GCSE Maths: Proportion: the unitary method and best buy (Higher)
Part of the GCSE Maths ratio and proportion chapter (Higher). Chapter overview · Foundation version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Proportion: the unitary method and best buy
Two quantities are in direct proportion when they increase or decrease at the same rate. If you buy twice as many pens you pay twice as much; if you buy half as many, you pay half as much. The cost of each pen stays the same. Many everyday problems can be solved by finding the value of one first. This is called the unitary method.
Suppose 5 pens cost £3.50 and we want the cost of 8 pens. Step 1: divide to find the cost of 1 pen: £3.50 ÷ 5 = £0.70. Step 2: multiply to find the cost of 8 pens: 8 × £0.70 = £5.60. A table helps you keep track of what you have done.
| Pens | Cost |
|---|---|
| 5 | £3.50 |
| ÷ 5 ↓ 1 | ↓ ÷ 5 £0.70 |
| × 8 ↓ 8 | ↓ × 8 £5.60 |
Best buy and value for money. A bigger pack is not always cheaper per gram, and a lower price does not always give better value. To compare, find the cost of one unit (for example the price of 1 g) for each option, in the same units. The lowest price per unit is the best buy. Another way is to find how much you get for the same money (for example per £1): the most is the best buy.
Offers. Offers such as “3 for the price of 2” are compared in the same way. Work out the total price with the offer, divide by the number of items, and compare the price per item with the normal price. Always check the units: change pounds to pence when that avoids decimals, and make sure both quantities use the same unit (g or kg, ml or litres).
Key ideas
- Direct proportion: if one quantity is multiplied by a number, the other is multiplied by the same number.
- Unitary method: divide to find the value of 1, then multiply to find the value of the amount you want.
- Best buy: find the price of 1 unit (or the amount for the same money) for every option, using the same units. Lowest price per unit (or most per £1) wins.
- Offers: find the cost per item with the offer, then compare.
Key words. Two quantities are in direct proportion when one is a fixed multiple of the other: if one doubles, so does the other. The unitary method means finding the value of one unit first.
Method. The unitary method: divide to find the value of one, then multiply to find the value of the number you want.
- Write down what we know. We know the value for one amount, and want it for a different amount.5 pens cost £3.50
- Divide to find the value of one. This is the unitary method: find the value for 1 pen first. Divide the total by the amount it was for.£3.50 ÷ 5 = £0.70 for 1 pen
- Multiply to find the value for 8. 8 is 8 times 1 pen, so multiply the value for 1 pen by 8.8 × £0.70 = £5.60
Check. A second way (the multiplier): 8 ÷ 5 = 1.6, so the value is £3.50 × 1.6 = £5.60. ✓
Watch out. Divide first to find the value of one, then multiply. Multiplying the original numbers together gives nonsense.
Key words. Two quantities are in direct proportion when one is a fixed multiple of the other: if one doubles, so does the other. The unitary method means finding the value of one unit first.
Method. The unitary method: divide to find the value of one, then multiply to find the value of the number you want.
- Write down what we know. We know the value for one amount, and want it for a different amount.90 km needs 6 litres
- Divide to find the value of one. This is the unitary method: find the value for 1 km first. Divide the total by the amount it was for.6 litres ÷ 90 = 115 litres for 1 km
- Multiply to find the value for 240 km. 240 km is 240 times 1 km, so multiply the value for 1 km by 240.240 × 115 litres = 16 litres
Check. A second way (the multiplier): 240 ÷ 90 = 223, so the value is 6 litres × 223 = 16 litres. ✓
Watch out. Divide first to find the value of one, then multiply. Multiplying the original numbers together gives nonsense.
Key words. Two quantities are in direct proportion when one is a fixed multiple of the other: if one doubles, so does the other. The unitary method means finding the value of one unit first.
Method. Work out the price of one unit (or the same quantity) for each option. The lowest price per unit is the best value.
