When two ratios share a group, make that group’s parts match before joining the ratios. If the question gives the sum of the other two groups, use only their combined parts to find the value of one part.
MOE’s Primary Six Mathematics syllabus includes equivalent ratios and ratios of two or three quantities. This original example uses a total that covers only two groups, even though three appear in the question.
- Match the two appearances of the shared group.
- Mark which groups are included in the given sum.
- Find the value of one part and use it for the group requested.
Worked example | Red and green beads total 57
The ratio of red to blue beads is 3:4. The ratio of blue to green beads is 6:5. There are 57 red and green beads altogether. How many blue beads are there?
Blue has four parts in the first ratio and six in the second. Rewrite red:blue as 9:12 and blue:green as 12:10. The combined ratio is red:blue:green = 9:12:10.
The 57 beads cover red and green, so they represent 9 + 10 = 19 parts. One part is 57 ÷ 19 = 3 beads. Blue has 12 parts, so there are 36 blue beads.
Check all three counts. Red has 27, blue has 36 and green has 30. Red plus green gives 57, and the total across all three colours is 93.
Which parts belong to the given number?
Dividing 57 by 31 would treat 57 as the total of all three colours. Underline the words “red and green” beside the number and circle just those two entries in the combined ratio.
If the same question gave 93 beads altogether, then 31 parts would be the correct starting point. The combined ratio is unchanged, but the group named by the given number determines the denominator.
Try a new pair of ratios
Cats:dogs = 2:3 and dogs:rabbits = 4:5. Cats and rabbits total 46 animals. How many dogs are there? The joined ratio is 8:12:15. Cats and rabbits make 23 parts, so each part is two animals. There are 24 dogs.
A parent can ask the pupil to name the shared group before joining the ratios and the groups covered by 46 before dividing. Those two questions check different steps.
Frequently asked questions
Do I need the smallest common multiple?
Any common multiple gives equivalent results. The smallest keeps the working short.
Why is the shared group counted once?
Both ratio statements refer to the same group. The combined ratio lists that group once.
Official information
MOE’s 2021 Primary Mathematics syllabus, October 2025 update, Primary Six Ratio. Checked 9 November 2026.