GCSE Maths: Ratio in context: recipes, mixing, concentration and conversion (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 in context: recipes, mixing, concentration and conversion
Ratios appear all around us. In each case the idea is the same: multiply or divide every quantity by the same number so that the ratios stay the same.
Recipes (scaling). A recipe for 4 people uses 200 g flour, 80 g sugar and 120 g butter. For 6 people, every amount is multiplied by the same number, 6 ÷ 4 = 1.5. Where the multiplier is awkward, use the unitary idea: work out the amounts for 1 person (divide by 4), then multiply by 6. The ratio flour : sugar : butter stays 5 : 2 : 3, so the recipe tastes the same.
Mixing. Squash is made by mixing cordial and water in the ratio 1 : 4. That is 1 part cordial and 4 parts water, so there are 1 + 4 = 5 parts in the drink. The drink is 15 cordial. It is not 14: the ratio compares cordial with the water, not with the whole drink.
Concentration. The strength or concentration of a mixture is the fraction of it that is the active ingredient. A larger fraction means a stronger mixture. A drink mixed 1 : 4 is 15 cordial and one mixed 1 : 5 is only 16 cordial, so the 1 : 4 drink is stronger. Conversion is also a ratio. If £1 = €1.15 then the ratio of pounds to euros is 1 : 1.15, so we multiply pounds by 1.15 to get euros and divide euros by 1.15 to get pounds.
Map scales. A scale of 1 : 25 000 means that 1 cm on the map is 25 000 cm in real life. Since 25 000 cm = 250 m, the real distance is 250 m for every 1 cm. (Scale drawings are studied again in Chapter 22.)
Key ideas
- To scale a recipe, multiply every ingredient by the same multiplier (new amount ÷ old amount).
- A mixing ratio a : b has a + b parts in the mixture. The fraction of the mixture that is a is a/(a + b).
- A larger fraction of the active ingredient means a stronger mixture.
- Conversion rates are ratios: multiply to change into the other unit, divide to change back.
- To join two mixtures, work out the actual amounts of each liquid, add them, then form the ratio.
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 amounts for one (the unitary method), then multiply by the number needed.
- Find the amounts for 1 people. The unitary method: divide every amount by the number it serves now.Flour: 200 g ÷ 4 = 50 g
Sugar: 80 g ÷ 4 = 20 g
Butter: 120 g ÷ 4 = 30 g
Eggs: 2 ÷ 4 = 0.5 - Multiply by 6 for 6 people. Multiply the amount for one by the number needed.Flour: 50 g × 6 = 300 g
Sugar: 20 g × 6 = 120 g
Butter: 30 g × 6 = 180 g
Eggs: 0.5 × 6 = 3 - Write the amounts needed. These are the new amounts, with the units.Flour: 300 g
Sugar: 120 g
Butter: 180 g
Eggs: 3
Check. Check the first ingredient in reverse: 300 ÷ 1.5 = 200, the original amount of flour. ✓
Watch out. Scale every ingredient, not just the main one, and keep the units.
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. Add the parts to find how many equal parts there are, find the value of one part, then multiply.
- 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.1 + 4 = 5 parts
- Find the value of one part. 750 ml is shared into 5 equal parts, so divide it by 5.750 ml ÷ 5 = 150 ml 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.Cordial: 1 × 150 ml = 150 ml
Water: 4 × 150 ml = 600 ml
Check. Add the shares: 150 ml + 600 ml = 750 ml, which is the total 750 ml. ✓
Watch out. Do not divide the total by one of the ratio numbers. Divide by the total number of parts.
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 rate tells us that each £1 is worth €1.15. To get euros, multiply the pounds by 1.15. To get pounds, divide the euros by 1.15.
- Write the rate. The ratio £ : € is 1 : 1.15.£1 = €1.15
- (a) Multiply the pounds by 1.15. There are 80 lots of £1, each worth €1.15.80 × 1.15 = €92
- (b) Divide the euros by 1.15. We want to know how many lots of €1.15 are in €138. Each lot is £1.138 ÷ 1.15 = £120
Check. £120 × 1.15 = €138, which takes us back to €138. ✓
Watch out. Multiply to go to the bigger-numbered currency here (£ to €) and divide to come back. Check that your answer is sensible: €92 is more than £80, as it should be.