- Write both prices in pence. Comparing is easier when both prices are in the same unit.£1.60 = 160p
£2.25 = 225p - Work out the cost of 1 g for each option. Dividing the price by the quantity gives the cost per single unit. Then they can be compared fairly.Cereal A: 160p ÷ 500 = 0.32p per 1 g
Cereal B: 225p ÷ 750 = 0.3p per 1 g - Compare and choose. Cost per unit: the lower the cost, the better the value.0.32p , 0.3p
The lowest is Cereal B at 0.3p per 1 g, so Cereal B is the best buy.
Check. A second way: how much do you get for £1 (100p)? Cereal A: 500 ÷ 160 × 100 = 312.5 g , Cereal B: 750 ÷ 225 × 100 = 33313 g. Cereal B gives the most for £1. ✓
Watch out. The biggest pack is not always the best value, and the lowest price is not always the best value. Compare the price per unit.
Key words. Two quantities are in direct proportion when one is a fixed multiple of the other: if one doubles, so does the other. The unitary method means finding the value of one unit first.
Method. Work out the price of one unit (or the same quantity) for each option. The lowest price per unit is the best value.
- Write both prices in pence. Comparing is easier when both prices are in the same unit.£1.80 = 180p
£2.80 = 280p
£4.95 = 495p - Work out the cost of 1 ml for each option. Dividing the price by the quantity gives the cost per single unit. Then they can be compared fairly.Small: 180p ÷ 250 = 0.72p per 1 ml
Medium: 280p ÷ 400 = 0.7p per 1 ml
Large: 495p ÷ 750 = 0.66p per 1 ml - Compare and choose. Cost per unit: the lower the cost, the better the value.0.72p , 0.7p , 0.66p
The lowest is Large at 0.66p per 1 ml, so Large is the best buy.
Check. A second way: how much do you get for £1 (100p)? Small: 250 ÷ 180 × 100 = 13889 ml , Medium: 400 ÷ 280 × 100 = 14267 ml , Large: 750 ÷ 495 × 100 = 1511733 ml. Large gives the most for £1. ✓
Watch out. The biggest pack is not always the best value, and the lowest price is not always the best value. Compare the price per unit.
Key words. Two quantities are in direct proportion when one is a fixed multiple of the other: if one doubles, so does the other. The unitary method means finding the value of one unit first.
Method. With the offer, 3 bottles cost the same as 2 bottles. Work out that price, divide by 3, and compare.
- Find the offer price for 3 bottles. You pay for 2 bottles only.2 × £2.40 = £4.80
- Find the price of 1 bottle with the offer. Divide by the 3 bottles that you get.£4.80 ÷ 3 = £1.60
- Find the saving on each bottle. Subtract the offer price from the normal price.£2.40 − £1.60 = £0.80
Check. Without the offer 3 bottles cost 3 × £2.40 = £7.20. With the offer they cost £4.80. The saving £2.40 ÷ 3 = £0.80 per bottle. ✓
Watch out. “3 for the price of 2” does not mean the bottles are 2/3 of the price each; work out the total first.
Watch out
Do not compare prices when the quantities are different. A pack that costs less may hold much less. Also, check that both quantities are in the same unit: 1 kg and 500 g are not “1 and 500”.Exercise 15G Unitary method and best buy
Calculator allowedKey words. Two quantities are in direct proportion when one is a fixed multiple of the other: if one doubles, so does the other. The unitary method means finding the value of one unit first.
Method. The unitary method: divide to find the value of one, then multiply to find the value of the number you want.
- Write down what we know. We know the value for one amount, and want it for a different amount.4 pens cost £2.80
- Divide to find the value of one. This is the unitary method: find the value for 1 pen first. Divide the total by the amount it was for.£2.80 ÷ 4 = £0.70 for 1 pen
- Multiply to find the value for 7. 7 is 7 times 1 pen, so multiply the value for 1 pen by 7.7 × £0.70 = £4.90
Check. A second way (the multiplier): 7 ÷ 4 = 1.75, so the value is £2.80 × 1.75 = £4.90. ✓
Watch out. Divide first to find the value of one, then multiply. Multiplying the original numbers together gives nonsense.
