Primary 6 Mathematics Learning Guide | Repeated Identity, Chained Ratios and Shared Middle Quantities

Wait, What? The Same Quantity Can Appear in Two Different Ratios

Some Primary 6 Mathematics problems do not give one complete ratio. Instead, they give two linked comparisons: A:B and B:C, or red:blue and blue:green. The middle quantity appears twice. That repeated identity is the bridge that lets the learner combine both relationships into one three-part ratio.

This is sometimes taught as a repeated identity or chained-ratio method. The key is to make the shared quantity carry the same number of ratio units in both comparisons before joining the chains.

Two ratios can be joined only after the shared quantity means the same number of units in both.

Quick Answer

A reliable routine is:

IDENTIFY THE REPEATED QUANTITY → WRITE BOTH RATIOS → SCALE THEM SO THE SHARED QUANTITY MATCHES → JOIN THE OUTER QUANTITIES → FORM THE THREE-PART RATIO → MAP TO ANY KNOWN TOTAL, DIFFERENCE OR ACTUAL VALUE → CHECK BOTH ORIGINAL RATIOS.

1. The Basic Structure

Suppose A:B = 2:3 and B:C = 4:5. The repeated identity is B. But B is 3 units in the first ratio and 4 units in the second, so the unit sizes are not aligned.

Use a common multiple for B: 12. Scale A:B by 4 to get 8:12. Scale B:C by 3 to get 12:15. Therefore A:B:C = 8:12:15.

2. Why Raw Ratios Cannot Be Joined

A common error is to write 2:3:5 from A:B=2:3 and B:C=4:5. That silently treats B=3 and B=4 as the same number of units, which is impossible unless the unit sizes differ.

Repeated identity problems are therefore alignment problems before they are arithmetic problems.

3. Worked Example: Chained Ratios

Red:Blue = 3:4 and Blue:Green = 6:5. Find Red:Blue:Green.

1. Repeated identity = Blue.
2. Blue is 4 units in the first ratio and 6 in the second.
3. LCM of 4 and 6 = 12.
4. Scale 3:4 by 3 → 9:12.
5. Scale 6:5 by 2 → 12:10.
6. Combined ratio = 9:12:10.

4. Shared Quantity as an Anchor

The repeated quantity acts like a hinge between two relationships. Once aligned, the two ratios become compatible descriptions of one system.

This is conceptually similar to aligning a constant part across before-and-after ratios, except here the shared identity connects different pairs rather than different time states.

5. Worked Example With a Known Total

A:B = 2:5 and B:C = 3:4. Altogether A+B+C = 126. Find C.

1. Align B: first B=5 units, second B=3 units.
2. LCM=15.
3. A:B = 6:15.
4. B:C = 15:20.
5. Combined ratio A:B:C = 6:15:20.
6. Total ratio units = 41.
7. 126÷41 is not whole, so the data produces non-integer unit value.

In a continuous context this can still be mathematically valid. In a whole-object context, the values should normally be chosen to make the unit value compatible. Feasibility remains part of world-class checking.

6. Worked Example With a Difference

A:B = 3:7 and B:C = 14:9. C is 18 more than A. Find A, B and C.

1. Align B: first ratio 3:7 becomes 6:14.
2. Second ratio is already 14:9.
3. Combined ratio = 6:14:9.
4. C−A = 3 units = 18.
5. 1 unit = 6.
6. A=36, B=84, C=54.

Check: 36:84 simplifies to 3:7, and 84:54 simplifies to 14:9.

7. More Than Two Ratios Can Be Chained

If A:B, B:C and C:D are given, the learner can align B first, form A:B:C, then align C with the C:D relationship and extend the chain.

At each stage, only one repeated identity needs to be controlled. Do not attempt to scale everything mentally at once.

8. Tables Reduce Cognitive Load

Relationship	A	B	C
A:B	2	3	—
B:C	—	4	5
Aligned	8	12	15

The table makes the repeated identity visible and reduces accidental ratio mixing.

