When a problem changes one group and leaves the other unchanged, make the unchanged group use the same number of units in both ratios. Then the difference in units for the changed group matches the amount added or removed.
This works even when the two ratios use different numbers for the group that stayed the same. Ratio terms describe relative sizes. Rescale one or both ratios before comparing the units.
- Identify the group whose amount stays the same.
- Rewrite both ratios with the same term for that group.
- Use the change in the other term to find one unit.
Worked example | Six beads are added to one box
Boxes A and B contain beads in the ratio 3:5. Six beads are added to Box A. The new ratio of A to B is 7:10. How many beads were in each box originally?
Box B stays the same. Rewrite 3:5 as 6:10 so B has ten units in both ratios. Box A goes from six units to seven units, so one unit is six beads.
Originally A had 6 × 6 = 36 beads and B had 10 × 6 = 60 beads. After the addition, 42:60 simplifies to 7:10.
Draw the unchanged bar first
Draw two B bars of equal length. Label each ten units. Next draw an A bar of six units for the start and seven units for the end. The extra block on the second A bar represents the six new beads.
The combined number rises from 96 to 102 because the beads come from outside the two boxes. In a transfer from one box to the other, both box amounts would change.
Try a removal instead
Two trays contain counters in the ratio 5:6. Eight counters are removed from the first tray. The new ratio is 2:3. Find the original numbers.
Answer. Rewrite 2:3 as 4:6 to align the second tray. The first tray loses one unit, so a unit is eight counters. The original numbers are 40 and 48.
Frequently asked questions
Does the unchanged group always appear second?
Either group can stay unchanged. Keep the same order for both ratios before matching the unchanged term.
What if neither term is a multiple of the other?
Use a common multiple. For example, if the unchanged group has four units in one ratio and six in the other, scale both to twelve units.
Official information
MOE, Primary Mathematics Syllabus (Primary 1 to 6), Primary Six, equivalent ratios. Syllabus checked 11 November 2026. The examples are original.