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. A scale of 1 : 25 000 means every 1 cm on the map is 25 000 cm in real life. Multiply, then change the units.
- Multiply the map length by the scale. Each 1 cm is 25 000 cm in real life.4 × 25 000 = 100 000 cm
- Change centimetres to metres. 100 cm = 1 m, so divide by 100.100 000 ÷ 100 = 1000 m
- Change metres to kilometres. 1000 m = 1 km, so divide by 1000.1000 ÷ 1000 = 1 km
Check. 1 km = 100 000 cm and 100 000 ÷ 25 000 = 4 cm. ✓
Watch out. The answer is in the unit of the map measurement (cm) until you convert it.
Watch out
A mixing ratio such as 1 : 4 means 1 part cordial and 4 parts water, so the cordial is 15 of the drink, not 14. And when you scale a recipe, scale every ingredient by the same number.Exercise 15F Ratio in context
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. Find the multiplier and multiply every ingredient by it.
- Find the multiplier from 4 to 8. To keep the recipe tasting the same, every ingredient is multiplied by the same number.8 ÷ 4 = 2
- Multiply every ingredient by the multiplier. Multiplying all the amounts by the same number keeps the ratios between them the same.Flour: 200 g × 2 = 400 g
Sugar: 80 g × 2 = 160 g
Butter: 120 g × 2 = 240 g
Eggs: 2 × 2 = 4
Check. Check the first ingredient in reverse: 400 ÷ 2 = 200, the original amount of flour. ✓
Watch out. Scale every ingredient, not just the main one, and keep the units.
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 multiplier and multiply every ingredient by it.
- Find the multiplier from 4 to 12. To keep the recipe tasting the same, every ingredient is multiplied by the same number.12 ÷ 4 = 3
- Multiply every ingredient by the multiplier. Multiplying all the amounts by the same number keeps the ratios between them the same.Flour: 200 g × 3 = 600 g
Sugar: 80 g × 3 = 240 g
Butter: 120 g × 3 = 360 g
Eggs: 2 × 3 = 6
Check. Check the first ingredient in reverse: 600 ÷ 3 = 200, the original amount of flour. ✓
Watch out. Scale every ingredient, not just the main one, and keep the units.
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 amounts for one (the unitary method), then multiply by the number needed.
- Find the amounts for 1 people. The unitary method: divide every amount by the number it serves now.Flour: 200 g ÷ 4 = 50 g
Sugar: 80 g ÷ 4 = 20 g
Butter: 120 g ÷ 4 = 30 g
Eggs: 2 ÷ 4 = 0.5 - Multiply by 2 for 2 people. Multiply the amount for one by the number needed.Flour: 50 g × 2 = 100 g
Sugar: 20 g × 2 = 40 g
Butter: 30 g × 2 = 60 g
Eggs: 0.5 × 2 = 1 - Write the amounts needed. These are the new amounts, with the units.Flour: 100 g
Sugar: 40 g
Butter: 60 g
Eggs: 1
Check. Check the first ingredient in reverse: 100 ÷ 0.5 = 200, the original amount of flour. ✓
Watch out. Scale every ingredient, not just the main one, and keep the units.
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 amounts for one (the unitary method), then multiply by the number needed.
- Find the amounts for 1 people. The unitary method: divide every amount by the number it serves now.Flour: 200 g ÷ 4 = 50 g
Sugar: 80 g ÷ 4 = 20 g
Butter: 120 g ÷ 4 = 30 g
Eggs: 2 ÷ 4 = 0.5 - Multiply by 6 for 6 people. Multiply the amount for one by the number needed.Flour: 50 g × 6 = 300 g
Sugar: 20 g × 6 = 120 g
Butter: 30 g × 6 = 180 g
Eggs: 0.5 × 6 = 3 - Write the amounts needed. These are the new amounts, with the units.Flour: 300 g
Sugar: 120 g
Butter: 180 g
Eggs: 3
Check. Check the first ingredient in reverse: 300 ÷ 1.5 = 200, the original amount of flour. ✓
Watch out. Scale every ingredient, not just the main one, and keep the units.
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. Add the parts to find how many equal parts there are, find the value of one part, then multiply.
