : 15
Check. Look back at the ratios: in 8 : 12 : 15, the first two numbers (8 and 12) simplify to 2 : 3, which equals 2 : 3. The last two (12 and 15) simplify to 4 : 5, which equals 4 : 5. ✓
Watch out. Match the linking quantity using the LCM. Do not just write the numbers side by side.
Key words. To share in a ratio means to split an amount into equal-sized parts, then give each person the number of parts shown in the ratio. The ratio 3 : 5 means 3 parts for the first person and 5 parts for the second.
Method. First combine the two ratios into one three-part ratio. Then use the known share (Lee's £45) to find the value of one part.
- Spot the quantity that is in both ratios. Kim appears in both ratios, so it links them together. To combine, Kim must stand for the same number of parts in both.James : Kim = 3 : 5
Kim : Lee = 2 : 3 - Find the lowest common multiple (LCM) of the two Kim numbers. We make both Kim numbers equal to the LCM of 5 and 2.Multiples of 5: 5, 10
Multiples of 2: 2, 4, 6, 8, 10
LCM = 10 - Multiply each ratio so that Kim becomes 10. Multiplying every part of a ratio by the same number gives an equivalent ratio.James : Kim = 3 : 5 (multiply by 2) = 6 : 10
Kim : Lee = 2 : 3 (multiply by 5) = 10 : 15 - Write the combined ratio James : Kim : Lee. Kim is now the same in both, so the three numbers can be joined.James : Kim : Lee = 6 : 10 : 15
- Write down what we know. We are told the value of one of the shares. That tells us how many equal parts it is worth.Lee has 15 parts, worth £45
- Find the value of one part. 15 parts are worth £45, so divide by 15.£45 ÷ 15 = £3 for one part
- Multiply to find what the question asks for. Multiply the value of one part by the number of parts needed.Kim: 10 × £3 = £30
Check. James has 6 × £3 = £18. Then James : Kim = 18 : 30, which simplifies to 3 : 5. ✓
Watch out. Kim's number is 10 in the combined ratio, not 5 or 2. Always use the number in the combined ratio.
Watch out
When one share is known, divide by that share's ratio number (3 for red in 3 : 5), not by the total number of parts (8). And when you are given a difference, subtract the ratio numbers; do not add them.Exercise 15D Ratio problems
Non-calculatorKey words. To share in a ratio means to split an amount into equal-sized parts, then give each person the number of parts shown in the ratio. The ratio 3 : 5 means 3 parts for the first person and 5 parts for the second.
Method. When one share is known, divide it by its number of parts to find one part. Then multiply to find any other share.
- Write down what we know. We are told the value of one of the shares. That tells us how many equal parts it is worth.2 parts has 2 parts, worth 10
- Find the value of one part. 2 parts are worth 10, so divide by 2.10 ÷ 2 = 5 for one part
- Multiply to find what the question asks for. Multiply the value of one part by the number of parts needed.3 parts: 3 × 5 = 15
Check. Check the share we were given: 2 × 5 = 10, which matches. ✓
Watch out. The amount you are given belongs to one share, not to the whole. Do not divide by the total number of parts.
Key words. To share in a ratio means to split an amount into equal-sized parts, then give each person the number of parts shown in the ratio. The ratio 3 : 5 means 3 parts for the first person and 5 parts for the second.
Method. When one share is known, divide it by its number of parts to find one part. Then multiply to find any other share.
- Write down what we know. We are told the value of one of the shares. That tells us how many equal parts it is worth.4 parts has 4 parts, worth 20
- Find the value of one part. 4 parts are worth 20, so divide by 4.20 ÷ 4 = 5 for one part
- Multiply to find what the question asks for. Multiply the value of one part by the number of parts needed.3 parts: 3 × 5 = 15
Check. Check the share we were given: 4 × 5 = 20, which matches. ✓
Watch out. The amount you are given belongs to one share, not to the whole. Do not divide by the total number of parts.
Key words. To share in a ratio means to split an amount into equal-sized parts, then give each person the number of parts shown in the ratio. The ratio 3 : 5 means 3 parts for the first person and 5 parts for the second.
Method. When one share is known, divide it by its number of parts to find one part. Then multiply to find any other share.
