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Ratio

Hello! I am Bea. Ratio is about fair shares and keeping things in proportion. Bees are good at that!

Cheat sheet: Ratio

Ratio

2 : 7 compares parts. For every 2 red there are 7 blue. In all there are 2 + 7 = 9 parts, so red is 2/9 of the whole.

Scaling a recipe

  • Multiply or divide every amount by the same number.
  • Go via one: ÷ to get 1, then ×.
  • Do not add the same amount to everything.

Sharing in a ratio

  1. Add the parts.
  2. Total ÷ parts = one part.
  3. Parts × one part for each share.
  4. Check: shares add to the total.

From one part or a difference

Red : blue = 3 : 5 and 12 red? One part = 12 ÷ 3 = 4, so blue = 20.

Difference of 3 parts = £18? One part = £6.

Best buys

Find the price for 1 (or 100 g). The lower unit price is the better buy. Same units first!

Speed, distance, time

  • Speed = distance ÷ time
  • Distance = speed × time
  • Time = distance ÷ speed
  • 150 km in 3 h = 50 km/h

Scale and maps

  • Scale factor 4: all lengths × 4.
  • 1 cm : 6 km: 5 cm is 30 km.
  • 1 : 50: real length = drawing × 50.

Words to know

ratio, proportion, part, share, scale, scale factor, rate, speed, unit price, equivalent ratio, simplify.

Mini lessons

Scaling recipes and quantities

Skills: RAT-01

When you make more or less of something, every amount changes by the same multiplier. A ratio table helps. A good trick is to go via one: divide to find the amount for 1, then multiply.

PeopleFlour(g)4200150÷ 410500× 10
Now 1 person → 10 people means × 10. So 50 g × 10 = 500 g. We multiplied; we did not just add.
Worked example

A pancake recipe for 4 people uses 200 g flour, 2 eggs and 300 ml milk. How much of each for 10 people?

  1. Find the amount for 1 person: 200 ÷ 4 = 50 g flour, 2 ÷ 4 = 0.5 egg, 300 ÷ 4 = 75 ml milk.
  2. Multiply by 10: 50 × 10 = 500 g flour.
  3. 0.5 × 10 = 5 eggs.
  4. 75 × 10 = 750 ml milk.

Answer: 500 g flour, 5 eggs, 750 ml milk

Sharing in a ratio

Skills: RAT-02

A ratio like 3 : 5 says how many parts each person gets. Add the parts to get the total number of equal parts. Divide the total by that number to find one part. Then multiply.

£6£6£6£6£6£6£6£42Isla: 3 partsKofi: 4 parts3 × £6 = £184 × £6 = £24
Isla: 3 × £6 = £18. Kofi: 4 × £6 = £24. Check: £18 + £24 = £42.
Worked example

Asha and Ben share 40 sweets in the ratio 3 : 5. How many does each get?

  1. Add the parts: 3 + 5 = 8.
  2. One part: 40 ÷ 8 = 5 sweets.
  3. Asha: 3 × 5 = 15. Ben: 5 × 5 = 25.
  4. Check: 15 + 25 = 40 ✓

Answer: Asha 15, Ben 25

Ratio from a difference or one part

Skills: RAT-03

Sometimes you are told one amount, not the total. Match that amount to its number of parts to find one part. A difference is the same: it matches the difference in parts.

TomZoe3 parts = £18, so 1 part = £65 × £6 = £302 × £6 = £12

Tom : Zoe = 5 : 2. The difference is 5 − 2 = 3 parts, and that is £18.

Worked example

Tom and Zoe have money in the ratio 5 : 2. Tom has £18 more than Zoe. How much does each have?

  1. The difference in parts is 5 − 2 = 3 parts.
  2. 3 parts = £18, so 1 part = £18 ÷ 3 = £6.
  3. Tom: 5 × £6 = £30. Zoe: 2 × £6 = £12.
  4. Check: £30 − £12 = £18 ✓

Answer: Tom has £30 and Zoe has £12

Best buys and unit price

Skills: RAT-04

To find the best buy, compare the price for the same amount. Work out the price of one item, or of 100 g. The smaller unit price is the better value.