- 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.2 + 5 = 7 parts
- Find the value of one part. 14 litres is shared into 7 equal parts, so divide it by 7.14 litres ÷ 7 = 2 litres 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.red: 2 × 2 litres = 4 litres
white: 5 × 2 litres = 10 litres
Check. Add the shares: 4 litres + 10 litres = 14 litres, which is the total 14 litres. ✓
Watch out. Do not divide the total by one of the ratio numbers. 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.concentrate has 1 part, worth 250 ml
- Find the value of one part. 1 part is worth 250 ml, so divide by 1.250 ml ÷ 1 = 250 ml for one part
- Multiply to find what the question asks for. Multiply the value of one part by the number of parts needed.water: 4 × 250 ml = 1000 ml
Total: 1 + 4 = 5 parts, so 5 × 250 ml = 1250 ml
Check. Check the share we were given: 1 × 250 ml = 250 ml, 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. Add the parts to find how many equal parts there are, find the value of one part, then multiply.
- 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.1 + 2 + 4 = 7 parts
- Find the value of one part. 56 kg is shared into 7 equal parts, so divide it by 7.56 kg ÷ 7 = 8 kg 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.cement: 1 × 8 kg = 8 kg
sand: 2 × 8 kg = 16 kg
gravel: 4 × 8 kg = 32 kg
Check. Add the shares: 8 kg + 16 kg + 32 kg = 56 kg, which is the total 56 kg. ✓
Watch out. Do not divide the total by one of the ratio numbers. Divide by the total number of parts.
Method. Multiply the pounds by 1.15 to get euros.
- Write the rate. £1 buys €1.15.£1 = €1.15
- Multiply. 80 pounds is 80 lots of €1.15.80 × 1.15 = 92
Check. 92 ÷ 1.15 = 80. ✓
Watch out. Check that the answer is bigger, since each pound buys more than 1 euro.
Method. Multiply the pounds by 1.15 to get euros.
- Write the rate. £1 buys €1.15.£1 = €1.15
- Multiply. 250 pounds is 250 lots of €1.15.250 × 1.15 = 287.5
Check. 287.5 ÷ 1.15 = 250. ✓
Watch out. Check that the answer is bigger.
Method. Divide the euros by 1.15 to get pounds.
- Write the rate. £1 is worth €1.15.£1 = €1.15
- Divide. We want the number of lots of €1.15 in €138.138 ÷ 1.15 = 120
Check. 120 × 1.15 = 138. ✓
Watch out. Divide when converting back to pounds.
Method. Divide the euros by 1.15 to get pounds.
- Write the rate. £1 is worth €1.15.£1 = €1.15
- Divide. We want the number of lots of €1.15 in €46.46 ÷ 1.15 = 40
Check. 40 × 1.15 = 46. ✓
Watch out. Divide when converting back to pounds.
Method. Multiply the map length by the scale, then change the units.
- Multiply by 50 000. 1 cm on the map is 50 000 cm in real life.6 × 50 000 = 300 000 cm
- Change to metres. Divide by 100.300 000 ÷ 100 = 3000 m
- Change to kilometres. Divide by 1000.3000 ÷ 1000 = 3 km
Check. 3 km = 300 000 cm and 300 000 ÷ 50 000 = 6 cm. ✓
Watch out. Do not forget to change centimetres into kilometres.
Method. Change the real length into centimetres, then divide by the scale.
- Change 2.5 km to metres. 1 km = 1000 m.2.5 × 1000 = 2500 m
- Change to centimetres. 1 m = 100 cm.2500 × 100 = 250 000 cm
- Divide by 50 000. Every 50 000 cm in real life is 1 cm on the map.250 000 ÷ 50 000 = 5 cm
Check. 5 cm on the map is 5 × 50 000 = 250 000 cm = 2.5 km. ✓
Watch out. To go from real life to the map, divide by the scale.
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. Add the parts to find how many equal parts there are, find the value of one part, then multiply.
- 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.1 + 9 = 10 parts
- Find the value of one part. 500 ml is shared into 10 equal parts, so divide it by 10.500 ml ÷ 10 = 50 ml 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.cleaner: 1 × 50 ml = 50 ml
water: 9 × 50 ml = 450 ml
Check. Add the shares: 50 ml + 450 ml = 500 ml, which is the total 500 ml. ✓
Watch out. Do not divide the total by one of the ratio numbers. Divide by the total number of parts.
Method. The stronger drink is the one with the larger fraction of squash. Work out the fraction of each drink that is squash.
