Scale by a multiplier (direct proportion)
Direct proportion means multiplying every quantity by the same scale factor. For a recipe for people made for , multiply by . On a map, represents .
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. \2403:53+5=8=240\div 8=30$90$150$.
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.
Drawn from real examiner reports.
Dividing by the wrong total
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.
A commonly reported examiner note records 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.
Forgetting the whole (total)
Finding a part but not the total (or using the wrong total) when a fraction or proportion of the whole is required.
A commonly reported examiner note records 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 a recent higher-tier paper some divided by the wrong total.
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.
A commonly reported examiner note records 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.
Multiply vs divide when reversing
When working backwards from one known share to find another, students apply the multiplier the wrong way round — multiplying when the relationship requires division.
A commonly reported examiner note records 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.
Ratio share vs fraction of the whole
In the ratio , the first share is of the whole (parts over TOTAL parts), not . Using the wrong denominator when a fraction of the total is asked for is a common slip.
Label shares; re-check they sum to 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.
Best value: compare per single unit
For best value, reduce each option to a common measure — price per single unit (or amount per dollar) — then compare. The smallest price-per-unit wins; state which option is best value and why.
Write "y = k ×" for proportion problems
For any proportion question, write an equation with a constant: (direct), (inverse). Substitute the given pair to find , then use the formula. A sign alone earns no answer.
Split in the ratio with this three-step routine:
Example — share \2403:5: $$\text{parts}=3+5=8,\qquad \text{one part}=\frac{240}{8}=30,$$ $$\text{Amy}=3\times 30=\90,\qquad \text{Ben}=5\times 30=$150.$$ Self-check: the shares must add back to the original total: . ✓
A recipe for 4 people is made for 10. What is the scale factor?
Priya and Quinn share \3604:5$. Work out how much each of them receives.