Key words. Two quantities are in direct proportion when one is a fixed multiple of the other: if one doubles, so does the other. The unitary method means finding the value of one unit first.
Method. The unitary method: divide to find the value of one, then multiply to find the value of the number you want.
- Write down what we know. We know the value for one amount, and want it for a different amount.6 buns cost £4.50
- Divide to find the value of one. This is the unitary method: find the value for 1 bun first. Divide the total by the amount it was for.£4.50 ÷ 6 = £0.75 for 1 bun
- Multiply to find the value for 10. 10 is 10 times 1 bun, so multiply the value for 1 bun by 10.10 × £0.75 = £7.50
Check. A second way (the multiplier): 10 ÷ 6 = 123, so the value is £4.50 × 123 = £7.50. ✓
Watch out. Divide first to find the value of one, then multiply. Multiplying the original numbers together gives nonsense.
Key words. Two quantities are in direct proportion when one is a fixed multiple of the other: if one doubles, so does the other. The unitary method means finding the value of one unit first.
Method. The unitary method: divide to find the value of one, then multiply to find the value of the number you want.
- Write down what we know. We know the value for one amount, and want it for a different amount.5 kg of apples cost £12
- Divide to find the value of one. This is the unitary method: find the value for 1 kg first. Divide the total by the amount it was for.£12 ÷ 5 = £2.40 for 1 kg
- Multiply to find the value for 8 kg. 8 kg is 8 times 1 kg, so multiply the value for 1 kg by 8.8 × £2.40 = £19.20
Check. A second way (the multiplier): 8 ÷ 5 = 1.6, so the value is £12 × 1.6 = £19.20. ✓
Watch out. Divide first to find the value of one, then multiply. Multiplying the original numbers together gives nonsense.
Key words. Two quantities are in direct proportion when one is a fixed multiple of the other: if one doubles, so does the other. The unitary method means finding the value of one unit first.
Method. The unitary method: divide to find the value of one, then multiply to find the value of the number you want.
- Write down what we know. We know the value for one amount, and want it for a different amount.12 eggs cost £3
- Divide to find the value of one. This is the unitary method: find the value for 1 egg first. Divide the total by the amount it was for.£3 ÷ 12 = £0.25 for 1 egg
- Multiply to find the value for 18. 18 is 18 times 1 egg, so multiply the value for 1 egg by 18.18 × £0.25 = £4.50
Check. A second way (the multiplier): 18 ÷ 12 = 1.5, so the value is £3 × 1.5 = £4.50. ✓
Watch out. Divide first to find the value of one, then multiply. Multiplying the original numbers together gives nonsense.
Key words. Two quantities are in direct proportion when one is a fixed multiple of the other: if one doubles, so does the other. The unitary method means finding the value of one unit first.
Method. The unitary method: divide to find the value of one, then multiply to find the value of the number you want.
- Write down what we know. We know the value for one amount, and want it for a different amount.3 books cost £14.25
- Divide to find the value of one. This is the unitary method: find the value for 1 book first. Divide the total by the amount it was for.£14.25 ÷ 3 = £4.75 for 1 book
- Multiply to find the value for 5. 5 is 5 times 1 book, so multiply the value for 1 book by 5.5 × £4.75 = £23.75
Check. A second way (the multiplier): 5 ÷ 3 = 123, so the value is £14.25 × 123 = £23.75. ✓
Watch out. Divide first to find the value of one, then multiply. Multiplying the original numbers together gives nonsense.
Key words. Two quantities are in direct proportion when one is a fixed multiple of the other: if one doubles, so does the other. The unitary method means finding the value of one unit first.
