Ratio & Proportion

Cambridge Stage 7 — Unit 12

What is Ratio?

A ratio compares quantities in the same units. The ratio 3:5:4 means for every 3 parts of one thing, there are 5 of another and 4 of a third. The total number of parts = 3 + 5 + 4 = 12.

To divide a quantity in a ratio: find one part (total ÷ sum of ratio) then multiply each ratio number.

Dividing into 3 (or more) Parts

Step 1: Add the ratio parts  →  3 + 5 + 4 = 12 parts
Step 2: One part = Total ÷ 12  →  £120 ÷ 12 = £10
Step 3: Multiply  →  3×£10 = £30,  5×£10 = £50,  4×£10 = £40
Step 4: Check: £30 + £50 + £40 = £120 ✓

The Unitary Method

Unitary method: find the value of one unit, then scale up or down.

Example: 8 tickets cost £52  →  1 ticket = £52 ÷ 8 = £6.50  →  5 tickets = 5 × £6.50 = £32.50

Use a ratio table to organise the working:

TicketsCost (£)
852
16.50
532.50

Best Value Comparisons

Find the unit cost for each option and compare. Smaller unit cost = better value.

3 for £2.40  →  1 costs £2.40 ÷ 3 = 80p
5 for £3.75  →  1 costs £3.75 ÷ 5 = 75p ← Better value!

Map Scales

A scale of 1 : 25 000 means 1 cm on the map = 25 000 cm in real life.

1 cm on map  →  25 000 cm real = 250 m = 0.25 km
Map distance × scale = Real distance (in same units)
Real distance ÷ scale = Map distance

Speed, Distance & Time

Speed = Distance ÷ Time  |  Distance = Speed × Time  |  Time = Distance ÷ Speed
D S T Cover what you want to find
Units matter! If Speed is in km/h, Distance must be in km and Time in hours.

Proportion in Recipes, Mixtures & Dilutions

If a recipe serves 4 people, scale all ingredients by the same multiplier.

Multiplier = (people you want) ÷ (people in recipe)
Dilution ratio 1:4 (concentrate:water) means 1 part concentrate per 5 parts total.

Try the interactive Recipe Scaler on the Visualizer tab!

Key Rules to Remember:

Worked Examples

Study these carefully before attempting the exercises!

Example 1 — Dividing £120 in the Ratio 3 : 5 : 4

Question: Three friends share £120 in the ratio 3 : 5 : 4. How much does each person receive?

Step 1 — Total parts: 3 + 5 + 4 = 12 parts
Step 2 — Value of 1 part: £120 ÷ 12 = £10 per part
Step 3 — Multiply:
Person A: 3 × £10 = £30
Person B: 5 × £10 = £50
Person C: 4 × £10 = £40
Check: £30 + £50 + £40 = £120 ✓

Example 2 — Unitary Method

Question: 8 theatre tickets cost £52. Using the unitary method, find the cost of 5 tickets.

Step 1 — Find 1 ticket: £52 ÷ 8 = £6.50
Step 2 — Scale to 5: 5 × £6.50 = £32.50
TicketsCost
8£52.00
1£6.50 (÷8)
5£32.50 (×5)

Answer: £32.50

Example 3 — Best Value Comparison

Question: Which is better value — Pack A: 400 g for £2.80, or Pack B: 650 g for £4.29?

Pack A unit cost: £2.80 ÷ 400 = £0.007 per gram = 0.7p/g
Pack B unit cost: £4.29 ÷ 650 = £0.0066 per gram ≈ 0.66p/g
Compare: 0.66p/g < 0.7p/g, so Pack B is better value.

Tip: always divide price by quantity to get cost per gram/litre/item.

Example 4 — Map Scale Problem (1 : 25 000)

Question: On a map with scale 1 : 25 000, two towns are 7.4 cm apart. How far apart are they in real life? Give your answer in km.

Step 1 — Real distance in cm: 7.4 × 25 000 = 185 000 cm
Step 2 — Convert to metres: 185 000 ÷ 100 = 1 850 m
Step 3 — Convert to km: 1 850 ÷ 1 000 = 1.85 km

Answer: 1.85 km

Recipe Scaler Visualizer

Adjust the slider to scale the recipe up or down. Watch all ingredient quantities change in real time!

Blueberry Muffin Recipe (base: 4 people)

Self-raising flour 200 g
Butter 100 g
Sugar 150 g
Eggs 2
Blueberries 120 g
Milk 80 ml
Vanilla extract 1 tsp
4 people

Slide left (fewer) or right (more)
Multiplier: 4 ÷ 4 = ×1 (no change)

Ratio of ingredients (base recipe): Flour : Butter : Sugar = 200 : 100 : 150 = 4 : 2 : 3

The ratio stays the same no matter how many people you scale to.

Dilution Visualizer

A cordial drink uses the ratio 1 part cordial : 4 parts water. How much of each do you need for different total volumes?

500 ml
Cordial
100 ml
Water
400 ml
Cordial
Water

Exercise 1 — Three-Part Ratio Division

Divide each quantity in the given ratio. Type all three parts. Check your answers when done!

