Ratios & Proportions

What is a Ratio?

A ratio compares two or more quantities and shows how much of one thing there is relative to another. Ratios do not have units — they describe a relationship.

If there are 3 apples and 5 oranges, the ratio of apples to oranges is 3:5 (read as "3 to 5").

Ratios can be written three ways:

• 3:5 (colon notation)
• 3/5 (fraction notation)
• "3 to 5" (word form)

Key Terms

• Ratio: a comparison of two or more quantities using division
• Equivalent ratio: ratios that express the same relationship (like equivalent fractions)
• Simplest form: a ratio where both numbers share no common factors other than 1
• Proportion: a statement that two ratios are equal
• Scale factor: the number you multiply or divide all parts of a ratio by

Simplifying Ratios

Like fractions, ratios can be simplified by dividing all parts by their Greatest Common Factor (GCF).

15:25 → GCF is 5 → 15÷5 : 25÷5 = 3:5

12:8 → GCF is 4 → 12÷4 : 8÷4 = 3:2

24:36:60 → GCF is 12 → 2:3:5

Equivalent Ratios

Ratios are equivalent if one can be obtained from the other by multiplying or dividing all parts by the same number.

3:5 and 9:15 → multiply 3:5 by 3 → equivalent ✓

3:5 and 6:15 → 3×2=6 but 5×2=10 ≠ 15 → not equivalent ✗

Cross-multiplying is a quick check: 3/5 = 9/15 → 3 × 15 = 5 × 9 → 45 = 45 ✓

Dividing a Quantity in a Ratio

This is one of the most common exam question types.

Method:

1. Add up all the parts to find the total number of parts
2. Divide the quantity by the total to find the value of one part
3. Multiply to find each share

Example: Share £40 between two people in the ratio 3:2.

• Total parts = 3 + 2 = 5
• One part = £40 ÷ 5 = £8
• Person A gets: 3 × £8 = £24
• Person B gets: 2 × £8 = £16
• Check: £24 + £16 = £40 ✓

Example: Three friends share £200 in the ratio 2:3:5.

• Total parts = 2 + 3 + 5 = 10
• One part = £200 ÷ 10 = £20
• Shares: £40 : £60 : £100
• Largest share = £100

Proportions and Scaling

A proportion states that two ratios are equal: 2/3 = 4/6.

Scaling up and down uses this idea. If a recipe for 4 people needs 300g of flour, how much for 10 people?

Method (unitary method):

• 4 people → 300g
• 1 person → 300 ÷ 4 = 75g
• 10 people → 75 × 10 = 750g

Map Scales

A map scale like 1:50,000 means 1cm on the map = 50,000cm in real life.

50,000cm = 500m = 0.5km

Example: A road is 4cm on a 1:50,000 map. Real length = 4 × 50,000 = 200,000cm = 2,000m = 2km

Worked Example — Three-Part Ratio

Concrete is made from cement, sand, and gravel in the ratio 1:2:3. If you use 6kg of sand, find the amount of cement and gravel.

• Sand = 2 parts = 6kg, so 1 part = 3kg
• Cement = 1 × 3 = 3kg
• Gravel = 3 × 3 = 9kg
• Total = 3 + 6 + 9 = 18kg

Ratios and Fractions

A ratio a:b means the total is a + b parts. Each part as a fraction of the whole:

• Part A = a/(a+b)
• Part B = b/(a+b)

In a 3:2 ratio: Part A = 3/5 of the total, Part B = 2/5.

Common Mistakes to Avoid

• Confusing ratio with fraction: the ratio 3:2 does NOT mean 3/2; it means 3 parts out of 5 total (3/5 and 2/5)
• Adding instead of multiplying when scaling: always multiply/divide by the same factor
• Forgetting to check: after dividing a quantity, add the shares to confirm they total the original amount
• Wrong order: ratio 3:5 (A to B) is different from 5:3 (B to A) — always match the order of the question

Tips and Tricks

• To find the value of one part, always divide the total by the sum of all ratio parts
• When simplifying ratios with decimals, multiply all parts to make whole numbers first (e.g. 0.5:1.5 → multiply by 2 → 1:3)
• Use the unitary method (find the value of 1 unit first) for proportion problems — it works for all scaling questions
• Keep the order of the ratio consistent with the order quantities are named in the question