Method. The unitary method: divide to find the value of one, then multiply to find the value of the number you want.
- Write down what we know. We know the value for one amount, and want it for a different amount.8 metres of cloth cost £34
- Divide to find the value of one. This is the unitary method: find the value for 1 metre first. Divide the total by the amount it was for.£34 ÷ 8 = £4.25 for 1 metre
- Multiply to find the value for 5 m. 5 m is 5 times 1 metre, so multiply the value for 1 metre by 5.5 × £4.25 = £21.25
Check. A second way (the multiplier): 5 ÷ 8 = 0.625, so the value is £34 × 0.625 = £21.25. ✓
Watch out. Divide first to find the value of one, then multiply. Multiplying the original numbers together gives nonsense.
Key words. Two quantities are in direct proportion when one is a fixed multiple of the other: if one doubles, so does the other. The unitary method means finding the value of one unit first.
Method. The unitary method: divide to find the value of one, then multiply to find the value of the number you want.
- Write down what we know. We know the value for one amount, and want it for a different amount.90 km needs 6 litres
- Divide to find the value of one. This is the unitary method: find the value for 1 km first. Divide the total by the amount it was for.6 litres ÷ 90 = 115 litres for 1 km
- Multiply to find the value for 240 km. 240 km is 240 times 1 km, so multiply the value for 1 km by 240.240 × 115 litres = 16 litres
Check. A second way (the multiplier): 240 ÷ 90 = 223, so the value is 6 litres × 223 = 16 litres. ✓
Watch out. Divide first to find the value of one, then multiply. Multiplying the original numbers together gives nonsense.
Key words. Two quantities are in direct proportion when one is a fixed multiple of the other: if one doubles, so does the other. The unitary method means finding the value of one unit first.
Method. The unitary method: divide to find the value of one, then multiply to find the value of the number you want.
- Write down what we know. We know the value for one amount, and want it for a different amount.In 3 minutes she types 150 words
- Divide to find the value of one. This is the unitary method: find the value for 1 minute first. Divide the total by the amount it was for.150 words ÷ 3 = 50 words for 1 minute
- Multiply to find the value for 8 minutes. 8 minutes is 8 times 1 minute, so multiply the value for 1 minute by 8.8 × 50 words = 400 words
Check. A second way (the multiplier): 8 ÷ 3 = 223, so the value is 150 words × 223 = 400 words. ✓
Watch out. Divide first to find the value of one, then multiply. Multiplying the original numbers together gives nonsense.
Key words. Two quantities are in direct proportion when one is a fixed multiple of the other: if one doubles, so does the other. The unitary method means finding the value of one unit first.
Method. The unitary method: divide to find the value of one, then multiply to find the value of the number you want.
- Write down what we know. We know the value for one amount, and want it for a different amount.120 pages take 8 minutes
- Divide to find the value of one. This is the unitary method: find the value for 1 page first. Divide the total by the amount it was for.8 minutes ÷ 120 = 115 minutes for 1 page
- Multiply to find the value for 300 pages. 300 pages is 300 times 1 page, so multiply the value for 1 page by 300.300 × 115 minutes = 20 minutes
Check. A second way (the multiplier): 300 ÷ 120 = 2.5, so the value is 8 minutes × 2.5 = 20 minutes. ✓
Watch out. Divide first to find the value of one, then multiply. Multiplying the original numbers together gives nonsense.
Key words. Two quantities are in direct proportion when one is a fixed multiple of the other: if one doubles, so does the other. The unitary method means finding the value of one unit first.
Method. Work out the price of one unit (or the same quantity) for each option. The lowest price per unit is the best value.