#AmountRatioPart APart BPart C

Exercise 2 — Unitary Method & Ratio Tables

Complete the ratio table for each problem. Find 1 unit first, then scale to the target.

Exercise 3 — Best Value Calculator

Click the better value option, then type its unit cost (in pence or £ as instructed).

Exercise 4 — Map Scale Problems

Use the given scale to find real distances or map distances. Type your answer in the box.

Exercise 5 — Speed, Distance & Time

Find the missing value. Type just the number; units are shown. Check all when done.

S = D ÷ T  |  D = S × T  |  T = D ÷ S
#SpeedDistanceTimeFindYour Answer

Exercise 6 — True or False Proportions

Click TRUE or FALSE for each statement. Get instant feedback!

Practice Questions

Grab your exercise book and work through all 20 questions. Show all workings!

  1. Divide £90 in the ratio 2 : 3 : 1.
  2. Share 360 stickers in the ratio 5 : 3 : 4.
  3. Divide 4.5 kg in the ratio 2 : 3 : 4.
  4. Three people invest in a business in the ratio 4 : 5 : 1. Total profit is £8 000. How much does each person earn?
  5. 6 identical books cost £22.50. Use the unitary method to find the cost of 9 books.
  6. A car travels 240 km in 3 hours. Find the speed in km/h.
  7. A train travels at 90 km/h for 2.5 hours. How far does it travel?
  8. A cyclist covers 54 km at 18 km/h. How long does the journey take? Give your answer in hours and minutes.
  9. A map has a scale of 1 : 50 000. Two lakes are 4.6 cm apart on the map. How far apart are they in km?
  10. On a map with scale 1 : 20 000, a road is actually 3.4 km long. How long is it on the map in cm?
  11. Pack A: 250 g for £1.75. Pack B: 400 g for £2.64. Which is better value? Show your working.
  12. Orange squash is mixed with water in the ratio 1 : 6. How much water is needed to mix with 150 ml of squash?
  13. A recipe for 8 people uses 320 g of rice. How much rice is needed for 5 people?
  14. Paint is mixed red : white : yellow in the ratio 3 : 5 : 2. How much of each colour is in 500 ml of paint?
  15. 12 workers take 15 days to complete a job. Using the unitary method, how many days would 9 workers take?
  16. A map scale is 1 : 25 000. A forest on the map measures 3 cm by 5 cm. What is the real area in km²?
  17. A car averages 60 km/h for the first 90 km, then 80 km/h for the next 160 km. What is the total journey time?
  18. A cordial recipe uses the ratio 2 : 9 (cordial : water). How much cordial is needed for 550 ml of drink?
  19. Three sisters share an inheritance of £27 000 in the ratio 2 : 3 : 4. How much more does the largest share receive than the smallest?
  20. A train journey of 420 km takes 3 hours 30 minutes. Find the average speed in km/h.

Answers

  1. £90 ÷ 6 = £15 per part → £30 : £45 : £15
  2. 360 ÷ 12 = 30 per part → 150 : 90 : 120
  3. 4.5 kg ÷ 9 = 0.5 kg per part → 1 kg : 1.5 kg : 2 kg
  4. £8000 ÷ 10 = £800 per part → £3200 : £4000 : £800
  5. 1 book = £22.50 ÷ 6 = £3.75; 9 books = 9 × £3.75 = £33.75
  6. Speed = 240 ÷ 3 = 80 km/h
  7. Distance = 90 × 2.5 = 225 km
  8. Time = 54 ÷ 18 = 3 hours = 3 hours 0 minutes
  9. 4.6 × 50 000 = 230 000 cm ÷ 100 000 = 2.3 km
  10. 3.4 km = 340 000 cm ÷ 20 000 = 17 cm
  11. Pack A: £1.75 ÷ 250 = 0.7p/g; Pack B: £2.64 ÷ 400 = 0.66p/g → Pack B is better value
  12. 1 part squash = 150 ml, 6 parts water = 6 × 150 ml = 900 ml
  13. 1 person: 320 ÷ 8 = 40 g; 5 people: 5 × 40 = 200 g
  14. 500 ÷ 10 = 50 ml per part → Red: 150 ml, White: 250 ml, Yellow: 100 ml
  15. 1 worker takes 12 × 15 = 180 days; 9 workers: 180 ÷ 9 = 20 days
  16. 3 cm × 25 000 = 750 m = 0.75 km; 5 cm = 1.25 km; Area = 0.75 × 1.25 = 0.9375 km²
  17. Time 1: 90 ÷ 60 = 1.5 h; Time 2: 160 ÷ 80 = 2 h; Total = 3.5 hours = 3 h 30 min
  18. Total parts = 11; 550 ml ÷ 11 = 50 ml; Cordial = 2 × 50 = 100 ml
  19. £27 000 ÷ 9 = £3 000; Largest = 4 × £3 000 = £12 000; Smallest = 2 × £3 000 = £6 000; Difference = £6 000
  20. 3h 30min = 3.5 h; Speed = 420 ÷ 3.5 = 120 km/h

Challenge — Hard Word Problems

Multi-step problems. Show all workings on paper before revealing!