- Write down what we know. We are told the value of one of the shares. That tells us how many equal parts it is worth.5 parts has 5 parts, worth £35
- Find the value of one part. 5 parts are worth £35, so divide by 5.£35 ÷ 5 = £7 for one part
- Multiply to find what the question asks for. Multiply the value of one part by the number of parts needed.2 parts: 2 × £7 = £14
Check. Check the share we were given: 5 × £7 = £35, which matches. ✓
Watch out. The amount you are given belongs to one share, not to the whole. Do not divide by the total number of parts.
Key words. To share in a ratio means to split an amount into equal-sized parts, then give each person the number of parts shown in the ratio. The ratio 3 : 5 means 3 parts for the first person and 5 parts for the second.
Method. When one share is known, divide it by its number of parts to find one part. Then multiply to find any other share.
- Write down what we know. We are told the value of one of the shares. That tells us how many equal parts it is worth.6 parts has 6 parts, worth 42
- Find the value of one part. 6 parts are worth 42, so divide by 6.42 ÷ 6 = 7 for one part
- Multiply to find what the question asks for. Multiply the value of one part by the number of parts needed.1 part: 1 × 7 = 7
Check. Check the share we were given: 6 × 7 = 42, which matches. ✓
Watch out. The amount you are given belongs to one share, not to the whole. Do not divide by the total number of parts.
Key words. To share in a ratio means to split an amount into equal-sized parts, then give each person the number of parts shown in the ratio. The ratio 3 : 5 means 3 parts for the first person and 5 parts for the second.
Method. When one share is known, divide it by its number of parts to find one part. Then multiply to find any other share.
- Write down what we know. We are told the value of one of the shares. That tells us how many equal parts it is worth.4 parts has 4 parts, worth 28
- Find the value of one part. 4 parts are worth 28, so divide by 4.28 ÷ 4 = 7 for one part
- Multiply to find what the question asks for. Multiply the value of one part by the number of parts needed.9 parts: 9 × 7 = 63
Check. Check the share we were given: 4 × 7 = 28, which matches. ✓
Watch out. The amount you are given belongs to one share, not to the whole. Do not divide by the total number of parts.
Key words. To share in a ratio means to split an amount into equal-sized parts, then give each person the number of parts shown in the ratio. The ratio 3 : 5 means 3 parts for the first person and 5 parts for the second.
Method. When one share is known, divide it by its number of parts to find one part. Then multiply to find any other share.
- Write down what we know. We are told the value of one of the shares. That tells us how many equal parts it is worth.5 parts has 5 parts, worth 45
- Find the value of one part. 5 parts are worth 45, so divide by 5.45 ÷ 5 = 9 for one part
- Multiply to find what the question asks for. Multiply the value of one part by the number of parts needed.7 parts: 7 × 9 = 63
Check. Check the share we were given: 5 × 9 = 45, which matches. ✓
Watch out. The amount you are given belongs to one share, not to the whole. Do not divide by the total number of parts.
Key words. To share in a ratio means to split an amount into equal-sized parts, then give each person the number of parts shown in the ratio. The ratio 3 : 5 means 3 parts for the first person and 5 parts for the second.
Method. When one share is known, divide it by its number of parts to find one part. Then multiply to find any other share.
- Write down what we know. We are told the value of one of the shares. That tells us how many equal parts it is worth.1 part has 1 part, worth 6
- Find the value of one part. 1 part is worth 6, so divide by 1.6 ÷ 1 = 6 for one part
- Multiply to find what the question asks for. Multiply the value of one part by the number of parts needed.2 parts: 2 × 6 = 12
Check. Check the share we were given: 1 × 6 = 6, which matches. ✓
Watch out. The amount you are given belongs to one share, not to the whole. Do not divide by the total number of parts.
Key words. To share in a ratio means to split an amount into equal-sized parts, then give each person the number of parts shown in the ratio. The ratio 3 : 5 means 3 parts for the first person and 5 parts for the second.
Method. When one share is known, divide it by its number of parts to find one part. Then multiply to find any other share.
- Write down what we know. We are told the value of one of the shares. That tells us how many equal parts it is worth.Red has 3 parts, worth 15
- Find the value of one part. 3 parts are worth 15, so divide by 3.15 ÷ 3 = 5 for one part
- Multiply to find what the question asks for. Multiply the value of one part by the number of parts needed.Total: 3 + 5 = 8 parts, so 8 × 5 = 40
Check. Check the share we were given: 3 × 5 = 15, which matches. ✓
Watch out. The amount you are given belongs to one share, not to the whole. Do not divide by the total number of parts.
