Ratio and Proportion Problems

Ratio problems in the real world

A ratio problem asks you to compare or share quantities fairly. You meet them constantly: splitting a bill between friends, mixing squash with water, scaling a recipe for more guests, or reading distances off a map. The good news is that almost every ratio and proportion problem comes down to two reliable techniques — sharing in parts and the unitary method. Learn those two and you can solve the rest.

If you want a refresher on what a ratio actually is and how to simplify one, read Ratios and Proportions first. This lesson focuses on solving problems.

Sharing a total in a given ratio

This is the classic exam question: "Share £40 between Ana and Ben in the ratio 3 : 5." The method has three steps that never change.

1. Add the parts. 3 + 5 = 8 parts in total.
2. Find one part. £40 ÷ 8 = £5 per part.
3. Multiply each share. Ana gets 3 × £5 = £15; Ben gets 5 × £5 = £25.

Check: £15 + £25 = £40. ✓ The shares add back to the total, so the answer is correct.

The same three steps work for any number of people. To split a 36 kg load of sand between three trucks in the ratio 2 : 3 : 4:

• Parts: 2 + 3 + 4 = 9 parts
• One part: 36 ÷ 9 = 4 kg
• Shares: 8 kg, 12 kg, 16 kg (check: 8 + 12 + 16 = 36 ✓)

The unitary method

The unitary method means "find the value of one, then scale." It is the engine behind recipes, prices, and rates.

Worked example 1 — shopping. If 6 apples cost £1.50, how much do 10 apples cost?

• Find the cost of one apple: £1.50 ÷ 6 = £0.25.
• Multiply by 10: 10 × £0.25 = £2.50.

Going from many to one you divide; going from one to many you multiply. The single unit in the middle keeps everything tidy.

Worked example 2 — scaling a recipe

A bread recipe for 4 loaves uses 600 g of flour and 350 ml of water. How much of each is needed for 6 loaves?

Find the amount per loaf first, then scale:

Ingredient	For 4 loaves	Per loaf (÷ 4)	For 6 loaves (× 6)
Flour	600 g	150 g	900 g
Water	350 ml	87.5 ml	525 ml

So you need 900 g of flour and 525 ml of water. Notice that both ingredients scale by the same factor (6 ÷ 4 = 1.5), which keeps the recipe tasting the same.

Worked example 3 — map scales

Map scales are pure proportion. A scale of 1 : 100 000 means 1 cm on the map equals 100 000 cm in real life.

Two towns are 6 cm apart on the map. How far apart are they really?

$$ 6 \times 100\,000 = 600\,000 \text{ cm} $$

Now convert: 600 000 cm ÷ 100 = 6000 m, and 6000 m ÷ 1000 = 6 km.

Working backwards is just as useful. If a real road is 9 km long, how long is it on the map? 9 km = 900 000 cm, and 900 000 ÷ 100 000 = 9 cm.

Worked example 4 — finding a quantity from one share

Sometimes you are told one part of the ratio and must find another.

The ratio of cordial to water in a drink is 1 : 4. A jug contains 200 ml of cordial. How much water, and how much total drink?

• One part = the cordial = 200 ml.
• Water = 4 parts = 4 × 200 = 800 ml.
• Total drink = 1 + 4 = 5 parts = 5 × 200 = 1000 ml (1 litre).

Comparing with unit rates

A unit rate lets you compare two offers fairly by reducing each to "per one." Which is better value: 3 chocolate bars for £1.20, or 5 for £1.75?

• Offer A: £1.20 ÷ 3 = £0.40 each
• Offer B: £1.75 ÷ 5 = £0.35 each

Offer B is cheaper per bar, so it is the better deal. Shoppers use price-per-100 g labels for exactly this reason.

Worked example 5 — keeping a mixture the right strength

Ratios matter most when getting them wrong has real consequences, such as mixing a drink, a cleaning fluid, or paint. Squash should be mixed with water in the ratio 1 : 5. You pour out 150 ml of squash. How much water keeps it tasting right?

• The squash is 1 part = 150 ml.
• Water is 5 parts = 5 × 150 = 750 ml.
• Total drink = 6 parts = 6 × 150 = 900 ml.

Now suppose you only have a 600 ml glass to fill, still at 1 : 5. Work from the total instead:

• Total parts = 1 + 5 = 6 parts.
• One part = 600 ÷ 6 = 100 ml.
• Squash = 1 × 100 = 100 ml; water = 5 × 100 = 500 ml.

Either direction — starting from one ingredient or from the total — uses the same idea of finding the value of one part. That single habit unlocks almost every mixture problem you will meet.

Practice activity

Solve these on paper. Use parts for sharing problems and the unitary method for rates. Check each answer.

1. Share 48 sweets between Mia and Sam in the ratio 5 : 3.
2. 8 pens cost £2.40. What do 11 pens cost?
3. A recipe for 2 people uses 300 g of pasta. How much for 7 people?
4. A map scale is 1 : 25 000. A path is 8 cm long on the map. How long is it in metres?
5. Which is better value: 4 oranges for £1.20, or 6 oranges for £1.62?

Answers: 1) Mia 30, Sam 18 2) £3.30 (one pen 30p) 3) 1050 g 4) 8 × 25 000 = 200 000 cm = 2000 m 5) A is 30p each, B is 27p each, so 6 for £1.62 is better.

Why this matters

Ratio and proportion problems are the maths of fair sharing and sensible scaling. Cooks adjust recipes, builders mix concrete, nurses calculate doses, and shoppers spot the best deal — all using the same two moves you practised here. Whenever a quantity changes "in proportion," find one part or one unit first, then scale. That single habit turns a confusing word problem into three calm steps, and it will keep paying off in science, cooking, and money for the rest of your life.