500 g£1.2024p per 100 g800 g£1.7622p per 100 g

Compare the price for the same amount. 120p ÷ 5 = 24p, and 176p ÷ 8 = 22p. The 800 g pack is the better buy.

Worked example

Rice costs £1.20 for 500 g or £1.76 for 800 g. Which is the better buy?

  1. Change to pence: £1.20 = 120p and £1.76 = 176p.
  2. Price per 100 g: 120 ÷ 5 = 24p, and 176 ÷ 8 = 22p.
  3. 22p is less than 24p.

Answer: The 800 g pack (22p per 100 g)

Rates: speed, distance and time

Skills: RAT-05. Stretch: usually taught in Year 7, but some 11+ tests include it.

Speed tells you how far something goes in one hour (or one minute). Distance = speed × time. To find speed, divide distance by time.

DistanceSpeedTimeS = D ÷ TD = S × TT = D ÷ S

Cover the one you want. Distance is speed × time. Speed is distance ÷ time.

Worked example

A train travels 150 km in 3 hours. What is its speed? How far does it go in 4½ hours at that speed?

  1. Speed = distance ÷ time = 150 ÷ 3 = 50 km/h.
  2. Distance = speed × time = 50 × 4.5.
  3. 50 × 4 = 200 and 50 × 0.5 = 25, so 200 + 25 = 225.

Answer: 50 km/h, then 225 km

Scale factors and map scales

Skills: RAT-06

A scale factor tells you how many times bigger or smaller every length gets. On a map, the scale says how real distances compare with the map. Always use the same units on both sides.

3 cm2 cm12 cm8 cm→× 4(drawn to scale: 3 by 2, then 12 by 8)

Scale factor 4: every length is × 4. 3 × 4 = 12 and 2 × 4 = 8. The shape stays the same, only bigger.

Worked example

A map scale is 1 cm : 6 km. Two villages are 5 cm apart on the map. How far apart are they really? A model is built at 1 : 50. The real boat is 4.5 m long. How long is the model?

  1. 1 cm is 6 km, so 5 cm is 5 × 6 = 30 km.
  2. For the model, change 4.5 m to centimetres: 4.5 × 100 = 450 cm.
  3. Divide by 50: 450 ÷ 50 = 9 cm.

Answer: 30 km, and 9 cm

Worked examples across the levels

Level 1 is the easiest. Level 5 is a real challenge. Try each one yourself before you read the steps.

Level 1 (Year 3): one step

One packet has 6 biscuits. How many biscuits are in 4 packets?

  1. 4 × 6 = 24.

Answer: 24 biscuits

Level 2 (Year 4): one or two steps

A recipe makes 12 muffins using 150 g of flour. How much flour is needed for 24 muffins?

  1. 24 is double 12.
  2. Double the flour: 150 × 2 = 300.

Answer: 300 g

Level 3 (Year 5): two steps

Share £60 between Kai and Lily in the ratio 1 : 4.

  1. Parts: 1 + 4 = 5.
  2. One part: £60 ÷ 5 = £12.
  3. Kai: £12. Lily: 4 × £12 = £48.

Answer: Kai £12, Lily £48

Level 4 (exam standard): several steps or working backwards

A garden has red and yellow tulips in the ratio 3 : 7. There are 40 tulips. How many more yellow than red are there?

  1. Parts: 3 + 7 = 10.
  2. One part: 40 ÷ 10 = 4.
  3. Red: 3 × 4 = 12. Yellow: 7 × 4 = 28.
  4. Difference: 28 − 12 = 16.

Answer: 16 more yellow tulips

Level 5 (challenge): three to five steps

A tin has blue and yellow buttons in the ratio 5 : 8. When 6 more blue buttons are added, there are the same number of blue and yellow buttons. How many buttons were in the tin at the start?

