Share a ratio: add, divide, multiply
To split a quantity in the ratio , add the parts to get the total number of parts, divide the quantity by that total to find the value of one part, then multiply each share by its number of parts. E.g. shared has parts, so one part , giving and .
Finding one part when another is known
If you are told the value of one share, find the value of a single part first (), then scale up to whatever is asked. The total quantity is the value of one part the total number of parts.
Scale recipes and maps by a multiplier
Direct proportion means multiplying every quantity by the same scale factor. For a recipe for people made for , multiply by . On a map, represents .
Drawn from real examiner reports.
Dividing by the wrong number of parts
When one part of a split is a multiple of another (e.g. one outcome is twice as likely as another), students divide by the wrong number of parts, or skip the division entirely, instead of forming the correct ratio first.
On June 2024 Paper 1H (Q2) the probability of travelling by car was twice that by bus, so the remaining probability had to be divided by $3$; many students did not divide by $3$ (some divided by $2$), and their work then became muddled.
Multiplying when you should divide
When working backwards from one known share to find another, students apply the multiplier the wrong way round — multiplying when the relationship requires division.
On June 2024 Paper 1H (Q14) a common error, after finding one person's amount, was to multiply by $1.5$ rather than divide by $1.5$ to find the next person's share.
Not converting units before forming a ratio
Comparing or combining quantities in different units (cm and m, g and kg, minutes and hours) without converting first gives a ratio in the wrong proportion.
On November 2024 Paper 1H (Q5) some students had to convert lengths (e.g. $70\text{ cm}$ and $18\text{ cm}$) before using them, and unit-conversion slips were noted; the report also stressed that $1\text{ m}^3 = 1000$ litres was not known by many.
Forgetting the whole in a proportion
Finding a part but not the total (or using the wrong total) when a fraction or proportion of the whole is required.
On November 2024 Paper 1H (Q18) students found a partial count but "failed to realise that they also needed the total number of students in order to form a proportion"; on November 2024 Paper 2H (Q8) some divided by the wrong total.
Ratio share is not a fraction of one part
In the ratio the first share is of the whole — parts over the TOTAL parts , not . Writing treats the second part as the whole and overstates the share.
Leaving the answer in the wrong unit
Give the answer in the unit the question names. Scaling flour to when kilograms are asked means converting to ; comparing with directly ignores that the two pack sizes differ.
Add shares back to the original total
Write the number of parts beside each share, and at the end add the shares back up — they must total the original quantity. This catches a wrong-total or wrong-multiplier slip before it costs the mark.
Find the value of one part first
Whenever you know one share, work out the value of ONE part first (share its part-count), then scale to whatever is asked. This single habit fixes both the "multiply not divide" and the "wrong total" slips.
Convert to a common unit before the ratio
Before writing any ratio, put both quantities in the SAME unit: to is , never . Convert first, then simplify.
Best value: compare one common measure
For "best value", reduce both options to the SAME measure — price per unit, or units per pound — then compare. The lowest price-per-unit (or the most units per pound) wins; comparing raw prices of different sizes proves nothing.
Split in the ratio with this three-step routine:
Example — share between Amy and Ben in the ratio : Self-check: the shares must add back to the original total: . ✓
What are the three steps to share a quantity in a given ratio?
Priya and Quinn share in the ratio . Work out how much each of them receives.
