Proportion

What you'll learn

What proportion means in recipe questions.
How to scale ingredients up or down using a multiplier.
How to find the maximum number you can make from the ingredients available.
How to spot the ingredient that runs out first.

What is proportion?

Proportion is about keeping the same balance between quantities.

In recipe questions, if you make more cakes, you need more of every ingredient. If you make fewer cakes, you need less of every ingredient.

Definition

Direct proportion

Two quantities are in direct proportion if they increase or decrease by the same multiplier. For example, doubling the number of cakes means doubling every ingredient.

Direct proportion means the number made and every ingredient are scaled by the same multiplier.

Key Idea

Recipe proportion

In a recipe, the ingredients and the number made must be scaled by the same multiplier.

Scaling up recipes

To scale up means to make a larger amount than the original recipe.

A multiplier tells you what to multiply by. For example, going from 10 flapjacks to 15 flapjacks uses the multiplier:

1510=1.5\frac{15}{10}=1.5

So every ingredient is multiplied by 1.5.

Example

Scaling a recipe up

A recipe makes 10 oat bars.

70 g oats
50 g butter
20 ml syrup
40 g sugar

Work out the ingredients needed to make 15 oat bars.

The recipe for 10 oat bars is scaled to 15 oat bars using the same multiplier for every ingredient.

Find the multiplier from 10 to 15:
$1510=1.5\frac{15}{10}=1.5$
Multiply each ingredient by 1.5:
$70×1.5=10550×1.5=7520×1.5=3040×1.5=60\begin{aligned} 70 \times 1.5 &= 105\\ 50 \times 1.5 &= 75\\ 20 \times 1.5 &= 30\\ 40 \times 1.5 &= 60 \end{aligned}$
Write the answer with units: 105 g oats, 75 g butter, 30 ml syrup, and 60 g sugar.

Common Mistake

Only changing one ingredient

If the number made changes, every ingredient must change by the same multiplier, not just one of them.

Scaling by using a simple fraction

Sometimes the new amount is easy to compare with the old amount.

For example, going from 30 biscuits to 45 biscuits is the same as multiplying by $4530\frac{45}{30}$ , which simplifies to $32\frac{3}{2}$ .

That means you can do:

Divide by 2.
Multiply by 3.

Example

Making more biscuits

A recipe makes 30 biscuits using:

240 g flour
150 g butter
90 g sugar
2 eggs

Work out the ingredients needed for 45 biscuits.

Going from 30 biscuits to 45 biscuits is multiplying the whole recipe by one and a half.

Find the multiplier:
$4530=32=1.5\frac{45}{30}=\frac{3}{2}=1.5$
Multiply the flour, butter and sugar by 1.5:
$240×1.5=360150×1.5=22590×1.5=135\begin{aligned} 240 \times 1.5 &= 360\\ 150 \times 1.5 &= 225\\ 90 \times 1.5 &= 135 \end{aligned}$
Multiply the eggs by 1.5:
$\times 1.5 = 3$
The ingredients are 360 g flour, 225 g butter, 135 g sugar, and 3 eggs.

Tip

Use the numbers that feel easiest

Multiplying by 1.5 is the same as finding one and a half lots. For example, 240 g becomes 240 g + 120 g = 360 g.

Scaling down recipes

To scale down means to make a smaller amount than the original recipe.

You still use the same idea:

made\text{new amount}=\text{old amount}\times\frac{\text{new number made}}{\text{old number made}}

Example

Scaling down flour

A recipe uses 200 g flour to make 12 small cakes. Work out how much flour is needed for 18 small cakes.

The amount of flour changes in the same ratio as the number of small cakes.

Find the multiplier from 12 to 18:
$1812=1.5\frac{18}{12}=1.5$
Multiply the flour by 1.5:
$200 \times 1.5 = 300$
So 300 g of flour is needed.

This example is technically scaling up, but the same method also works when the multiplier is less than 1. For example, from 12 cakes to 6 cakes, the multiplier would be $612=12\frac{6}{12}=\frac{1}{2}$ .

Finding how many can be made

Sometimes you are told how much of one ingredient was used, and you need to work out how many items were made.

Here, compare the ingredient amount used with the ingredient amount in the recipe.

Example

Using milk to find the number made

A recipe makes 12 shortcakes using 15 ml of milk. A baker uses 40 ml of milk. How many shortcakes does the baker make?

Compare the milk used with the recipe milk, then apply the same multiplier to the number of shortcakes.