  1. Blue is 5 parts and yellow is 8 parts. Yellow is 3 parts more than blue.
  2. Adding 6 blue makes them equal, so 3 parts = 6 buttons and 1 part = 2 buttons.
  3. Blue: 5 × 2 = 10. Yellow: 8 × 2 = 16.
  4. At the start: 10 + 16 = 26. Check: 10 + 6 = 16 ✓

Answer: 26 buttons

Common mistakes

Adding instead of multiplying

Mistake code: RATIO_ADDITIVE

The mistake: Because 6 is 2 more than 4, adding 2 to 200 g of flour to get 202 g for 6 people.

Why it is wrong: Ratios work by multiplying. If you add the same amount to everything, the proportions change and the recipe tastes different.

How to avoid it: Use a ratio table or a bar model. Find one, then multiply.

4 people, 200 g. 1 person is 50 g. 6 people: 6 × 50 = 300 g. (Not 202 g.)

Reading the ratio as fractions of the whole

The mistake: Thinking that in the ratio 2 : 7, red is 2/7 of the marbles.

Why it is wrong: 2 : 7 compares red with blue. The whole is 2 + 7 = 9 parts, so red is 2/9.

How to avoid it: Always add the parts to find the total number of parts first.

Red : blue = 2 : 7. Red is 2/9 of all marbles, blue is 7/9.

Comparing total prices, not unit prices

The mistake: Saying the cheaper pack (£1.20) is the better buy without checking how much you get.

Why it is wrong: A cheaper pack can have much less in it. You need the price for the same amount.

How to avoid it: Work out the price of 1 item (or 100 g) for each deal. Compare those.

500 g for 120p is 24p per 100 g. 800 g for 176p is 22p per 100 g, so 800 g wins.

Mixing units in a scale

The mistake: A model is 1 : 50 and the boat is 4.5 m. Writing 4.5 ÷ 50 = 0.09 and calling it 0.09 cm.

Why it is wrong: The answer is in the same unit you started with. 0.09 is in metres, not centimetres.

How to avoid it: Change to one unit first (4.5 m = 450 cm), or label your answer carefully.

450 cm ÷ 50 = 9 cm (and 0.09 m = 9 cm).

Quick check

Five questions, easy to hard. Use paper and have a go before you open the answer.

  1. 5 biscuits cost 60p. How much do 8 biscuits cost?

    Show answer

    96p

    One biscuit is 60 ÷ 5 = 12p. Eight biscuits cost 8 × 12 = 96p.

  2. Share 54 stickers between Dev and Eve in the ratio 2 : 7. How many does Eve get?

    Show answer

    42

    Parts: 2 + 7 = 9. One part: 54 ÷ 9 = 6. Eve: 7 × 6 = 42 (Dev gets 12).

  3. A box has green and orange pencils in the ratio 4 : 3. There are 28 green pencils. How many orange pencils are there?

    Show answer

    21

    4 parts = 28, so 1 part = 7. Orange: 3 × 7 = 21.

  4. Pack A has 6 pens for £1.50. Pack B has 10 pens for £2.30. How much is one pen in the better buy?

    Show answer

    23p

    A: 150 ÷ 6 = 25p a pen. B: 230 ÷ 10 = 23p a pen. B is cheaper per pen.

  5. On a map, 1 cm stands for 4 km. Two towns are 7.5 cm apart on the map. Dan walks the real distance at 5 km/h. How many hours does he take?

    Show answer

    6 hours

    Real distance: 7.5 × 4 = 30 km. Time = 30 ÷ 5 = 6 hours.

For grown-ups

  • Cook together and change the recipe: “this is for 4, we need it for 6. What changes?” Ask your child to work out each amount.
  • At the shop, ask which pack is the best value and why. Let them find the price per 100 g or per item.
  • Encourage drawing a bar model for ratio problems. Draw the parts as boxes first and only then do the sums.