Ratio and Proportion

British & IGCSE › Checkpoint 8 › Checkpoint 8 Maths — Complete Course

A ratio compares the sizes of two or more quantities. Ratios are used in recipes, maps, mixing paint or concrete, sharing money and comparing prices. In Stage 8 you need to simplify ratios (even when the units are different), compare two ratios, share an amount in a ratio, and see how ratio is connected to direct proportion.

Key words

1. What a ratio tells you

A bag has 3 red beads and 2 blue beads. The ratio red : blue is 3 : 2 (order matters — blue : red is 2 : 3). There are 3 + 2 = 5 beads in each 'group', so the fraction of beads that are red is 3/5 and the fraction that are blue is 2/5.

Do not confuse them: the ratio 3 : 2 does not mean 3/2 of the beads are red. A ratio compares parts with parts; a fraction compares a part with the whole.

2. Simplifying ratios

1. If the units are different, change them to the same unit first (usually the smaller unit). Then you can remove the units.
2. If there are decimals, multiply every part by 10 (or 100) to get whole numbers. If there are fractions, multiply every part by the common denominator.
3. Divide every part by the highest common factor (HCF). Check that no number except 1 divides into all the parts.

Units you must know: 1 m = 100 cm, 1 cm = 10 mm, 1 km = 1000 m, 1 kg = 1000 g, 1 l = 1000 ml, 1 h = 60 min, 1 min = 60 s.

The form 1 : n. Divide both parts by the first part. Example: 4 : 10 = 4/4 : 10/4 = 1 : 2.5. Here n can be a decimal. This form is very useful for comparing ratios.

3. Comparing ratios

You cannot compare 2 : 7 and 3 : 10 just by looking. Make them comparable in one of these ways:

• Write both in the form 1 : n. Orange drink A is concentrate : water = 2 : 7 = 1 : 3.5. Drink B is 3 : 10 = 1 : 3.33… Drink B has less water for each part of concentrate, so B is stronger.
• Write the fraction of the whole. A: concentrate is 2/9 = 0.222… of the drink. B: 3/13 = 0.230… of the drink. B has a bigger fraction of concentrate, so B is stronger.
• Make one part the same (equivalent ratios). 2 : 7 = 6 : 21 and 3 : 10 = 6 : 20. For the same 6 parts of concentrate, B uses less water — B is stronger.

4. Dividing an amount in a ratio

1. Add the numbers in the ratio to find the total number of parts.
2. Divide the amount by the total number of parts to find one part.
3. Multiply one part by each number in the ratio.
4. Check: the shares must add up to the original amount.

Three types of ratio question. Read carefully which number you are given:

5. Ratio and direct proportion

Two quantities are in direct proportion when they always increase or decrease by the same multiplier: double one and the other doubles; take a third of one and the other is a third. Another way to say it: the ratio between them never changes.

Mass : cost is 1 : 6 = 2 : 12 = 3 : 18 = 4 : 24 — the same ratio every time, so cost is in direct proportion to mass. On a graph, direct proportion is always a straight line through (0, 0).

Testing for direct proportion: divide the second quantity by the first for each pair. If you always get the same number, they are in direct proportion. If even one pair is different, they are not.

Solving proportion problems — two methods.

• Unitary method: 5 pens cost $3.40. One pen costs 3.40 ÷ 5 = $0.68. So 8 pens cost 8 × 0.68 = $5.44.
• Multiplier (scaling) method: a recipe for 4 people uses 300 g of flour. For 6 people: 6 ÷ 4 = 1.5, so use 300 × 1.5 = 450 g. Here the ratio people : flour stays 4 : 300 = 6 : 450.

Worked examples

Worked example 1 — Write each ratio in its simplest form. (a) 24 : 36 (b) 1.2 kg : 450 g (c) 2.5 : 4 [3]

• (a) The HCF of 24 and 36 is 12. 24 ÷ 12 = 2, 36 ÷ 12 = 3 → 2 : 3.
• (b) Same units: 1.2 kg = 1200 g, so 1200 : 450. Divide by 10 → 120 : 45; divide by 15 → 8 : 3.
• (c) Multiply by 2 to clear the decimal: 5 : 8 (5 and 8 have no common factor).