Key words. To share in a ratio means to split an amount into equal-sized parts, then give each person the number of parts shown in the ratio. The ratio 3 : 5 means 3 parts for the first person and 5 parts for the second.
Method. When one share is known, divide it by its number of parts to find one part. Then multiply to find any other share.
- Write down what we know. We are told the value of one of the shares. That tells us how many equal parts it is worth.White has 4 parts, worth 24 litres
- Find the value of one part. 4 parts are worth 24 litres, so divide by 4.24 litres ÷ 4 = 6 litres for one part
- Multiply to find what the question asks for. Multiply the value of one part by the number of parts needed.Blue: 3 × 6 litres = 18 litres
Total: 4 + 3 = 7 parts, so 7 × 6 litres = 42 litres
Check. Check the share we were given: 4 × 6 litres = 24 litres, which matches. ✓
Watch out. The amount you are given belongs to one share, not to the whole. Do not divide by the total number of parts.
Key words. To share in a ratio means to split an amount into equal-sized parts, then give each person the number of parts shown in the ratio. The ratio 3 : 5 means 3 parts for the first person and 5 parts for the second.
Method. Subtract the ratio numbers to find the difference in parts. The known difference is worth that many parts, so divide to find one part.
- Find the difference in parts. The difference between two shares is the number of extra parts that the bigger share has.5 − 2 = 3 parts
- Find the value of one part. We are told the difference is £12. That is worth 3 parts, so divide by 3.£12 ÷ 3 = £4 for one part
- Multiply to find what the question asks for. Multiply the value of one part by the number of parts needed.5 parts: 5 × £4 = £20
Check. Check the difference: 5 × £4 − 2 × £4 = £20 − £8 = £12, which is the difference we were given. ✓
Watch out. Subtract the ratio numbers (e.g. 7 − 4 = 3 parts). Do not add them. The sum gives the total, the subtraction gives the difference.
Key words. To share in a ratio means to split an amount into equal-sized parts, then give each person the number of parts shown in the ratio. The ratio 3 : 5 means 3 parts for the first person and 5 parts for the second.
Method. Subtract the ratio numbers to find the difference in parts. The known difference is worth that many parts, so divide to find one part.
- Find the difference in parts. The difference between two shares is the number of extra parts that the bigger share has.7 − 3 = 4 parts
- Find the value of one part. We are told the difference is 20 kg. That is worth 4 parts, so divide by 4.20 kg ÷ 4 = 5 kg for one part
- Multiply to find what the question asks for. Multiply the value of one part by the number of parts needed.3 parts: 3 × 5 kg = 15 kg
Check. Check the difference: 7 × 5 kg − 3 × 5 kg = 35 kg − 15 kg = 20 kg, which is the difference we were given. ✓
Watch out. Subtract the ratio numbers (e.g. 7 − 4 = 3 parts). Do not add them. The sum gives the total, the subtraction gives the difference.
Key words. To share in a ratio means to split an amount into equal-sized parts, then give each person the number of parts shown in the ratio. The ratio 3 : 5 means 3 parts for the first person and 5 parts for the second.
Method. Subtract the ratio numbers to find the difference in parts. The known difference is worth that many parts, so divide to find one part.
- Find the difference in parts. The difference between two shares is the number of extra parts that the bigger share has.9 − 5 = 4 parts
- Find the value of one part. We are told the difference is 28 m. That is worth 4 parts, so divide by 4.28 m ÷ 4 = 7 m for one part
- Multiply to find what the question asks for. Multiply the value of one part by the number of parts needed.Total: 9 + 5 = 14 parts, so 14 × 7 m = 98 m
Check. Check the difference: 9 × 7 m − 5 × 7 m = 63 m − 35 m = 28 m, which is the difference we were given. ✓
Watch out. Subtract the ratio numbers (e.g. 7 − 4 = 3 parts). Do not add them. The sum gives the total, the subtraction gives the difference.
Key words. To share in a ratio means to split an amount into equal-sized parts, then give each person the number of parts shown in the ratio. The ratio 3 : 5 means 3 parts for the first person and 5 parts for the second.
Method. Subtract the ratio numbers to find the difference in parts. The known difference is worth that many parts, so divide to find one part.
