4.1 Ratio, Rates & Proportion

Ratio and rate questions appear in every ICAS Mathematics paper from Paper A upwards, and by Papers E-H they are often the hardest multi-step items on the paper. The good news is that every ratio question uses the same small toolkit: simplify, find the value of one part, and scale up or down. Because personal calculators are not allowed, ICAS chooses numbers that divide neatly — if your arithmetic is producing ugly decimals, check whether you have set the ratio up correctly.

What a Ratio Means

A ratio compares two or more quantities of the same kind. We write it with a colon, like 3:2, read as "three to two". If a class has 18 boys and 12 girls, the ratio of boys to girls is 18:12.

The order matters. Boys to girls is 18:12, but girls to boys is 12:18. ICAS multiple-choice options almost always include the reversed ratio as a distractor, so underline which quantity is named first in the question.

A ratio compares part to part. A fraction compares part to whole. If the ratio of red to blue counters is 3:2, there are 5 parts altogether, so the fraction of red counters is 3/5, not 3/2. Converting between these two views is one of the most-tested ideas in this strand.

Simplifying Ratios

Simplify a ratio by dividing every term by the same common factor, just like simplifying a fraction:

• 18:12 = 9:6 = 3:2 (divide by 2, then by 3)
• 45:30 = 3:2 (divide by 15 in one step)

For three-part ratios, find a factor common to all three: 24:36:60 = 2:3:5 (divide all by 12). A ratio is fully simplified when no whole number greater than 1 divides every term.

If the quantities have different units, convert them to the same unit first. The ratio 50 cm to 2 m is not 50:2 — it is 50 cm : 200 cm = 1:4.

Dividing a Quantity in a Given Ratio

This is the classic ICAS multi-step question. The method has three steps:

1. Add the parts to find the total number of shares.
2. Find one share by dividing the total quantity by the total parts.
3. Multiply one share by each part.

Worked example. $84 is shared between Ava and Ben in the ratio 3:4. How much does each receive?

• Total parts: 3 + 4 = 7
• One part: $84 ÷ 7 = $12
• Ava: 3 × $12 = $36; Ben: 4 × $12 = $48
• Check: $36 + $48 = $84 ✓

Always check that the shares add back to the original total — it takes five seconds and catches the most common error, which is dividing the total by just one of the parts.

Harder variant. The ratio of fiction to non-fiction books on a shelf is 5:3 and there are 30 fiction books. How many books are there altogether?

• Fiction is 5 parts = 30 books, so 1 part = 30 ÷ 5 = 6 books.
• Total parts: 5 + 3 = 8, so total books = 8 × 6 = 48.

Here you work from one share rather than the whole, but the unitary idea is the same: find one part first, then scale. Notice the numbers divide exactly — ICAS questions are engineered to give whole-number answers, so if your one-part value comes out as a messy decimal, that is your cue to re-read the question and re-check the setup before going further.

Equivalent Ratios, Recipes and Scaling

Equivalent ratios are formed by multiplying every term by the same number: 2:3 = 4:6 = 10:15. This is the engine behind recipe and mixing questions.

Worked example. A cordial is mixed using concentrate and water in the ratio 1:6. How much water is needed for 250 mL of concentrate?

• The scale factor from 1 part to 250 mL is 250.
• Water: 6 × 250 = 1500 mL = 1.5 L.

Worked example. A recipe for 4 people uses 300 g of flour. How much flour is needed for 10 people?

• Flour per person: 300 ÷ 4 = 75 g
• For 10 people: 75 × 10 = 750 g

This "divide down to one, multiply up to many" pattern is called the unitary method and it solves most proportion questions ICAS sets.

Rates and Best Value

A rate compares quantities measured in different units: kilometres per hour, dollars per kilogram, beats per minute. A unit rate states the amount per single unit, and it is the fair basis for comparisons.

Best value example. Brand A sells 250 g of coffee for $6.00; Brand B sells 400 g for $8.80. Which is better value?

• Brand A: 600 ÷ 250 = 2.4 cents per gram
• Brand B: 880 ÷ 400 = 2.2 cents per gram

Brand B is cheaper per gram, so it is the better value — even though it costs more overall. ICAS loves this trap: the cheapest ticket price is not always the best value.

A quick mental alternative is to scale to a common amount. Brand A at 400 g would cost 600 × 400/250 = 960 cents = $9.60, which is more than $8.80, so Brand B wins. Both methods work; use whichever keeps the numbers friendlier.

Direct Proportion

Two quantities are in direct proportion when they scale by the same factor: double one and the other doubles. Graphically this is a straight line through the origin. Typical ICAS contexts are currency conversion, map scales, and pay for hours worked.

Worked example. 5 identical notebooks cost $12.50. What do 8 notebooks cost?

• One notebook: $12.50 ÷ 5 = $2.50
• Eight notebooks: 8 × $2.50 = $20.00

Watch for questions that are not direct proportion: "3 workers build a wall in 8 days, how long for 6 workers?" More workers means less time — that is inverse proportion, and the unitary method flips (total work = 3 × 8 = 24 worker-days, so 6 workers need 24 ÷ 6 = 4 days). ICAS includes these in the harder papers to test whether you actually understand proportion or are just pattern-matching.

How ICAS Phrases These Questions

• "The ratio of X to Y is..." — check the order before you compute.
• "Which is the best value?" — compute a unit rate for every option; do not trust the headline price.
• "In the same ratio..." or "at this rate..." — signals direct proportion and the unitary method.
• Answers are usually whole or friendly decimals; ugly arithmetic means re-check the setup.