- Drink A: add the parts. There are 1 + 4 parts.1 + 4 = 5
- Drink A: fraction of squash. Squash is 1 part out of 5.15
- Drink B: add the parts. There are 1 + 5 parts.1 + 5 = 6
- Drink B: fraction of squash. Squash is 1 part out of 6.16
- Compare. Unit fractions with a smaller denominator are bigger (the pieces are bigger).15 > 16 so Drink A is stronger.
Check. As decimals: 1/5 = 0.2 and 1/6 = 0.1666... 0.2 is bigger. ✓
Watch out. Compare the fractions of the whole drink, not the water numbers 4 and 5 alone.
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 multiplier for the butter, then multiply the number of cupcakes by it.
- Find the multiplier. The amount of butter has been multiplied by the same number as the number of cupcakes.450 ÷ 150 = 3
- Multiply the cupcakes by 3. The recipe is made 3 times bigger.12 × 3 = 36
Check. 36 cupcakes need 36 ÷ 12 = 3 lots of the recipe, and 3 × 150 = 450 g of butter. ✓
Watch out. Both quantities are multiplied by the same number.
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.plant food has 1 part, worth 100 ml
- Find the value of one part. 1 part is worth 100 ml, so divide by 1.100 ml ÷ 1 = 100 ml for one part
- Multiply to find what the question asks for. Multiply the value of one part by the number of parts needed.water: 24 × 100 ml = 2400 ml
Total: 1 + 24 = 25 parts, so 25 × 100 ml = 2500 ml
Check. Check the share we were given: 1 × 100 ml = 100 ml, 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.
Method. Change the US price into pounds so that both prices are in the same currency, then compare.
- Write the rate. Each £1 is worth $1.25.£1 = $1.25
- Change $30 to pounds. Divide the dollars by 1.25, because we are going from dollars to pounds.30 ÷ 1.25 = £24
- Compare. £24 is more than £22, so the UK shirt is cheaper.£24 − £22 = £2
Check. £22 is $22 × 1.25 = $27.50, which is less than $30. ✓
Watch out. Never compare $30 with £22 directly. Change to one currency first.
Method. Check which colour runs out first. Find how much of the other colour is needed to match each tin.
- Try using all the red. Red is 3 parts, so 3 parts = 600 ml and one part = 200 ml.600 ÷ 3 = 200 ml per part
- Yellow needed. Yellow is 2 parts.2 × 200 = 400 ml
- Is there enough yellow? Ella has 500 ml of yellow, which is more than 400 ml. So the red runs out first.500 ml > 400 ml: yes
- Total paint made. Red 600 ml and yellow 400 ml.600 + 400 = 1000 ml
Check. If she tried to use all the yellow, she would need 3 × 250 = 750 ml of red, but she has only 600 ml. ✓
Watch out. Do not add 600 and 500. She cannot use all of the yellow paint.
Method. Find the total volume, change it to the same unit as the answer, then find the share that is concentrate.
- Find the total volume in ml. 24 glasses each hold 250 ml.24 × 250 = 6000 ml
- Add the parts. The ratio 1 : 4 has 5 parts.1 + 4 = 5
- Find one part. Share the 6000 ml into 5 equal parts.6000 ÷ 5 = 1200 ml
- Concentrate is 1 part. Then change to litres: 1000 ml = 1 litre.1200 ml = 1200 ÷ 1000 = 1.2 litres
Check. Water is 4 × 1200 = 4800 ml, and 1200 + 4800 = 6000 ml. ✓
Watch out. The ratio 1 : 4 means 5 parts altogether; do not divide by 4.
Method. The amount of concentrate stays the same. Find it, then work out the water needed for the new ratio.
- Change to millilitres. 1 litre = 1000 ml.1.2 × 1000 = 1200 ml
- Find the concentrate. The ratio 1 : 5 has 6 parts, so concentrate is one sixth of 1200 ml.1200 ÷ 6 = 200 ml concentrate
Water = 5 × 200 = 1000 ml - Find the water needed for 1 : 8. The concentrate is still 200 ml, which is 1 part, so water must be 8 parts.8 × 200 = 1600 ml
- Find the water to add. New water amount minus the water already in the jug.1600 − 1000 = 600 ml
Check. New jug: 200 ml concentrate and 1600 ml water, ratio 200 : 1600 = 1 : 8. ✓
Watch out. Only the water changes. The concentrate stays at 200 ml.