Sharing in a Ratio

Many students see the “2” in the ratio 2 : 3 and calculate \( \frac{2}{3} \) of £60 = £40. This is the “ratio number as denominator” misconception. The ratio 2 : 3 means there are 2 + 3 = 5 equal parts in total.

We can visualize this using a bar model. The total bar is worth £60. We split it into 5 pieces. Each piece is worth £60 ÷ 5 = £12.

The smaller share is 2 × £12 = £24. The correct fraction is \( \frac{2}{5} \) of the total, not \( \frac{2}{3} \).

For 4 : 6, there are 4 + 6 = 10 parts. Each part is £120 ÷ 10 = £12. The shares are 4 × £12 = £48 and 6 × £12 = £72. For 2 : 3, there are 2 + 3 = 5 parts. Each part is £120 ÷ 5 = £24. The shares are 2 × £24 = £48 and 3 × £24 = £72. Identical results.

This works because 4 : 6 simplifies to 2 : 3. These are Equivalent Ratios. Just as equivalent fractions represent the same value, equivalent ratios represent the same proportional split.

A common mistake is to treat the ratio as describing the cash difference: “4 is 3 more than 1, so the second person gets £3 more.” This is the “ratio means difference” misconception.

The ratio 1 : 4 tells us the multiplicative relationship: the second person gets 4 times as much as the first person. It also tells us there are 5 parts in total. If the total is £100, the shares are £20 and £80 — a difference of £60. The actual difference depends entirely on the total amount shared.

Ben’s share corresponds to the “5” in the ratio 3 : 5. If 5 parts = £40, then 1 part = £40 ÷ 5 = £8. The total number of parts is 3 + 5 = 8, so the total amount = 8 × £8 = £64. Check: Ali gets 3 × £8 = £24, Ben gets 5 × £8 = £40, and £24 + £40 = £64 ✓

Example: £50 (10 parts → £5 per part → shares of £15 and £35)

Another: £100 (10 parts → £10 per part → shares of £30 and £70)

Creative: £10 (10 parts → £1 per part → shares of £3 and £7) — the smallest whole-pound amount that works

Trap: £37 — a student might pick this because they see the digits 3 and 7 in the ratio and think the total should contain those numbers. But 3 + 7 = 10 parts, and £37 ÷ 10 = £3.70. Any valid amount must be a multiple of £10.

Example: 3 : 1 (4 parts → £12 per part → first gets £36, second gets £12)

Another: 5 : 3 (8 parts → £6 per part → first gets £30, second gets £18)

Creative: 47 : 1 (48 parts → £1 per part → first gets £47, second gets £1) — an extreme but valid ratio

Trap: 2 : 4 — a student might think “2 is the first number and it comes first, so the first person gets more.” But 2 : 4 means 2 parts vs 4 parts, so the first person actually gets less.

Example: £100 (10 parts → £10 per part → shares of £20, £30, and £50)

Another: £50 (10 parts → £5 per part → shares of £10, £15, and £25)

Creative: £200 (10 parts → £20 per part → shares of £40, £60, and £100) — works because the total parts is still 10

Trap: £235 — a student might combine the ratio digits (2, 3, 5) to make a total. But 2 + 3 + 5 = 10 parts, and £235 ÷ 10 = £23.50. The total must be a multiple of £10.

Example: 1 : 3 (4 parts → £15 per part → shares of £15 and £45)

Another: 3 : 9 (this simplifies to 1 : 3 — 12 parts → £5 per part → shares of £15 and £45)

Creative: 3 : 1 (the ratio reversed — 4 parts, £15 per part, shares of £45 and £15). Students often forget that the person receiving £45 can appear as either the first or second number in the ratio.

Trap: 4 : 5 — a student might reason “£45 out of £60 simplifies to something with a 5.” But in ratio 4 : 5, there are 9 parts, and £60 ÷ 9 = £6.67. This confuses the ratio between shares (4:5) with the fraction of the total (4/9 and 5/9).

If \( a < b \), then \( b \) is more than half of \( a + b \). The “\( b \)” share is \( \frac{b}{a+b} \) of the total. Since \( a < b \), we know \( a + b < 2b \), so \( \frac{b}{a+b} > \frac{b}{2b} = \frac{1}{2} \). This holds for all positive values of \( a \) and \( b \) where \( a < b \).

True case: Share £50 in the ratio 2 : 3 → 5 parts → £10 per part → shares of £20 and £30. Both whole numbers ✓

False case: Share £13 in the ratio 2 : 3 → 5 parts → £2.60 per part → shares of £5.20 and £7.80. Not whole numbers ✗. It works only when the total is a multiple of 5 (the total number of parts).

Doubling gives \( 2a : 2b \), which simplifies back to \( a : b \). With the doubled ratio, there are twice as many parts, but each part is worth half as much, so these effects exactly cancel out.

True case: 1 : 2 : 8. Largest part is 8. Others sum to 3. 8 > 3 ✓

False case: 3 : 4 : 5. Largest part is 5. Others sum to 7. 5 < 7 ✗.

This is the “divide by each ratio number” misconception. The student divided the whole amount by each number in the ratio separately.

Total Check: £20 + £13.33 = £33.33. This does not add up to £40, so the method must be wrong.

The student is using the “adjust from the midpoint” misconception. This method is mathematically unsound, even though it produced the correct answer in this specific case.

Why did it work here? By coincidence! The difference between the ratio parts ($5 – 3 = 2$) corresponds exactly to the 2 “steps” of £5 the student took from the center. If we tried this with a ratio of 1:3 (difference of 2) for £40, this method would give £15 and £25, but the correct answer is £10 and £30.

This is the “ratio as fraction — wrong denominator” misconception. The student used the other ratio number (3) as the denominator, calculating \( \frac{2}{3} \) instead of \( \frac{2}{5} \).

Total Check: £30 + £15 = £45. The total adds up correctly, which makes this error harder to spot! The real check is the order: in 2:3, the second person should get more, but here the second person only got £15.

This is the “ignoring parts of a three-part ratio” misconception. The student treated 1 : 2 : 3 as if only the first two numbers mattered, dividing by 1 + 2 = 3 parts instead of 1 + 2 + 3 = 6 parts.

The student’s error was treating the third number as “leftover” rather than as a share instruction. All parts of the ratio must be included in the total number of parts.