Ratio Mixture Problems

Use a ratio as a recipe: find which ingredient runs out, change a mixture to a new ratio, and combine two mixtures.

In the exam for Cambridge IGCSE Maths 0580, Edexcel IGCSE Maths 4MA1 · 4 real past-paper questions below

A Ratio Is a Recipe

A mixture ratio tells you how much of each thing goes in. Squash in the ratio juice to water $2 : 7$ means 2 cups of juice for every 7 cups of water.

Juice	Water
2 litres	7 litres
4 litres	14 litres
6 litres	21 litres

Rule

To make more, multiply both numbers by the same amount. The drink then tastes the same.

Key idea

Gold is measured in 24 parts called carats. 18-carat gold is gold to other metals in the ratio $18 : 6 = 3 : 1$ , so it is $\frac{3}{4}$ or 75% gold. 21-carat gold, a common choice in the Gulf, is $21 : 3 = 7 : 1$ .

Which Ingredient Runs Out?

The 2-mark question: you have some of each ingredient. Find the most mixture you can make, or say if someone is right.

A drink is juice to water $3 : 5$ . You have 12 litres of juice and 25 litres of water. What is the most drink you can make?

Worked example

1
$12 \div 3 = 4$ — Juice is 3 parts. 12 litres of juice is enough for parts of 4 litres.
2
$25 \div 5 = 5$ — Water is 5 parts. 25 litres of water is enough for parts of 5 litres.
3
$one part = 4 litres$ — Take the smaller. Parts of 5 litres would need 3 × 5 = 15 litres of juice, and you only have 12. The juice runs out first.
4
$3 + 5 = 8 parts$ — The whole drink is all the parts added together.
5
$8 \times 4 = 32 litres$ — 8 parts of 4 litres each. This is the most drink you can make.
6
$5 \times 4 = 20 litres of water$ — Water used is 5 parts of 4 litres.
7
$25 - 20 = 5 litres spare$ — You had 25 litres of water, so 5 litres are left over.

Common mistake

Giving $12 + 25 = 37$ litres. The drink would be too weak. The spare water cannot be used.

Key idea

If asked "Is she correct?", work it out, then write a reason with your numbers: "No, the juice only makes 32 litres."

Key idea

Show the division for every ingredient, even the ones that do not run out. Most of the marks are for these lines, so an answer with no working can score 0.

Recipes: The Most You Can Make

The 4-mark question: a recipe makes a set number of items, and you have some of each ingredient. Work out the most items you can make.

Cookies for 8: 150 g butter, 100 g sugar, 250 g flour. You have 400 g of butter, 0.5 kg of sugar and 1 kg of flour. How many cookies can you make?

Worked example

1
$0.5 kg = 500 g, 1 kg = 1000 g$ — Change everything to grams first, so you divide grams by grams.
2
$butter: 400 \div 150 = 2.66 \dots$ — Enough butter for 2.66… batches of the recipe. Keep the full decimal.
3
$sugar: 500 \div 100 = 5$ — Enough sugar for 5 batches.
4
$flour: 1000 \div 250 = 4$ — Enough flour for 4 batches.
5
$batches = 2.66 \dots$ — Take the smallest. The butter runs out first, so it decides.
6
$2.66 \dots \times 8 = 21.3 \dots$ — Each batch makes 8 cookies.
7
$21 cookies$ — Round down. 22 cookies would need 412.5 g of butter, and you only have 400 g.

Common mistake

Rounding 21.3 up to 22, or using the ingredient you have most of. The one that runs out first decides, and you cannot make part of a cookie.

Changing the Ratio

The 3-mark question: add more of one ingredient to reach a new ratio. The one you do not add stays the same.

40 litres of syrup to water in the ratio $1 : 4$ . How much water must you add to make it $1 : 6$ ?

Worked example

1
$1 + 4 = 5 parts$ — Syrup is 1 part and water is 4 parts.
2
$40 \div 5 = 8$ — 40 litres shared into 5 parts, so one part is 8 litres.
3
$syrup = 1 \times 8 = 8$ — Syrup is 1 part, so there are 8 litres now.
4
$water = 4 \times 8 = 32$ — Water is 4 parts, so there are 32 litres now.
5
$syrup stays 8$ — You only add water, so the syrup does not change. It is the fixed amount you build on.
6
$new water = 6 \times 8 = 48$ — In 1 : 6 the water is 6 times the syrup.
7
$48 - 32 = 16 litres$ — New water take away the water already there gives the water to add.
8
$8 : 48 = 1 : 6$ — Check: divide both by 8. It is the ratio we wanted.

