When items move from one group to another, the two group sizes change and the combined total stays the same. Rewrite both ratios so their parts add to the same number. Then compare one group before and after the transfer.
MOE’s Primary Six Mathematics syllabus includes equivalent ratios and dividing a quantity in a given ratio. The examples below show how to use the unchanged total when the ratios originally have different numbers of parts.
- Check that the items stay within the two groups.
- Rewrite the ratios with the same total number of parts.
- Match the transfer to the change in one group.
Worked example | Six counters move between boxes
Box A and Box B contain counters in the ratio 5:3. Six counters move from A to B, and the new ratio is 3:2. Find the total number of counters.
The first ratio has eight parts and the second has five. Both describe the same total, so express both in 40 equal parts. Before the transfer, A:B = 25:15. After the transfer, A:B = 24:16.
| Stage | Box A | Box B | Combined parts |
|---|---|---|---|
| Before | 25 | 15 | 40 |
| After | 24 | 16 | 40 |
A lost one of those 40 parts. That part represents six counters, so the total is 40 × 6 = 240. The original counts were 150 and 90. The new counts are 144 and 96, which simplify to 3:2.
Why 5 parts minus 3 parts does not give the transfer
One original part is one eighth of the total. One new part is one fifth. Those parts have different sizes. Comparing 5 and 3 directly treats them as equal.
The rewritten ratios use one fortieth of the total for every part. You can check either group. A decreases by one part and B increases by one part, so both calculations describe the same transfer.
Try a transfer that makes the groups equal
Two bags contain beads in the ratio 7:5. Moving nine beads from the larger bag to the smaller bag makes their counts equal. Write the final ratio as 6:6 so both ratios total 12 parts.
The larger bag loses one part, worth nine beads. The combined total is 108 beads. Check the original counts of 63 and 45 become 54 and 54.
A parent can ask what quantity stays fixed before the child draws any bars. If beads were added from outside the two bags, the combined total would change and this matching method would need to be reconsidered.
Frequently asked questions
Do I have to use 40 parts in the first example?
Any common multiple of eight and five works. Forty is the smallest.
Can I compare the gains in the second group?
Yes. If the transfer stays within the two groups, the gain in one group equals the loss in the other.
Official information
MOE’s 2021 Primary Mathematics syllabus, October 2025 update, Primary Six Ratio. Checked 9 November 2026.