Ratio and Proportion

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1. 1

   Add the parts to find the number of shares

   Share £60 in the ratio 3 : 2 → 3 + 2 = 5 shares

   The ratio tells you how many equal shares the total splits into. Skipping this step and dividing by one of the numbers instead is the most common error in the topic, and it produces an answer that does not add back to the total.

2. 2

   Find the value of one share

   £60 ÷ 5 = £12 per share

   This is the unitary method, and it is the single most useful technique in the whole of proportion. Once one share is known, every question about any number of shares becomes a multiplication.

3. 3

   Multiply out each part

   3 × £12 = £36 · 2 × £12 = £24

   Each part of the ratio takes that many shares. Doing both parts rather than only the one asked for costs seconds and gives you the check in the next step for free.

4. 4

   Check the parts add back to the total

   £36 + £24 = £60 ✓

   The parts must reconstruct the original amount. This catches division by the wrong number instantly — if you had divided by 3 or by 2, the parts would not add up, and no amount of re-reading the working would tell you as quickly.

5. 5

   Simplify ratios like fractions

   12 : 18 → divide both by 6 → 2 : 3

   Ratios simplify by dividing every part by a common factor, exactly as fractions do. Units must match first: 50 cm to 2 m is not 50 : 2 but 50 : 200, which is 1 : 4.

A ratio is parts, a fraction is a share of the whole

In the ratio 3 : 2, the first quantity is three parts and the second is two — but the first is three fifths of the total, not three halves and not three of anything obvious. Students who read 3 : 2 as the fraction 3/2 get consistent, confident, wrong answers, and the working looks fine.

The reliable translation is to convert to fractions explicitly: 3 : 2 means 3/5 and 2/5. Written that way the total is visible, and the question of whether an answer should be bigger or smaller than half answers itself.

• Ratio compares parts to parts
• Fraction compares a part to the whole
• 3 : 2 means 3/5 and 2/5 of the total
• Add the parts first, always
• Match units before simplifying

The unitary method solves most of these questions

Find the value of one, then scale to however many you need. If 7 pens cost £2.10, one costs 30p, so 12 cost £3.60. It sounds too simple to be a named technique, and it handles recipe scaling, currency conversion, best-buy comparisons, map scales and speed questions without modification.

Its advantage over cross-multiplying is that every intermediate number means something. A student who has computed 'one pen costs 30p' can sanity-check that figure; a student who has cross-multiplied has a number with no interpretation and no way to tell whether it is plausible.

Direct and inverse proportion are opposite

In direct proportion, doubling one quantity doubles the other — more pens, more cost. In inverse proportion, doubling one halves the other — more workers, less time. Questions about taps filling a tank or people painting a fence are almost always inverse, and treating them as direct gives an answer that is wrong in the obvious direction.

The check is to ask whether more of the first thing should mean more or less of the second. It takes a second and it decides the entire method, which makes it the highest-value question in the topic.

How we teach ratio

We require the total check on every sharing question. Students who verify that the parts add back to the original amount catch the divide-by-the-wrong-number error themselves, and that error accounts for most of the marks lost in this topic.

We also teach ratio and fractions together rather than as separate units. The 3 : 2 becomes 3/5 and 2/5 translation is what stops the two ideas colliding, and introducing them weeks apart is what makes them collide in the first place.