Find the multiplier for the milk:
$4015=83\frac{40}{15}=\frac{8}{3}$
Multiply the number of shortcakes by the same multiplier:
$12 \times \frac{8}{3}=32$
The baker makes 32 shortcakes.

Common Mistake

Using the wrong way round

If you are finding the multiplier, put new amount divided by old amount. For example, used milk divided by recipe milk.

Maximum number from one ingredient

Sometimes you only need to check one ingredient.

Example

Maximum cakes from flour

A recipe needs 225 g flour to make 9 cakes. A student has 500 g flour. Work out the maximum number of cakes they can make.

Finding the flour needed for one cake helps work out the maximum number of whole cakes possible.

Find how much flour is needed for 1 cake:
$225 \div 9 = 25$
Divide the flour available by the flour needed per cake:
$500 \div 25 = 20$
The maximum number of cakes is 20.

Tip

Per one method

Finding the amount for 1 item is often the clearest method when the question asks for the maximum number.

Checking if there is enough

Some questions ask whether someone has enough of an ingredient.

You should work out how much is needed first, then compare it with how much they have.

Example

Does she have enough flour?

A recipe needs 180 g flour to make 8 cakes. Aisha wants to make 20 cakes. She has 430 g flour. Does she have enough?

First scale the flour needed for 20 cakes, then compare it with the flour Aisha has.

Find the multiplier from 8 cakes to 20 cakes:
$208=2.5\frac{20}{8}=2.5$
Work out the flour needed:
$180 \times 2.5 = 450$
Compare 450 g needed with 430 g available.
She does not have enough flour, because she is short by 20 g.

Several ingredients: which one runs out first?

When you are given several ingredients available, check each one separately.

The limiting ingredient is the ingredient that runs out first. It decides the greatest number you can make.

Definition

Limiting ingredient

The limiting ingredient is the ingredient that allows you to make the fewest complete items.

Example

Greatest number of shortcakes

A recipe makes 10 shortcakes using:

50 g sugar
200 g butter
150 g flour
25 ml milk

Sam has:

180 g sugar
900 g butter
600 g flour
120 ml milk

Work out the greatest number of shortcakes Sam can make.

The limiting ingredient is the one that allows the fewest complete shortcakes.

Find how many shortcakes the sugar allows. The recipe has enough sugar for 10 shortcakes:
$180 \div 50 = 3.6$
So sugar allows 3.6 lots of the recipe:
$3.6 \times 10 = 36$
Check butter:
$900 \div 200 = 4.5$
Butter allows 45 shortcakes.
Check flour:
$600 \div 150 = 4$
Flour allows 40 shortcakes.
Check milk:
$120 \div 25 = 4.8$
Milk allows 48 shortcakes.
The smallest number is 36, so Sam can make at most 36 shortcakes.

Common Mistake

Complete items only

If the answer is not a whole number, round down for the maximum number you can make. You cannot make part of a biscuit in this type of question.

Unit conversions

You may need to convert units before comparing.

Common conversions:

1 kg = 1000 g
1 litre = 1000 ml

Example

Changing kilograms to grams

A recipe uses 250 g flour to make 20 biscuits. Ben has 1 kg of flour. What is the maximum number of biscuits he can make?

Convert 1 kg into grams:
$g1\text{ kg}=1000\text{ g}$
Find how many lots of the recipe he can make:
$1000 \div 250 = 4$
Multiply by the number of biscuits in one recipe:
$\times 20 = 80$
Ben can make 80 biscuits.

Exam technique

In the exam

Write down the multiplier clearly, such as $number\frac{\text{new number}}{\text{old number}}$ .
Keep units with your answers: g, ml, kg, or number of eggs.
For “greatest number” questions, check every ingredient and choose the smallest possible number.

Self review

Check yourself

If a recipe for 10 cakes is changed to 25 cakes, what multiplier should you use?
Why must every ingredient be multiplied by the same number?
In a maximum-number question, why do you choose the smallest result from the ingredients?

Proportion

What you'll learn

What proportion means in recipe questions.
How to scale ingredients up or down using a multiplier.
How to find the maximum number you can make from the ingredients available.
How to spot the ingredient that runs out first.

What is proportion?

Proportion is about keeping the same balance between quantities.

In recipe questions, if you make more cakes, you need more of every ingredient. If you make fewer cakes, you need less of every ingredient.

Definition

Direct proportion

Two quantities are in direct proportion if they increase or decrease by the same multiplier. For example, doubling the number of cakes means doubling every ingredient.