Common mistake in (b): writing 1.2 : 450 or 12 : 450. You must change the units first.

Worked example 2 — Two shades of green paint are made by mixing blue and yellow paint. Shade A: blue : yellow = 3 : 5. Shade B: blue : yellow = 4 : 7. Which shade has the greater proportion of blue? Show how you decide. [2]

• Fraction of blue in A = 3/(3 + 5) = 3/8 = 0.375.
• Fraction of blue in B = 4/(4 + 7) = 4/11 = 0.3636…
• 0.375 > 0.364, so shade A has the greater proportion of blue.
• Another way: A = 1 : 1.67 and B = 1 : 1.75 (blue : yellow). A has less yellow for each part of blue, so A is bluer.

Worked example 3 — Omar, Hana and Youssef share $360 in the ratio 2 : 3 : 4. How much does each person get? [2]

• Total parts: 2 + 3 + 4 = 9. One part: 360 ÷ 9 = $40.
• Omar 2 × 40 = $80, Hana 3 × 40 = $120, Youssef 4 × 40 = $160.
• Check: 80 + 120 + 160 = 360 ✓ (see the bar model above).

Common mistake: dividing 360 by 3 because 'there are three people'. Divide by the number of parts, not the number of people.

Worked example 4 — Sami and Nour share some stickers in the ratio 7 : 4. Sami gets 45 more stickers than Nour. How many stickers are there altogether? [2]

• The difference is 7 − 4 = 3 parts, and 3 parts = 45, so 1 part = 15.
• Total = 7 + 4 = 11 parts = 11 × 15 = 165 stickers (Sami 105, Nour 60; check 105 − 60 = 45 ✓).

Worked example 5 — 6 notebooks cost $8.70. How much do 10 of the same notebooks cost? [2]

• Unitary method: 1 notebook costs 8.70 ÷ 6 = $1.45.
• 10 notebooks cost 10 × 1.45 = $14.50.
• Check with a different route: 3 notebooks cost $4.35, so 9 cost $13.05, plus one more $1.45 = $14.50 ✓

Worked example 6 — The table shows how much Malak is paid for different numbers of hours of work. Hours | 2 | 3 | 5 Pay ($) | 30 | 45 | 80 Is her pay in direct proportion to the number of hours? Explain your answer. [1]

• Pay ÷ hours: 30 ÷ 2 = 15, 45 ÷ 3 = 15, 80 ÷ 5 = 16.
• No — the pay per hour is not the same for every pair (15, 15, 16), so the ratio hours : pay changes.
• Exam technique: 'Yes' or 'No' alone gets no mark. You must show the numbers that prove it.

Practice

All pages in this course

Start here

• Stage 8 Maths: The Tests and How to Prepare

Number

• Order of Operations and Calculating with Integers
• Factors, Multiples, Primes, HCF and LCM
• Types of Number and Index Laws
• Squares, Cubes and Roots (Positive and Negative)
• Multiplying and Dividing by 0.1 and 0.01, and Significant Figures
• Fractions: Recurring Decimals, Mixed Numbers and Order of Operations
• Percentage Increase, Decrease and Absolute Change
• Ordering and Calculating with Decimals
• Ratio and Proportion

Algebra

• Algebraic Expressions: Simplifying, Expanding and Factorising
• Writing Expressions and Formulae, and Changing the Subject
• Equations and Inequalities
• Sequences and the nth Term
• Functions and Straight-line Graphs (y = mx + c)
• Reading and Interpreting Real-life Graphs

Geometry and measure

• Quadrilaterals and Regular Polygons
• Circumference of a Circle, and Miles and Kilometres
• Area of Parallelograms and Trapezia
• 3D Shapes: Euler's Formula and Plans and Elevations
• Volume of Prisms and Surface Area
• Angles in Triangles and on Parallel Lines
• Constructions with Ruler and Compasses
• Bearings and Midpoints
• Translation, Reflection, Rotation and Enlargement

Statistics and probability

• Data Collection and Sampling
• Representing Data
• Averages, Range and Comparing Distributions
• Probability: Complementary and Combined Events

Practice tests

• Stage 8 Maths — Practice Test 1
• Stage 8 Maths — Practice Test 2
• Stage 8 Maths — Practice Test 3