9. Repeated Identity and Fractions

If A is 2/3 of B and B is 4/5 of C, then A:B=2:3 and B:C=4:5. The fraction statements can therefore be converted into chained ratios.

This builds a bridge between fraction-of-a-quantity language and three-part comparison models.

10. Repeated Identity and Percentage

If A is 60% of B, then A:B=3:5. If B is 75% of C, then B:C=3:4. Align the repeated B and combine the chain.

Percentage conversion can therefore feed directly into repeated-identity reasoning.

11. Repeated Identity and Algebra

A:B=2:3 can be written A=2u, B=3u. B:C=4:5 can be written B=4v, C=5v. Because both expressions describe the same B, 3u=4v. Choosing u=4k and v=3k gives A=8k, B=12k, C=15k.

The algebra reveals why common-multiple alignment is valid.

12. Shared Middle Quantity Can Be a Total, Not a Category

The repeated identity need not be a named group like B. It may be a common total, common remainder, common distance or common amount at an intermediate stage.

The deeper skill is identity tracking: recognising that two statements refer to the same underlying quantity.

13. When Repeated Identity Is Not Enough

If the repeated quantity changes between the two statements, they cannot be aligned as though they describe the same state. For example, B before a transfer and B after a transfer are different quantities unless the problem explicitly makes them equal.

Identity requires both same label and same state.

14. Common Error Families

Error	What it looks like	Repair
Raw joining	Writes 2:3:5 from 2:3 and 4:5	Align the repeated quantity first
Wrong identity	Aligns two quantities that only look similar	Check label and state
One-sided scaling	Changes B without scaling its partner	Scale both terms of a ratio together
LCM overkill	Uses huge multiples unnecessarily	Use the lowest useful common multiple
No original check	Gets a three-part ratio but never verifies source ratios	Simplify back to both original comparisons
State mixing	Links before-B to after-B	Separate time states explicitly

15. A First-Weak-Link Diagnostic

1. Identity: Can the repeated quantity be found?
2. State control: Is it truly the same quantity in both relationships?
3. Alignment: Can a common unit count be created?
4. Combination: Can the chained ratio be formed?
5. Mapping: Can totals, gaps or actual values be attached?
6. Verification: Can the chain simplify back to every source ratio?
7. Transfer: Can fraction and percentage statements be converted into chained ratios?

16. Examination Control

• Circle the repeated identity.
• Write the two ratios on separate lines.
• Align the repeated quantity before combining.
• Use a table for three or more linked quantities.
• Attach the actual known total or difference only after the combined ratio is stable.
• Check by reducing back to the original ratios.

17. What Parents Can Ask

• “Which quantity appears in both relationships?”
• “Does it have the same number of units yet?”
• “What common multiple would align it?”
• “Can you build one three-part ratio now?”
• “Can you check both original ratios from your answer?”

18. What Tutors Should Protect

• Identity fidelity. Same name must mean same state.
• Equivalent-ratio control. Scale complete ratios.
• Low-load representation. Use tables for chained systems.
• Cross-topic transfer. Convert fractions and percentages into ratios.
• Verification. Reduce the combined ratio back to each source.
• Prompt reduction. Let learners find the shared middle quantity independently.

19. Official Process Connection

Repeated-identity problems strengthen proportional reasoning, representation, mathematical connections and algebraic thinking within the Singapore Primary Mathematics framework. The named method is an instructional structure for combining linked relationships.

Official reference: MOE Mathematics Syllabus — Primary One to Six.

Continue the Primary 6 Mathematics Series

• Primary 6 Mathematics Learning Hub
• Equal Stage Problems and End-State Reconstruction
• Stack Model, Split Model and Two-Dimensional Visual Reasoning
• Three-Quantity Comparison Models and Multi-Bar Alignment

The Quiet Return

Repeated identity turns two incomplete ratios into one connected system by forcing the shared quantity to mean exactly the same thing in both.

The mature Primary 6 habit is to ask: what quantity appears twice, and have I aligned it before I join the ratios?