Direct proportion means the number made and every ingredient are scaled by the same multiplier.

Key Idea

Recipe proportion

In a recipe, the ingredients and the number made must be scaled by the same multiplier.

Scaling up recipes

To scale up means to make a larger amount than the original recipe.

A multiplier tells you what to multiply by. For example, going from 10 flapjacks to 15 flapjacks uses the multiplier:

1510=1.5\frac{15}{10}=1.5

So every ingredient is multiplied by 1.5.

Example

Scaling a recipe up

A recipe makes 10 oat bars.

70 g oats
50 g butter
20 ml syrup
40 g sugar

Work out the ingredients needed to make 15 oat bars.

The recipe for 10 oat bars is scaled to 15 oat bars using the same multiplier for every ingredient.

Find the multiplier from 10 to 15:
$1510=1.5\frac{15}{10}=1.5$
Multiply each ingredient by 1.5:
$70×1.5=10550×1.5=7520×1.5=3040×1.5=60\begin{aligned} 70 \times 1.5 &= 105\\ 50 \times 1.5 &= 75\\ 20 \times 1.5 &= 30\\ 40 \times 1.5 &= 60 \end{aligned}$
Write the answer with units: 105 g oats, 75 g butter, 30 ml syrup, and 60 g sugar.

Common Mistake

Only changing one ingredient

If the number made changes, every ingredient must change by the same multiplier, not just one of them.

Scaling by using a simple fraction

Sometimes the new amount is easy to compare with the old amount.

For example, going from 30 biscuits to 45 biscuits is the same as multiplying by $4530\frac{45}{30}$ , which simplifies to $32\frac{3}{2}$ .

That means you can do:

Divide by 2.
Multiply by 3.

Example

Making more biscuits

A recipe makes 30 biscuits using:

240 g flour
150 g butter
90 g sugar
2 eggs

Work out the ingredients needed for 45 biscuits.

Going from 30 biscuits to 45 biscuits is multiplying the whole recipe by one and a half.

Find the multiplier:
$4530=32=1.5\frac{45}{30}=\frac{3}{2}=1.5$
Multiply the flour, butter and sugar by 1.5:
$240×1.5=360150×1.5=22590×1.5=135\begin{aligned} 240 \times 1.5 &= 360\\ 150 \times 1.5 &= 225\\ 90 \times 1.5 &= 135 \end{aligned}$
Multiply the eggs by 1.5:
$\times 1.5 = 3$
The ingredients are 360 g flour, 225 g butter, 135 g sugar, and 3 eggs.

Tip

Use the numbers that feel easiest

Multiplying by 1.5 is the same as finding one and a half lots. For example, 240 g becomes 240 g + 120 g = 360 g.

Scaling down recipes

To scale down means to make a smaller amount than the original recipe.

You still use the same idea:

made\text{new amount}=\text{old amount}\times\frac{\text{new number made}}{\text{old number made}}

Example

Scaling down flour

A recipe uses 200 g flour to make 12 small cakes. Work out how much flour is needed for 18 small cakes.

The amount of flour changes in the same ratio as the number of small cakes.

Find the multiplier from 12 to 18:
$1812=1.5\frac{18}{12}=1.5$
Multiply the flour by 1.5:
$200 \times 1.5 = 300$
So 300 g of flour is needed.

Finding how many can be made

Sometimes you are told how much of one ingredient was used, and you need to work out how many items were made.

Here, compare the ingredient amount used with the ingredient amount in the recipe.

Example

Using milk to find the number made

A recipe makes 12 shortcakes using 15 ml of milk. A baker uses 40 ml of milk. How many shortcakes does the baker make?

Compare the milk used with the recipe milk, then apply the same multiplier to the number of shortcakes.

Find the multiplier for the milk:
$4015=83\frac{40}{15}=\frac{8}{3}$
Multiply the number of shortcakes by the same multiplier:
$12 \times \frac{8}{3}=32$
The baker makes 32 shortcakes.

Common Mistake

Using the wrong way round

If you are finding the multiplier, put new amount divided by old amount. For example, used milk divided by recipe milk.

Maximum number from one ingredient

Sometimes you only need to check one ingredient.

Example

Maximum cakes from flour

A recipe needs 225 g flour to make 9 cakes. A student has 500 g flour. Work out the maximum number of cakes they can make.

Finding the flour needed for one cake helps work out the maximum number of whole cakes possible.