- Write both prices in pence. Comparing is easier when both prices are in the same unit.£1.60 = 160p
£2.25 = 225p - Work out the cost of 1 g for each option. Dividing the price by the quantity gives the cost per single unit. Then they can be compared fairly.A: 160p ÷ 500 = 0.32p per 1 g
B: 225p ÷ 750 = 0.3p per 1 g - Compare and choose. Cost per unit: the lower the cost, the better the value.0.32p , 0.3p
The lowest is B at 0.3p per 1 g, so B is the best buy.
Check. A second way: how much do you get for £1 (100p)? A: 500 ÷ 160 × 100 = 312.5 g , B: 750 ÷ 225 × 100 = 33313 g. B gives the most for £1. ✓
Watch out. The biggest pack is not always the best value, and the lowest price is not always the best value. Compare the price per unit.
Key words. Two quantities are in direct proportion when one is a fixed multiple of the other: if one doubles, so does the other. The unitary method means finding the value of one unit first.
Method. Work out the price of one unit (or the same quantity) for each option. The lowest price per unit is the best value.
- Write both prices in pence. Comparing is easier when both prices are in the same unit.£3.20 = 320p
£7.50 = 750p - Work out the cost of 1 g for each option. Dividing the price by the quantity gives the cost per single unit. Then they can be compared fairly.A: 320p ÷ 2000 = 0.16p per 1 g
B: 750p ÷ 5000 = 0.15p per 1 g - Compare and choose. Cost per unit: the lower the cost, the better the value.0.16p , 0.15p
The lowest is B at 0.15p per 1 g, so B is the best buy.
Check. A second way: how much do you get for £1 (100p)? A: 2000 ÷ 320 × 100 = 625 g , B: 5000 ÷ 750 × 100 = 66623 g. B gives the most for £1. ✓
Watch out. The biggest pack is not always the best value, and the lowest price is not always the best value. Compare the price per unit.
Key words. Two quantities are in direct proportion when one is a fixed multiple of the other: if one doubles, so does the other. The unitary method means finding the value of one unit first.
Method. Work out the price of one unit (or the same quantity) for each option. The lowest price per unit is the best value.
- Write both prices in pence. Comparing is easier when both prices are in the same unit.£1.20 = 120p
£1.65 = 165p - Work out the cost of 1 ml for each option. Dividing the price by the quantity gives the cost per single unit. Then they can be compared fairly.A: 120p ÷ 1000 = 0.12p per 1 ml
B: 165p ÷ 1500 = 0.11p per 1 ml - Compare and choose. Cost per unit: the lower the cost, the better the value.0.12p , 0.11p
The lowest is B at 0.11p per 1 ml, so B is the best buy.
Check. A second way: how much do you get for £1 (100p)? A: 1000 ÷ 120 × 100 = 83313 ml , B: 1500 ÷ 165 × 100 = 909111 ml. B gives the most for £1. ✓
Watch out. The biggest pack is not always the best value, and the lowest price is not always the best value. Compare the price per unit.
Key words. Two quantities are in direct proportion when one is a fixed multiple of the other: if one doubles, so does the other. The unitary method means finding the value of one unit first.
Method. Work out the price of one unit (or the same quantity) for each option. The lowest price per unit is the best value.
- Write both prices in pence. Comparing is easier when both prices are in the same unit.£2.40 = 240p
£6.60 = 660p - Work out the cost of 1 tea bag for each option. Dividing the price by the quantity gives the cost per single unit. Then they can be compared fairly.A: 240p ÷ 80 = 3p per 1 tea bag
B: 660p ÷ 240 = 2.75p per 1 tea bag - Compare and choose. Cost per unit: the lower the cost, the better the value.3p , 2.75p
The lowest is B at 2.75p per 1 tea bag, so B is the best buy.
Check. A second way: how much do you get for £1 (100p)? A: 80 ÷ 240 × 100 = 3313 tea bag , B: 240 ÷ 660 × 100 = 36411 tea bag. B gives the most for £1. ✓
Watch out. The biggest pack is not always the best value, and the lowest price is not always the best value. Compare the price per unit.