- Find the difference in parts. The difference between two shares is the number of extra parts that the bigger share has.5 − 2 = 3 parts
- Find the value of one part. We are told the difference is £18. That is worth 3 parts, so divide by 3.£18 ÷ 3 = £6 for one part
- Multiply to find what the question asks for. Multiply the value of one part by the number of parts needed.Total: 5 + 2 = 7 parts, so 7 × £6 = £42
Check. Check the difference: 5 × £6 − 2 × £6 = £30 − £12 = £18, which is the difference we were given. ✓
Watch out. Subtract the ratio numbers (e.g. 7 − 4 = 3 parts). Do not add them. The sum gives the total, the subtraction gives the difference.
Key words. A ratio compares the sizes of two or more quantities, written with a colon, for example 2 : 3 (read “2 to 3”). The numbers in a ratio are called parts. The simplest form of a ratio has no whole number (other than 1) that divides into every part.
Method. To combine ratios, make the shared quantity (B) the same number in both ratios, using the LCM, then join them.
- Spot the quantity that is in both ratios. B appears in both ratios, so it links them together. To combine, B must stand for the same number of parts in both.A : B = 1 : 2
B : C = 3 : 4 - Find the lowest common multiple (LCM) of the two B numbers. We make both B numbers equal to the LCM of 2 and 3.Multiples of 2: 2, 4, 6
Multiples of 3: 3, 6
LCM = 6 - Multiply each ratio so that B becomes 6. Multiplying every part of a ratio by the same number gives an equivalent ratio.A : B = 1 : 2 (multiply by 3) = 3 : 6
B : C = 3 : 4 (multiply by 2) = 6 : 8 - Write the combined ratio A : B : C. B is now the same in both, so the three numbers can be joined.A : B : C = 3 : 6 : 8
Check. Look back at the ratios: in 3 : 6 : 8, the first two numbers (3 and 6) simplify to 1 : 2, which equals 1 : 2. The last two (6 and 8) simplify to 3 : 4, which equals 3 : 4. ✓
Watch out. Match the linking quantity using the LCM. Do not just write the numbers side by side.
Key words. A ratio compares the sizes of two or more quantities, written with a colon, for example 2 : 3 (read “2 to 3”). The numbers in a ratio are called parts. The simplest form of a ratio has no whole number (other than 1) that divides into every part.
Method. To combine ratios, make the shared quantity (B) the same number in both ratios, using the LCM, then join them.
- Spot the quantity that is in both ratios. B appears in both ratios, so it links them together. To combine, B must stand for the same number of parts in both.A : B = 2 : 3
B : C = 3 : 5 - Find the lowest common multiple (LCM) of the two B numbers. We make both B numbers equal to the LCM of 3 and 3.Multiples of 3: 3
Multiples of 3: 3
LCM = 3 - Multiply each ratio so that B becomes 3. Multiplying every part of a ratio by the same number gives an equivalent ratio.A : B = 2 : 3 (already 3)
B : C = 3 : 5 (already 3) - Write the combined ratio A : B : C. B is now the same in both, so the three numbers can be joined.A : B : C = 2 : 3 : 5
Check. Look back at the ratios: in 2 : 3 : 5, the first two numbers (2 and 3) simplify to 2 : 3, which equals 2 : 3. The last two (3 and 5) simplify to 3 : 5, which equals 3 : 5. ✓
Watch out. Match the linking quantity using the LCM. Do not just write the numbers side by side.
Key words. A ratio compares the sizes of two or more quantities, written with a colon, for example 2 : 3 (read “2 to 3”). The numbers in a ratio are called parts. The simplest form of a ratio has no whole number (other than 1) that divides into every part.
Method. To combine ratios, make the shared quantity (B) the same number in both ratios, using the LCM, then join them.
- Spot the quantity that is in both ratios. B appears in both ratios, so it links them together. To combine, B must stand for the same number of parts in both.A : B = 4 : 5
B : C = 10 : 7 - Find the lowest common multiple (LCM) of the two B numbers. We make both B numbers equal to the LCM of 5 and 10.Multiples of 5: 5, 10
Multiples of 10: 10
LCM = 10 - Multiply each ratio so that B becomes 10. Multiplying every part of a ratio by the same number gives an equivalent ratio.A : B = 4 : 5 (multiply by 2) = 8 : 10
B : C = 10 : 7 (already 10) - Write the combined ratio A : B : C. B is now the same in both, so the three numbers can be joined.A : B : C = 8 : 10 : 7
Check. Look back at the ratios: in 8 : 10 : 7, the first two numbers (8 and 10) simplify to 4 : 5, which equals 4 : 5. The last two (10 and 7) simplify to 10 : 7, which equals 10 : 7. ✓
Watch out. Match the linking quantity using the LCM. Do not just write the numbers side by side.