Common mistake

Sharing 40 in the new ratio. Adding water changes the total, so start from the part that stays fixed.

When Both Amounts Change

Sometimes both amounts change: you add a known number to one and an unknown number to the other. Find the new amount of the one you know first.

A box has 56 pens, blue to black $3 : 5$ . 6 more blue pens and $x$ more black pens go in. Now blue to black is $1 : 2$ . Find $x$ .

Worked example

1
$3 + 5 = 8 parts$ — Blue is 3 parts and black is 5 parts.
2
$56 \div 8 = 7$ — 56 pens shared into 8 parts, so one part is 7 pens.
3
$blue = 3 \times 7 = 21$ — Blue is 3 parts.
4
$black = 5 \times 7 = 35$ — Black is 5 parts, so there are 35 black pens before any are added.
5
$new blue = 21 + 6 = 27$ — You know exactly how many blue pens were added, so blue is fixed at 27.
6
$new black = 2 \times 27 = 54$ — In 1 : 2 black is twice blue.
7
$x = 54 - 35 = 19$ — New black take away the old black is the number added.
8
$27 : 54 = 1 : 2$ — Check: divide both by 27.

Key idea

The lines 21 and 35 each earn a mark for working, even if a later step goes wrong. Write them down.

Same Amount Added (Extended/Higher)

When the same unknown amount is added to both, write the new ratio as an equation and solve it.

A club has 24 girls and 40 boys. The same number, $x$ , of girls and of boys join. Now girls to boys is $5 : 7$ . Find $x$ .

Worked example

1
$(24 + x) : (40 + x) = 5 : 7$ — Write the new amounts. x is added to both.
2
$\frac{24 + x}{40 + x} = \frac{5}{7}$ — A ratio a : b can be written as the fraction a ÷ b, which gives an equation.
3
$7 (24 + x) = 5 (40 + x)$ — Cross-multiply: multiply both sides by 7 and by (40 + x) to clear the fractions.
4
$168 + 7 x = 200 + 5 x$ — Expand both brackets: 7 × 24 = 168 and 5 × 40 = 200.
5
$2 x + 168 = 200$ — Take 5x from both sides.
6
$2 x = 32$ — Take 168 from both sides.
7
$x = 16$ — Divide both sides by 2.
8
$40 : 56 = 5 : 7$ — Check: 24 + 16 = 40 and 40 + 16 = 56. Divide both by 8.

Key idea

A shortcut: adding the same amount to both never changes the gap. The gap is 40 − 24 = 16, and in 5 : 7 the gap is 2 parts, so one part is 8. Girls end at 5 × 8 = 40, so x = 40 − 24 = 16.

Combining Two Mixtures

The 3-mark question: pour two mixtures together and find the new ratio. Turn each ratio into amounts first.

Jug A has 30 litres of juice to water $1 : 2$ . Jug B has 20 litres of $3 : 1$ . Both go in one bowl.

Worked example

1
$A: 1 + 2 = 3 parts$ — Jug A is 1 part juice and 2 parts water.
2
$A: 30 \div 3 = 10$ — 30 litres shared into 3 parts, so one part of jug A is 10 litres.
3
$A: juice 10, water 20$ — Juice is 1 × 10 and water is 2 × 10.
4
$B: 3 + 1 = 4 parts$ — Jug B is 3 parts juice and 1 part water.
5
$B: 20 \div 4 = 5$ — 20 litres shared into 4 parts, so one part of jug B is 5 litres.
6
$B: juice 15, water 5$ — Juice is 3 × 5 and water is 1 × 5.
7
$juice = 10 + 15 = 25$ — Add the juice from both jugs.
8
$water = 20 + 5 = 25$ — Add the water from both jugs.
9
$25 : 25 = 1 : 1$ — Divide both by 25 to simplify. The bowl is half juice, half water.

Common mistake

Adding the ratios: $1 : 2$ and $3 : 1$ make $4 : 3$ . Ratios are not amounts. Find the litres first.

Which Method? Read the Question

Look at what you are given: amounts you have, a new ratio, or two mixtures.

You see	You do
You have some of each ingredient	one part from each, use the smaller
A recipe for a number of items	same units, batches from each, smallest, round down
Add more to make a new ratio	keep the other ingredient fixed
Amounts added to both	fix the one you know, or write an equation
Two mixtures poured together	find the amounts, add, then simplify
Share a total in a ratio	Dividing in a Given Ratio

Key idea

Always turn ratios into real amounts before you add or compare them.

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