How do you write 6 : 2 in the form n : 1?

n = 3. To turn the 2 into a 1 you divide by 2, so you divide the 6 by 2 as well: 6 ÷ 2 = 3, giving 3 : 1. Equivalent ratios keep the same shape, so you scale both parts by the same factor. The common wrong answer 4 subtracts (6 − 2), which changes the proportion.

How do you write 4 : 9 in the form 1 : n?

n = 2.25, so 4 : 9 = 1 : 2.25. To turn the 4 into a 1 you divide by 4, so divide the 9 by 4 too: 9 ÷ 4 = 2.25. A ratio is allowed to contain a decimal. The common wrong answer 5 subtracts (9 − 4).

A recipe uses flour and sugar in the ratio 2 : 3. If you use 8 units of flour, how much sugar?

12 units. The flour grew from 2 to 8, a factor of 8 ÷ 2 = 4, so multiply the sugar by the same factor: 3 × 4 = 12. The common wrong answer 9 adds the difference (3 + 6), which changes the recipe.

GCSE Maths Foundation · AQA · Ratio

Ratio: scale by multiplying, not adding

Equivalent-ratio questions trip students who add or subtract the same amount to each part. Asked to write $6 : 2$ in the form $n : 1$ , they answer 4 — doing $6 - 2$ — instead of dividing both parts by the same factor. But equivalent ratios keep the same shape: to turn the 2 into a 1 you divide by 2, so you divide the 6 by 2 as well, giving $6 : 2 = 3 : 1$ .

The thirty-second fix: multiply or divide both parts by the same number, never add or subtract. And a ratio is allowed to contain a decimal, so $4 : 9 = 1 : 2.25$ .

Ready to fix this? The Ratio lesson works through this misconception and the others in Ratio, one altitude at a time.

How to spot it in your own work

You subtracted to rewrite a ratio — writing $6 : 2 = 4 : 1$ (doing $6 - 2$ ) instead of $3 : 1$ .
You added to rewrite a ratio — writing $6 : 2 = 8 : 1$ (doing $6 + 2$ ).
You scaled by adding a constant — taking $2 : 3$ with 8 flour and answering 9 sugar ( $3 + 6$ ) instead of $3 \times 4 = 12$ .
You rejected or rounded a decimal answer — refusing $1 : 2.25$ because “a ratio can’t have a decimal”.
You added a scale factor instead of multiplying by it — a 3 cm length at scale $1 : 4$ answered 7 ( $3 + 4$ ) instead of 12.

An exam question that triggers it

Here is the canonical AQA Foundation trigger (Nov24 P1 Q12 shape, non-calculator):

Write the ratio 6 : 2 in the form n : 1.

The misconception answer is 4 — subtracting, $6 - 2 = 4$ . But to turn the 2 into a 1 you divide by 2, and you must divide the 6 by 2 as well.

Scale both parts by the same factor: $6 : 2 = (6 \div 2) : (2 \div 2) = 3 : 1$ , so n = 3.

Why students fall for this

A ratio is built from addition-friendly whole numbers, so under pressure students reach for the most familiar operation and add or subtract the same amount to each part. So $6 : 2$ in the form $n : 1$ becomes 4 (the 2 was turned into a 1 by subtracting 1, and 1 was subtracted from the 6 too — or the gap $6 - 2$ is read off directly). But a ratio records a proportion, not a difference: $6 : 2$ says the first part is three times the second. Only multiplying or dividing both parts by the same number keeps that relationship.

The same belief scales a recipe by adding a constant. Asked to grow $2 : 3$ so the flour is 8, students add 6 to the flour and add 6 to the sugar, answering 9 — but adding 6 makes the smaller part grow proportionally more, so $8 : 9$ is a different recipe. The flour grew by a factor of $8 \div 2 = 4$ , so the sugar must too: $3 \times 4 = 12$ .