Find how much flour is needed for 1 cake:
$225 \div 9 = 25$
Divide the flour available by the flour needed per cake:
$500 \div 25 = 20$
The maximum number of cakes is 20.

Tip

Per one method

Finding the amount for 1 item is often the clearest method when the question asks for the maximum number.

Checking if there is enough

Some questions ask whether someone has enough of an ingredient.

You should work out how much is needed first, then compare it with how much they have.

Example

Does she have enough flour?

A recipe needs 180 g flour to make 8 cakes. Aisha wants to make 20 cakes. She has 430 g flour. Does she have enough?

First scale the flour needed for 20 cakes, then compare it with the flour Aisha has.

Find the multiplier from 8 cakes to 20 cakes:
$208=2.5\frac{20}{8}=2.5$
Work out the flour needed:
$180 \times 2.5 = 450$
Compare 450 g needed with 430 g available.
She does not have enough flour, because she is short by 20 g.

Several ingredients: which one runs out first?

When you are given several ingredients available, check each one separately.

The limiting ingredient is the ingredient that runs out first. It decides the greatest number you can make.

Definition

Limiting ingredient

The limiting ingredient is the ingredient that allows you to make the fewest complete items.

Example

Greatest number of shortcakes

A recipe makes 10 shortcakes using:

50 g sugar
200 g butter
150 g flour
25 ml milk

Sam has:

180 g sugar
900 g butter
600 g flour
120 ml milk

Work out the greatest number of shortcakes Sam can make.

The limiting ingredient is the one that allows the fewest complete shortcakes.

Find how many shortcakes the sugar allows. The recipe has enough sugar for 10 shortcakes:
$180 \div 50 = 3.6$
So sugar allows 3.6 lots of the recipe:
$3.6 \times 10 = 36$
Check butter:
$900 \div 200 = 4.5$
Butter allows 45 shortcakes.
Check flour:
$600 \div 150 = 4$
Flour allows 40 shortcakes.
Check milk:
$120 \div 25 = 4.8$
Milk allows 48 shortcakes.
The smallest number is 36, so Sam can make at most 36 shortcakes.

Common Mistake

Complete items only

If the answer is not a whole number, round down for the maximum number you can make. You cannot make part of a biscuit in this type of question.

Unit conversions

You may need to convert units before comparing.

Common conversions:

1 kg = 1000 g
1 litre = 1000 ml

Example

Changing kilograms to grams

A recipe uses 250 g flour to make 20 biscuits. Ben has 1 kg of flour. What is the maximum number of biscuits he can make?

Convert 1 kg into grams:
$g1\text{ kg}=1000\text{ g}$
Find how many lots of the recipe he can make:
$1000 \div 250 = 4$
Multiply by the number of biscuits in one recipe:
$\times 20 = 80$
Ben can make 80 biscuits.

Exam technique

In the exam

Write down the multiplier clearly, such as $number\frac{\text{new number}}{\text{old number}}$ .
Keep units with your answers: g, ml, kg, or number of eggs.
For “greatest number” questions, check every ingredient and choose the smallest possible number.

Self review

Check yourself

If a recipe for 10 cakes is changed to 25 cakes, what multiplier should you use?
Why must every ingredient be multiplied by the same number?
In a maximum-number question, why do you choose the smallest result from the ingredients?

Proportion

What you'll learn

What is proportion?

Scaling up recipes

Scaling by using a simple fraction

Scaling down recipes

Finding how many can be made

Maximum number from one ingredient

Checking if there is enough

Several ingredients: which one runs out first?

Unit conversions

Recap questions

You've reached the end

Proportion

What you'll learn

What is proportion?

Scaling up recipes

Scaling by using a simple fraction

Scaling down recipes

Finding how many can be made

Maximum number from one ingredient

Checking if there is enough

Several ingredients: which one runs out first?

Unit conversions

Recap questions

You've reached the end

Proportion

What you'll learn

What is proportion?

Scaling up recipes

Scaling by using a simple fraction

Scaling down recipes

Finding how many can be made

Maximum number from one ingredient

Checking if there is enough

Several ingredients: which one runs out first?

Unit conversions

Recap questions

You've reached the end

Proportion Revision Guide

Proportion

What you'll learn

What is proportion?

Scaling up recipes

Scaling by using a simple fraction

Scaling down recipes

Finding how many can be made

Maximum number from one ingredient

Checking if there is enough

Several ingredients: which one runs out first?

Unit conversions

Recap questions

You've reached the end

Proportion Revision Guide