Key words. A ratio compares the sizes of two or more quantities, written with a colon, for example 2 : 3 (read “2 to 3”). The numbers in a ratio are called parts. The simplest form of a ratio has no whole number (other than 1) that divides into every part.
Method. To combine ratios, make the shared quantity (B) the same number in both ratios, using the LCM, then join them.
- Spot the quantity that is in both ratios. B appears in both ratios, so it links them together. To combine, B must stand for the same number of parts in both.A : B = 3 : 4
B : C = 6 : 5 - Find the lowest common multiple (LCM) of the two B numbers. We make both B numbers equal to the LCM of 4 and 6.Multiples of 4: 4, 8, 12
Multiples of 6: 6, 12
LCM = 12 - Multiply each ratio so that B becomes 12. Multiplying every part of a ratio by the same number gives an equivalent ratio.A : B = 3 : 4 (multiply by 3) = 9 : 12
B : C = 6 : 5 (multiply by 2) = 12 : 10 - Write the combined ratio A : B : C. B is now the same in both, so the three numbers can be joined.A : B : C = 9 : 12 : 10
Check. Look back at the ratios: in 9 : 12 : 10, the first two numbers (9 and 12) simplify to 3 : 4, which equals 3 : 4. The last two (12 and 10) simplify to 6 : 5, which equals 6 : 5. ✓
Watch out. Match the linking quantity using the LCM. Do not just write the numbers side by side.
Key words. To share in a ratio means to split an amount into equal-sized parts, then give each person the number of parts shown in the ratio. The ratio 3 : 5 means 3 parts for the first person and 5 parts for the second.
Method. Combine the two ratios so that pears have the same number in both, then share the 75 pieces of fruit in the combined ratio.
- Spot the quantity that is in both ratios. Pears appears in both ratios, so it links them together. To combine, Pears must stand for the same number of parts in both.Apples : Pears = 2 : 3
Pears : Bananas = 6 : 5 - Find the lowest common multiple (LCM) of the two Pears numbers. We make both Pears numbers equal to the LCM of 3 and 6.Multiples of 3: 3, 6
Multiples of 6: 6
LCM = 6 - Multiply each ratio so that Pears becomes 6. Multiplying every part of a ratio by the same number gives an equivalent ratio.Apples : Pears = 2 : 3 (multiply by 2) = 4 : 6
Pears : Bananas = 6 : 5 (already 6) - Write the combined ratio Apples : Pears : Bananas. Pears is now the same in both, so the three numbers can be joined.Apples : Pears : Bananas = 4 : 6 : 5
- Add up the parts of the ratio. Each number in the ratio is a number of equal parts. The total number of parts is the sum of the ratio numbers.4 + 6 + 5 = 15 parts
- Find the value of one part. 75 is shared into 15 equal parts, so divide it by 15.75 ÷ 15 = 5 for one part
- Multiply to find each share. Each person gets as many parts as their ratio number, so multiply the value of one part by that number.Apples: 4 × 5 = 20
Pears: 6 × 5 = 30
Bananas: 5 × 5 = 25
Check. Apples 20 + pears 30 + bananas 25 = 75. ✓
Watch out. Share in the combined three-part ratio, not in 2 : 3 or 6 : 5.
Key words. To share in a ratio means to split an amount into equal-sized parts, then give each person the number of parts shown in the ratio. The ratio 3 : 5 means 3 parts for the first person and 5 parts for the second.
Method. When one share is known, divide it by its number of parts to find one part. Then multiply to find any other share.
- Write down what we know. We are told the value of one of the shares. That tells us how many equal parts it is worth.Green has 3 parts, worth 21
- Find the value of one part. 3 parts are worth 21, so divide by 3.21 ÷ 3 = 7 for one part
- Multiply to find what the question asks for. Multiply the value of one part by the number of parts needed.Total: 2 + 3 + 7 = 12 parts, so 12 × 7 = 84
Check. Check the share we were given: 3 × 7 = 21, which matches. ✓
Watch out. The amount you are given belongs to one share, not to the whole. Do not divide by the total number of parts.