Key words. Two quantities are in direct proportion when one is a fixed multiple of the other: if one doubles, so does the other. The unitary method means finding the value of one unit first.
Method. Work out the price of one unit (or the same quantity) for each option. The lowest price per unit is the best value.
- Write both prices in pence. Comparing is easier when both prices are in the same unit.£2.70 = 270p
£4.20 = 420p - Work out the cost of 1 can for each option. Dividing the price by the quantity gives the cost per single unit. Then they can be compared fairly.the 6-pack: 270p ÷ 6 = 45p per 1 can
the 10-pack: 420p ÷ 10 = 42p per 1 can - Compare and choose. Cost per unit: the lower the cost, the better the value.45p , 42p
The lowest is the 10-pack at 42p per 1 can, so the 10-pack is the best buy.
Check. A second way: how much do you get for £1 (100p)? the 6-pack: 6 ÷ 270 × 100 = 229 can , the 10-pack: 10 ÷ 420 × 100 = 2821 can. the 10-pack gives the most for £1. ✓
Watch out. The biggest pack is not always the best value, and the lowest price is not always the best value. Compare the price per unit.
Key words. Two quantities are in direct proportion when one is a fixed multiple of the other: if one doubles, so does the other. The unitary method means finding the value of one unit first.
Method. Work out the price of one unit (or the same quantity) for each option. The lowest price per unit is the best value.
- Write both prices in pence. Comparing is easier when both prices are in the same unit.£1.20 = 120p
£1.80 = 180p
£4.20 = 420p - Work out the cost of 1 g for each option. Dividing the price by the quantity gives the cost per single unit. Then they can be compared fairly.A: 120p ÷ 250 = 0.48p per 1 g
B: 180p ÷ 400 = 0.45p per 1 g
C: 420p ÷ 1000 = 0.42p per 1 g - Compare and choose. Cost per unit: the lower the cost, the better the value.0.48p , 0.45p , 0.42p
The lowest is C at 0.42p per 1 g, so C is the best buy.
Check. A second way: how much do you get for £1 (100p)? A: 250 ÷ 120 × 100 = 20813 g , B: 400 ÷ 180 × 100 = 22229 g , C: 1000 ÷ 420 × 100 = 238221 g. C gives the most for £1. ✓
Watch out. The biggest pack is not always the best value, and the lowest price is not always the best value. Compare the price per unit.
Key words. Two quantities are in direct proportion when one is a fixed multiple of the other: if one doubles, so does the other. The unitary method means finding the value of one unit first.
Method. Work out how much you get for the same money. The more you get, the better the value.
- Write both prices in pence. Comparing is easier when both prices are in the same unit.£2.50 = 250p
- Work out how much you get for £1 (100p) for each option. Divide the quantity by the price to find the amount per penny, then multiply by 100 for £1.the 200 g bar: 200 g ÷ 90p × 100 = 22229 g per £1
the 500 g bar: 500 g ÷ 250p × 100 = 200 g per £1 - Compare and choose. Amount per £1: the more you get, the better the value.22229 g , 200 g
The most is the 200 g bar with 22229 g per £1, so the 200 g bar is the best buy.
Check. A second way, the price of 1 g: the 200 g bar: 90p ÷ 200 = 0.45p , the 500 g bar: 250p ÷ 500 = 0.5p. the 200 g bar is cheapest per g. ✓
Watch out. The biggest pack is not always the best value, and the lowest price is not always the best value. Compare the price per unit.
Key words. Two quantities are in direct proportion when one is a fixed multiple of the other: if one doubles, so does the other. The unitary method means finding the value of one unit first.
Method. The unitary method: divide to find the value of one, then multiply to find the value of the number you want.
