Example 1
Share $84 between Amir and Bea in the ratio 3:4.
- There are 3 + 4 = 7 equal parts.
- One part is $84 ÷ 7 = $12.
- Amir receives 3 × $12 = $36; Bea receives 4 × $12 = $48. Check: $36 + $48 = $84.
Home › Subjects › O-Level Elementary Maths (E-Maths) › Ratio and proportion
Ratio and proportion: sharing, scaling and inverse proportion
A ratio compares quantities using equal-sized parts. Before calculating, identify what stays fixed: the total, the size of one part, or a product such as workers × time. These are different problems even when the numbers look similar.
Choose the method
Write both quantities in the same units before simplifying a ratio. In 2:3, the first quantity is 2/5 of the total, not 2/3 of it.
For direct proportion y = kx, doubling x doubles y. For inverse proportion y = k/x, doubling x halves y. Find k from one complete pair before using a new value.
For a changed ratio, draw a before-and-after table. Track the quantity that did not change; the part size may be different in the two ratios.
Worked examples
Example 2
Six identical pumps empty a tank in 10 hours. How long would eight pumps take, assuming a constant rate per pump?
- The fixed work is 6 × 10 = 60 pump-hours.
- Time is inversely proportional to the number of pumps: t = 60/8.
- t = 7.5 hours, or 7 hours 30 minutes. More pumps should take less time.
Try it yourself
A map scale is 1:25,000. Two points are 6.4 cm apart on the map. Find the actual distance in km.
Show the worked answer
- Actual distance = 6.4 × 25,000 = 160,000 cm.
- There are 100,000 cm in 1 km, so the distance is 1.6 km.
Common mistakes
- Adding a fixed amount to both sides does not preserve a ratio.
- Inverse proportion needs a reason: the fixed amount of work and constant individual rates are assumptions, not automatic facts.
Sources and review
SEAB 2026 syllabus: O-Level Mathematics 4052: ratio and proportion.
Original Rae practice, prepared with AI assistance. Selected numerical results and their displayed working are automatically checked at publication; this does not verify every explanation. Curriculum references checked on 5 September 2026. No teacher review or SEAB endorsement is claimed.
What the syllabus expects
- ratios whose terms are rational numbers
- reducing a ratio to its simplest form
- map scales for both distance and area
- proportion of the direct and inverse kinds
Continue revising
- Percentages: reverse percentages and successive changes
- Speed and rates: total distance, total time and unit conversions
- Probability: replacement, tree diagrams and checking outcomes
More O-Level Elementary Maths (E-Maths) topics
Whole numbers and the operations performed on them · Percentage · Rate and speed · Algebraic expressions and formulae · Functions together with their graphs · Equations and inequalities · all of O-Level Elementary Maths (E-Maths)