Key words. To share in a ratio means to split an amount into equal-sized parts, then give each person the number of parts shown in the ratio. The ratio 3 : 5 means 3 parts for the first person and 5 parts for the second.
Method. Combine the ratios, then use Cat's £48 to find the value of one part.
- Spot the quantity that is in both ratios. Ben appears in both ratios, so it links them together. To combine, Ben must stand for the same number of parts in both.Amy : Ben = 2 : 3
Ben : Cat = 5 : 4 - Find the lowest common multiple (LCM) of the two Ben numbers. We make both Ben numbers equal to the LCM of 3 and 5.Multiples of 3: 3, 6, 9, 12, 15
Multiples of 5: 5, 10, 15
LCM = 15 - Multiply each ratio so that Ben becomes 15. Multiplying every part of a ratio by the same number gives an equivalent ratio.Amy : Ben = 2 : 3 (multiply by 5) = 10 : 15
Ben : Cat = 5 : 4 (multiply by 3) = 15 : 12 - Write the combined ratio Amy : Ben : Cat. Ben is now the same in both, so the three numbers can be joined.Amy : Ben : Cat = 10 : 15 : 12
- Write down what we know. We are told the value of one of the shares. That tells us how many equal parts it is worth.Cat has 12 parts, worth £48
- Find the value of one part. 12 parts are worth £48, so divide by 12.£48 ÷ 12 = £4 for one part
- Multiply to find what the question asks for. Multiply the value of one part by the number of parts needed.Amy: 10 × £4 = £40
Ben: 15 × £4 = £60
Check. Amy : Ben = 40 : 60 = 2 : 3. ✓
Watch out. Match the Ben numbers first using the LCM (3 and 5 give 15).
Key words. To share in a ratio means to split an amount into equal-sized parts, then give each person the number of parts shown in the ratio. The ratio 3 : 5 means 3 parts for the first person and 5 parts for the second.
Method. Subtract the ratio numbers to find the difference in parts. The known difference is worth that many parts, so divide to find one part.
- Find the difference in parts. The difference between two shares is the number of extra parts that the bigger share has.8 − 3 = 5 parts
- Find the value of one part. We are told the difference is £35. That is worth 5 parts, so divide by 5.£35 ÷ 5 = £7 for one part
- Multiply to find what the question asks for. Multiply the value of one part by the number of parts needed.Eve: 8 × £7 = £56
Fay: 3 × £7 = £21
Check. Check the difference: 8 × £7 − 3 × £7 = £56 − £21 = £35, which is the difference we were given. ✓
Watch out. Subtract the ratio numbers (e.g. 7 − 4 = 3 parts). Do not add them. The sum gives the total, the subtraction gives the difference.
Key words. To share in a ratio means to split an amount into equal-sized parts, then give each person the number of parts shown in the ratio. The ratio 3 : 5 means 3 parts for the first person and 5 parts for the second.
Method. Subtract the ratio numbers to find the difference in parts. The known difference is worth that many parts, so divide to find one part.
- Find the difference in parts. The difference between two shares is the number of extra parts that the bigger share has.5 − 3 = 2 parts
- Find the value of one part. We are told the difference is 6. That is worth 2 parts, so divide by 2.6 ÷ 2 = 3 for one part
- Multiply to find what the question asks for. Multiply the value of one part by the number of parts needed.Sopranos: 5 × 3 = 15
Altos: 3 × 3 = 9
Total: 5 + 3 = 8 parts, so 8 × 3 = 24
Check. Check the difference: 5 × 3 − 3 × 3 = 15 − 9 = 6, which is the difference we were given. ✓
Watch out. Subtract the ratio numbers (e.g. 7 − 4 = 3 parts). Do not add them. The sum gives the total, the subtraction gives the difference.
Method. Combine the ratios so that y is the same in both, then use z = 30 to find the value of one part.