  1. Ali, Ben and Cara share money in the ratio 3 : 4 : 5. Ali receives £240. How much do Ben and Cara receive, and what is the total amount shared?
  2. A car travels from Town A to Town B, a distance of 315 km. For the first 135 km the car averages 90 km/h. For the remaining distance it averages 60 km/h. What is the total journey time, and what is the overall average speed for the whole journey?
  3. A recipe for 6 servings needs flour, butter and sugar in the ratio 5 : 2 : 3. If you use 250 g of flour, how much butter and sugar do you need, and how many servings does this make?
  4. A map has a scale of 1 : 40 000. A park on the map is a rectangle 4.5 cm by 3 cm. What is the actual area of the park in hectares? (1 hectare = 10 000 m²)
  5. Three taps fill a tank in the following times: Tap A in 4 hours, Tap B in 6 hours, Tap C in 12 hours. If all three run together, how long does it take to fill the tank?
  6. A photographer enlarges a photo using the scale 3 : 8. The original photo is 15 cm × 10 cm. What are the dimensions of the enlarged photo, and what is the ratio of the areas (original : enlarged)?
  7. Two brands of honey: Brand X is 340 g for £4.25; Brand Y is 454 g for £5.49. Find the unit cost of each brand in pence per gram (to 2 d.p.). Which is better value? How much cheaper per 100 g is it?
  8. A train covers the first third of a journey at 60 km/h, the second third at 90 km/h, and the final third at 120 km/h. What is the average speed for the whole journey?
  9. Orange, mango and pineapple juice are mixed in the ratio 4 : 3 : 1 to make 2.4 litres of fruit punch. How many ml of each juice is used? If mango juice costs £1.80 per litre, what does the mango portion cost?
  10. A factory produces red, blue and green widgets in the ratio 7 : 4 : 1 per hour. In 8 hours it produces 1 440 widgets in total. How many of each colour are produced? If the ratio changes to 5 : 4 : 3 for a special order of 1 200 widgets, how many more green widgets are produced than in the original batch?

Full Workings

  1. Ali's 3 parts = £240, so 1 part = £80.
    Ben = 4 × £80 = £320; Cara = 5 × £80 = £400; Total = 12 × £80 = £960
  2. Remaining distance = 315 − 135 = 180 km.
    Time 1 = 135 ÷ 90 = 1.5 h; Time 2 = 180 ÷ 60 = 3 h; Total time = 4.5 hours.
    Average speed = 315 ÷ 4.5 = 70 km/h
  3. Flour is 5 parts = 250 g, so 1 part = 50 g.
    Butter = 2 × 50 = 100 g; Sugar = 3 × 50 = 150 g.
    Original 6 servings uses 5 parts = 250 g flour. New recipe uses same 250 g, so still 6 servings.
  4. Real dimensions: 4.5 × 40 000 = 180 000 cm = 1 800 m; 3 × 40 000 = 120 000 cm = 1 200 m.
    Area = 1 800 × 1 200 = 2 160 000 m² ÷ 10 000 = 216 hectares
  5. Rates: A fills 1/4 per hour, B fills 1/6, C fills 1/12.
    Combined rate = 3/12 + 2/12 + 1/12 = 6/12 = 1/2 per hour.
    Time = 1 ÷ (1/2) = 2 hours
  6. Scale factor = 8 ÷ 3.
    Enlarged width = 15 × (8/3) = 40 cm; Enlarged height = 10 × (8/3) ≈ 26.67 cm.
    Area ratio = 3² : 8² = 9 : 64
  7. Brand X: 425 ÷ 340 = 1.25p/g; Brand Y: 549 ÷ 454 ≈ 1.21p/g.
    Brand Y is better value. Difference per 100 g = (1.25 − 1.21) × 100 = 4p cheaper per 100 g.
  8. Let total distance = 3d km. Time for each third = d/60 + d/90 + d/120.
    LCM of 60, 90, 120 = 360. Times: 6d/360 + 4d/360 + 3d/360 = 13d/360.
    Average speed = 3d ÷ (13d/360) = 3d × 360/(13d) = 1080/13 ≈ 83.1 km/h
  9. Total parts = 4 + 3 + 1 = 8. 2.4 litres = 2 400 ml. 1 part = 300 ml.
    Orange = 4 × 300 = 1 200 ml; Mango = 3 × 300 = 900 ml; Pineapple = 1 × 300 = 300 ml.
    Mango cost: 900 ml = 0.9 litres; 0.9 × £1.80 = £1.62
  10. Total ratio = 7 + 4 + 1 = 12. Widgets per hour = 1440 ÷ 8 = 180/h. Per batch: Red = 7/12 × 1440 = 840; Blue = 4/12 × 1440 = 480; Green = 1/12 × 1440 = 120.
    Special order 5:4:3, total parts = 12. Green = 3/12 × 1200 = 300.
    Extra green = 300 − 120 = 180 more green widgets
Created by Mr Josh