- Write down what we know. We know the value for one amount, and want it for a different amount.7 hours earn £17.50
- Divide to find the value of one. This is the unitary method: find the value for 1 hour first. Divide the total by the amount it was for.£17.50 ÷ 7 = £2.50 for 1 hour
- Multiply to find the value for 12 hours. 12 hours is 12 times 1 hour, so multiply the value for 1 hour by 12.12 × £2.50 = £30
Check. A second way (the multiplier): 12 ÷ 7 = 157, so the value is £17.50 × 157 = £30. ✓
Watch out. Divide first to find the value of one, then multiply. Multiplying the original numbers together gives nonsense.
Key words. Two quantities are in direct proportion when one is a fixed multiple of the other: if one doubles, so does the other. The unitary method means finding the value of one unit first.
Method. Find the mass of butter for 40 biscuits, then find what that mass costs.
- Find the butter for one biscuit. Divide the mass by the number of biscuits (unitary method).90 ÷ 12 = 7.5 g for 1 biscuit
- Find the butter for 40 biscuits. Multiply by 40.7.5 × 40 = 300 g
- Find the cost of 1 g of butter. A 250 g block costs £1.80, so divide.£1.80 ÷ 250 = £0.0072
- Find the cost of 300 g. Multiply the cost of 1 g by the mass.300 × £0.0072 = £2.16
Check. 300 g is 1.2 blocks of 250 g, and 1.2 × £1.80 = £2.16. ✓
Watch out. Two steps of proportion: biscuits to butter, then butter to cost.
| Tickets | 2 | 5 | 8 | ? |
|---|---|---|---|---|
| Cost | £7 | ? | ? | £42 |
Key words. Two quantities are in direct proportion when one is a fixed multiple of the other: if one doubles, so does the other. The unitary method means finding the value of one unit first.
Method. The unitary method: divide to find the value of one, then multiply to find the value of the number you want.
- Write down what we know. We know the value for one amount, and want it for a different amount.2 tickets cost £7
- Divide to find the value of one. This is the unitary method: find the value for 1 ticket first. Divide the total by the amount it was for.£7 ÷ 2 = £3.50 for 1 ticket
- Multiply to find the value for 5. 5 is 5 times 1 ticket, so multiply the value for 1 ticket by 5.5 × £3.50 = £17.50
Check. A second way (the multiplier): 5 ÷ 2 = 2.5, so the value is £7 × 2.5 = £17.50. ✓
Watch out. Divide first to find the value of one, then multiply. Multiplying the original numbers together gives nonsense.
Key words. Two quantities are in direct proportion when one is a fixed multiple of the other: if one doubles, so does the other. The unitary method means finding the value of one unit first.
Method. The unitary method: divide to find the value of one, then multiply to find the value of the number you want.
- Write down what we know. We know the value for one amount, and want it for a different amount.2 tickets cost £7
- Divide to find the value of one. This is the unitary method: find the value for 1 ticket first. Divide the total by the amount it was for.£7 ÷ 2 = £3.50 for 1 ticket
- Multiply to find the value for 8. 8 is 8 times 1 ticket, so multiply the value for 1 ticket by 8.8 × £3.50 = £28
Check. A second way (the multiplier): 8 ÷ 2 = 4, so the value is £7 × 4 = £28. ✓
Watch out. Divide first to find the value of one, then multiply. Multiplying the original numbers together gives nonsense.
Key words. Two quantities are in direct proportion when one is a fixed multiple of the other: if one doubles, so does the other. The unitary method means finding the value of one unit first.
Method. Find the cost of 1 ticket, then find how many of those £42 buys.
- Find the cost of 1 ticket. Divide £7 by the 2 tickets.£7 ÷ 2 = £3.50
- Divide £42 by the cost of 1 ticket. This tells us how many tickets £42 pays for.£42 ÷ £3.50 = 12
Check. 12 × £3.50 = £42. ✓
Watch out. To find the number of items, divide the money by the price of one item.
Key words. Two quantities are in direct proportion when one is a fixed multiple of the other: if one doubles, so does the other. The unitary method means finding the value of one unit first.
