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Foundations · Numbers

Ratio, rates and proportion

Sharing in a ratio, simplifying ratios, unit rates, scaling a recipe, and the difference between direct and inverse proportion.

First, the rules

Parts, and what one part is worth

A ratio compares amounts in parts. Almost every ratio question is answered by one move: find what a single part is worth, then count parts.

★ Ratio
Example 1
2 : 3 means 2 parts to 3 parts. Simplify like a fraction: 12 : 18 = 2 : 3. Same units before you compare (2 m : 50 cm = 200 : 50 = 4 : 1).
★ Sharing in a ratio
Example 1
Total parts = sum; one part = total ÷ parts; each share = its parts × one part. 60 in 2 : 3 → 5 parts, one part 12, shares 24 and 36.
Ratio ↔ fraction
Example 1
In 2 : 3 the first is 25 of the total, not 23.
★ Rate and unit rate
Example 2
A rate compares different units (km per hour, price per kg). Divide to get "per one": 240 km in 3 h → 80 km/h. Best buy = lowest price per unit.
★ Direct proportion
Example 2
Double one, the other doubles; the ratio stays fixed. Use a multiplier or "one unit first". 5 pens cost 3.50 → 1 pen 0.70 → 8 pens 5.60.
★ Inverse proportion
Example 3
Double one, the other halves; the product stays fixed. 6 workers × 8 days = 48 worker-days → 4 workers need 12 days.

See it

Cut the total into parts

Set the ratio and a total. The bar is cut into a + b equal parts; the double number line underneath pairs parts with amounts, which is the whole method.

Step by step

Worked examples

1

Sharing, and a difference

share in a ratio

Three friends share 450 in the ratio 2 : 3 : 4. (a) How much does each get? (b) How much more does the largest share get than the smallest?

Ask yourself: How many parts in total, and what is one part worth?

Show the solution
  1. Parts: 2 + 3 + 4 = 9. One part: 450 ÷ 9 = 50.
  2. Shares: 100, 150, 200.
  3. Difference: 4 parts − 2 parts = 2 parts = 100 (no need to compute the shares first).

Answer 100 · 150 · 200; difference 100

2

Best buy and scaling

unit rate · direct proportion

(a) 750 g of rice costs 2.40 and 1.2 kg costs 3.60. Which is better value? (b) A recipe for 4 people needs 300 g of flour. How much for 10 people?

Ask yourself: Compare per kilogram; scale by the multiplier 10 ÷ 4.

Show the solution
  1. (a) 2.40 ÷ 0.75 = 3.20 per kg; 3.60 ÷ 1.2 = 3.00 per kg. The 1.2 kg bag is better value.
  2. (b) Multiplier 10 ÷ 4 = 2.5: 300 × 2.5 = 750 g (or 75 g per person × 10).

Answer 1.2 kg bag · 750 g

3

Direct or inverse?

proportion

(a) A car uses 6 L of fuel for 90 km. How far on 14 L? (b) A tank fills in 40 minutes with 3 identical pumps. How long with 5 pumps?

Ask yourself: In (a) more fuel means more distance (direct). In (b) more pumps means less time (inverse).

Show the solution
  1. (a) 90 ÷ 6 = 15 km per litre; 15 × 14 = 210 km.
  2. (b) Pumps × minutes is constant: 3 × 40 = 120; 120 ÷ 5 = 24 minutes.

Answer 210 km · 24 minutes

Practice: ratio and proportion

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Simplifying ratios, sharing, unit rates, scaling, and direct versus inverse proportion. Your best score is kept in this browser only.

About this lesson

Ratio, rates and proportion: writing and simplifying a ratio, dividing a quantity in a given ratio, unit rates and better value, scaling with a multiplier, and direct versus inverse proportion with the table that tells them apart. A double number line you can move, worked examples and practice questions. Everything on this page runs in your browser; nothing you type is sent or stored. Spotted an error? Tell us.