- Spot the quantity that is in both ratios. y appears in both ratios, so it links them together. To combine, y must stand for the same number of parts in both.x : y = 3 : 4
y : z = 2 : 5 - Find the lowest common multiple (LCM) of the two y numbers. We make both y numbers equal to the LCM of 4 and 2.Multiples of 4: 4
Multiples of 2: 2, 4
LCM = 4 - Multiply each ratio so that y becomes 4. Multiplying every part of a ratio by the same number gives an equivalent ratio.x : y = 3 : 4 (already 4)
y : z = 2 : 5 (multiply by 2) = 4 : 10 - Write the combined ratio x : y : z. y is now the same in both, so the three numbers can be joined.x : y : z = 3 : 4 : 10
- Write down what we know. We are told the value of one of the shares. That tells us how many equal parts it is worth.z has 10 parts, worth 30
- Find the value of one part. 10 parts are worth 30, so divide by 10.30 ÷ 10 = 3 for one part
- Multiply to find what the question asks for. Multiply the value of one part by the number of parts needed.x: 3 × 3 = 9
Check. y = 12: x : y = 9 : 12 = 3 : 4. ✓
Watch out. Use the combined ratio with the numbers for x, y and z all in it.
Method. A ratio only tells us how the quantities compare. Different pairs of actual numbers give the same ratio but different differences.
- Find one pair of numbers in the ratio 2 : 3. Multiply both parts by 1.2 cats and 3 dogs: the difference is 3 − 2 = 1
- Find another pair in the same ratio. Multiply both parts by 5.10 cats and 15 dogs: the difference is 15 − 10 = 5
- Find another pair. Multiply both parts by 20.40 cats and 60 dogs: the difference is 20
- Conclude. The difference in the numbers can be 1, 5, 20 or anything else. The 5 in Jo's statement came from adding 2 + 3, which is the number of parts in total, not the difference.Jo cannot be sure.
Check. In each case the ratio is the same: 2 : 3, 10 : 15 = 2 : 3 and 40 : 60 = 2 : 3. ✓
Watch out. The ratio numbers are numbers of parts. The difference in actual numbers is (difference in parts) × (value of one part).
Method. Ben has just written the numbers one after another. B is 3 in the first ratio and 4 in the second, so B has two different values in his answer.
- Spot the mistake. In 2 : 3 : 5 the B-to-C part is 3 : 5, but we were told B : C = 4 : 5. Also, 3 : 5 is not equal to 4 : 5.Ben's ratio does not match B : C = 4 : 5.
- Find the lowest common multiple (LCM) of the two B numbers. We make both B numbers equal to the LCM of 3 and 4.Multiples of 3: 3, 6, 9, 12
Multiples of 4: 4, 8, 12
LCM = 12 - Multiply each ratio so that B becomes 12. Multiplying every part of a ratio by the same number gives an equivalent ratio.A : B = 2 : 3 (multiply by 4) = 8 : 12
B : C = 4 : 5 (multiply by 3) = 12 : 15 - Write the combined ratio A : B : C. B is now the same in both, so the three numbers can be joined.A : B : C = 8 : 12 : 15
Check. Look at A : B and B : C in the answer: 8 : 12 = 2 : 3 and 12 : 15 = 4 : 5. ✓
Watch out. Make the middle quantity the same number in both ratios, using the LCM, before joining.
Key words. A ratio compares the sizes of two or more quantities, written with a colon, for example 2 : 3 (read “2 to 3”). The numbers in a ratio are called parts. The simplest form of a ratio has no whole number (other than 1) that divides into every part.
Method. Combine the ratios, then use the fact that A and C together are (8 + 15) parts to find the value of one part.
- Spot the quantity that is in both ratios. B appears in both ratios, so it links them together. To combine, B must stand for the same number of parts in both.A : B = 2 : 3
B : C = 4 : 5 - Find the lowest common multiple (LCM) of the two B numbers. We make both B numbers equal to the LCM of 3 and 4.Multiples of 3: 3, 6, 9, 12
Multiples of 4: 4, 8, 12
LCM = 12 - Multiply each ratio so that B becomes 12. Multiplying every part of a ratio by the same number gives an equivalent ratio.A : B = 2 : 3 (multiply by 4) = 8 : 12
B : C = 4 : 5 (multiply by 3) = 12 : 15 - Write the combined ratio A : B : C. B is now the same in both, so the three numbers can be joined.A : B : C = 8 : 12 : 15
- Add the parts for A and C. The sum of A and C is the number of parts for A plus the number of parts for C.8 + 15 = 23 parts
- Find the value of one part. 23 parts are worth 46.46 ÷ 23 = 2
- Find B. B is 12 parts.12 × 2 = 24
Check. A = 16, C = 30, and 16 + 30 = 46. ✓
Watch out. A and C together are the sum of their parts. B is not included in the sum.