Method. Work out the price of one unit (or the same quantity) for each option. The lowest price per unit is the best value.
- Write both prices in pence. Comparing is easier when both prices are in the same unit.£2.50 = 250p
- Work out the cost of 1 g for each option. Dividing the price by the quantity gives the cost per single unit. Then they can be compared fairly.the small pack: 90p ÷ 200 = 0.45p per 1 g
the large pack: 250p ÷ 500 = 0.5p per 1 g - Compare and choose. Cost per unit: the lower the cost, the better the value.0.45p , 0.5p
The lowest is the small pack at 0.45p per 1 g, so the small pack is the best buy.
Check. A second way: how much do you get for £1 (100p)? the small pack: 200 ÷ 90 × 100 = 22229 g , the large pack: 500 ÷ 250 × 100 = 200 g. the small pack gives the most for £1. ✓
Watch out. The biggest pack is not always the best value, and the lowest price is not always the best value. Compare the price per unit.
Key words. Two quantities are in direct proportion when one is a fixed multiple of the other: if one doubles, so does the other. The unitary method means finding the value of one unit first.
Method. In direct proportion we multiply both quantities by the same number. We do not add the same number to each.
- Find how many times bigger 15 is than 5. The number of notebooks is multiplied by this number.15 ÷ 5 = 3
- Multiply the cost by 3. The cost must be multiplied by the same number.£7.50 × 3 = £22.50
- Explain Raj's mistake. Raj added the number of extra notebooks (10) to the money. But 10 notebooks cost far more than £10: each notebook costs £1.50.10 notebooks cost 10 × £1.50 = £15, so £7.50 + £15 = £22.50
Check. £7.50 ÷ 5 = £1.50 for 1 notebook, and 15 × £1.50 = £22.50. ✓
Watch out. Never add pounds to a number of items. Multiply or divide both quantities by the same number.
Key words. Two quantities are in direct proportion when one is a fixed multiple of the other: if one doubles, so does the other. The unitary method means finding the value of one unit first.
Method. Equal ratios are multiples of each other. Find the multiplier on the flour, then multiply the sugar by the same number.
- Write the ratio flour : sugar. The two amounts for the original recipe.34 : 23
- Write 214 as an improper fraction. To divide, it is easier to have a single fraction.214 = 94
- Find the multiplier. Divide the new flour by the old flour. Dividing by a fraction is multiplying by its reciprocal: keep, change, flip.94 ÷ 34 = 94 × 43 = 3
- Multiply the sugar. Both amounts are multiplied by the same multiplier.23 × 3 = 2 cups
Check. Flour has been multiplied by 3 (3/4 × 3 = 9/4) and sugar by 3 (2/3 × 3 = 2). ✓
Watch out. The multiplier does not have to be a whole number, but here it works out to be 3.
Key words. Two quantities are in direct proportion when one is a fixed multiple of the other: if one doubles, so does the other. The unitary method means finding the value of one unit first.
Method. First find which pack is better value per pen. Then check that the cheaper way works for exactly 30 pens.
- Cost per pen: Pack A. Divide the price in pence by the number of pens.£2.10 = 210p, and 210 ÷ 6 = 35p per pen
- Cost per pen: Pack B. Do the same.£3.20 = 320p, and 320 ÷ 10 = 32p per pen
- Compare. Pack B is cheaper per pen.32p < 35p
- Check that Pack B works for 30 pens. 30 ÷ 10 = 3, a whole number of packs.3 packs of 10 = 30 pens
- Cost of the cheapest way. Multiply the price of a pack by 3.3 × £3.20 = £9.60
- Compare with the other way. 30 ÷ 6 = 5 packs of A.5 × £2.10 = £10.50, which is more than £9.60
Check. 30 pens at 32p each = 960p = £9.60. ✓
Watch out. Check that the cheaper pack gives exactly the number of items you need. Here it does (30 is a multiple of 10).