A further face rejects decimals. Writing $4 : 9$ as $1 : n$ needs $9 \div 4 = 2.25$ , but students who believe “ratios must be whole numbers” subtract instead ( $9 - 4 = 5$ ) or round 2.25 to 2. A ratio is perfectly allowed to contain a decimal. AQA Foundation papers exploit every face — write 6 : 2 as n : 1 (Nov24 P1 Q12), write 4 : 9 as 1 : n (Jun24 P3 Q17c), a scale drawing to an actual length (Jun22 P2 Q15), and recomputing a ratio after a change (Nov24 P3 Q24a).

The fix: Equivalent ratios keep the same shape — multiply or divide both parts by the same factor, never add or subtract, and accept a decimal

Writing n : 1 or 1 : n: divide both parts by the same number. $6 : 2 = 3 : 1$ (÷2 on both); $4 : 9 = 1 : 2.25$ (÷4 on both). The decimal 2.25 is correct, not a mistake.

Scaling up: find the factor, multiply both parts by it. Recipe $2 : 3$ with 8 flour: factor $8 \div 2 = 4$ , so sugar $= 3 \times 4 = 12$ — never $3 + 6 = 9$ .

Scale drawings: multiply by the scale factor, do not add it. At $1 : 4$ a 3 cm length is $3 \times 4 = 12$ m, not $3 + 4 = 7$ .

Worked example

A recipe uses flour and sugar in the ratio 2 : 3. You use 8 units of flour. How much sugar?

Find the factor that scales the known part. The flour grew from 2 to 8, so the factor is $8 \div 2 = 4$ .
Multiply the other part by the same factor.
$\text{sugar} = 3 \times 4 = 12$
Check the shape is kept. $2 : 3$ scaled by 4 is $8 : 12$ , and $8 : 12$ simplifies back to $2 : 3$ (÷4) — same recipe.

The same rule simplifies and writes the n : 1 form: $6 : 2 = 3 : 1$ (÷2), and $4 : 9 = 1 : 2.25$ (÷4) — a decimal is fine. Adding the same amount to each part, like 3 + 6 = 9, would change the ratio.

Find out if this is costing you marks

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Common questions

Why can’t I subtract to write 6 : 2 as n : 1?: Because a ratio records a proportion, not a difference. To turn the 2 into a 1 you divide by 2, so you divide the 6 by 2 as well: $6 \div 2 = 3$ , giving $3 : 1$ . Subtracting ( $6 - 2 = 4$ ) would give $4 : 1$ , a different ratio.
Can a ratio contain a decimal?: Yes. Writing $4 : 9$ as $1 : n$ needs $9 \div 4 = 2.25$ , so $4 : 9 = 1 : 2.25$ . The decimal just records how many times bigger the second part is. Rounding 2.25 to 2, or subtracting to avoid the decimal, gives the wrong ratio.
How do I scale a recipe in the ratio 2 : 3 to use 8 units of flour?: Find the factor: the flour grew $2 \to 8$ , so $8 \div 2 = 4$ . Multiply the sugar by the same factor: $3 \times 4 = 12$ . The wrong answer 9 adds the difference ( $3 + 6$ ), which changes the recipe.
On a scale drawing, do I add or multiply by the scale factor?: Multiply. At a scale of $1 : 4$ a 3 cm length on the drawing is $3 \times 4 = 12$ m in real life. Adding the scale factor ( $3 + 4 = 7$ ) treats it as a constant to add, which breaks the proportion.

Related misconceptions

Sharing in a ratio: divide by the sum of the partsThe neighbouring ratio skill — to share in 1 : 3 you divide the total by 1 + 3 = 4, then multiply each share, scaling by the same factor.
Turning a ratio into a fractionA ratio part is a fraction of the whole (the sum), not of the other part — the same care about what the numbers stand for.
Direct and inverse proportionProportion is multiplicative too — quantities scale by a constant factor, the same idea that makes ratios multiply, not add.

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