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Repeated Identity | Linking Ratios Through the Same Quantity in PSLE Mathematics

Repeated Identity is a PSLE Mathematics method for problems where the same actual quantity appears inside two different ratios or relationships. That repeated quantity acts as an identity bridge: its units can be equalised so separate ratios become one connected structure.

Students searching for repeated identity method, linked ratios, three-person ratio problems or PSLE ratio heuristics often try to combine two ratios directly. That fails when the shared quantity is represented by different numbers of units. The method is to identify the repeated quantity, equalise its units, then merge the relationships.

This page is the canonical Repeated Identity owner under the PSLE Mathematics Heuristics hub.

Quick answer: what is the repeated identity?

It is the same real quantity appearing in more than one ratio.

Example: A:B = 2:3 and B:C = 4:5. B is the repeated identity.

Because B is 3 units in the first ratio and 4 units in the second, the ratios cannot yet be merged. Equalise B first.

Worked example 1: combine two ratios

Given: A:B = 2:3 and B:C = 4:5.

LCM of 3 and 4 = 12.

Scale A:B by 4 → 8:12.
Scale B:C by 3 → 12:15.

Therefore A:B:C = 8:12:15.

The shared B now represents 12 equal units in both relationships.

Worked example 2: total known

Problem: A:B = 2:3 and B:C = 4:5. Altogether A, B and C have 350 marbles. How many marbles does C have?

Combined ratio = 8:12:15.
Total units = 35.
1 unit = 350 ÷ 35 = 10.
C = 15 units = 150 marbles.

Worked example 3: difference known

Problem: A:B = 3:5 and B:C = 2:7. C has 108 more than A. Find B.

Repeated identity B is 5 units in the first ratio and 2 in the second. LCM = 10.

A:B = 6:10.
B:C = 10:35.
Combined A:B:C = 6:10:35.

Difference C − A = 29 units = 108.
1 unit = 108 ÷ 29.

B = 10 units. The arithmetic produces a non-integer here; in a school problem, values are usually chosen for a cleaner result. The structural lesson is the equalisation step.

When constructing practice, choose data that make the final quantities sensible in context.

Why equalisation is necessary

A ratio tells us relative parts, not absolute unit size. If B is 3 units in one ratio and 4 in another, those unit counts describe the same actual B from different comparisons. The only safe merge is to rescale until B’s unit count matches.

The four-step method

  1. Identify the repeated quantity.
  2. Find the least common multiple of its unit counts.
  3. Scale both ratios so the repeated quantity matches.
  4. Merge the ratios and solve using the known total, difference or actual value.

Three linked ratios

The method can extend across more than two relationships.

Example structure: A:B = 2:3, B:C = 4:5, C:D = 6:7.

Equalise B first to connect A:B:C, then equalise C between that combined ratio and C:D. Work step by step rather than trying to find one giant common multiple immediately.

Repeated Identity with fractions

The same idea appears when one actual quantity is described as a fraction of two different wholes.

Example: B is 3/5 of A and 4/7 of C. B is the repeated identity. Represent B with a common number of units so A, B and C can be compared.

This is one reason units and fractions belong to the same larger heuristic family.

Repeated Identity versus Constant Part

Both methods use a common quantity as a bridge. The difference is context.

  • Repeated Identity: the same quantity appears in two simultaneous relationships.
  • Constant Part: the same quantity remains unchanged across two time states.

Mathematically, both depend on preserving identity before scaling.

Repeated Identity versus Equal Fractions

Equal Fractions problems often create a shared actual amount expressed as different fractions of different wholes. Repeated Identity is the broader ratio idea: one common quantity links relationships. Atlas treats Equal Fractions separately because the fraction interpretation creates its own recurring problem structures.

A bar-model view

Draw B once conceptually, then show how each ratio partitions it differently. Rescale the bars until B has the same number of equal subdivisions in both diagrams. Once B matches, the other quantities inherit a common unit scale.

Common errors

  • combines ratios without equalising the shared quantity
  • equalises the wrong letter
  • uses addition instead of multiplication to scale ratios
  • changes one side of a ratio without scaling the other
  • finds the combined ratio but uses the wrong total units
  • confuses a repeated identity with a quantity that merely has the same numerical value by coincidence

Transfer practice

Practice 1

Ratios: A:B = 2:5 and B:C = 3:4.

Combined: Equalise B to 15 units → A:B:C = 6:15:20.

Practice 2

Ratios: P:Q = 4:7 and Q:R = 2:9.

Combined: Equalise Q to 14 units → P:Q:R = 8:14:63.

Practice 3

Ratios: X:Y = 3:8 and Y:Z = 6:5.

Combined: Equalise Y to 24 units → X:Y:Z = 9:24:20.

Practice 4

Ratios: M:N = 5:6 and N:P = 9:2.

Combined: Equalise N to 18 units → M:N:P = 15:18:4.

Frequently asked questions

What is a repeated identity?

The same actual quantity appearing in two or more mathematical relationships.

Why use the LCM?

It gives a convenient common unit count for the repeated quantity so the ratios can be merged.

Can I just multiply the two ratios together?

No. Scale each entire ratio so the shared quantity matches first.

Is this only for three-person ratios?

No. It can connect prices, groups, fractions, rates and other linked relationships.

How is this related to Constant Part?

Both use a common unchanged quantity as a bridge, but Repeated Identity usually links simultaneous relationships rather than before-and-after states.

Where does this sit in Atlas?

This is the canonical Repeated Identity owner under PSLE Mathematics Heuristics.

The final Repeated Identity rule

When two relationships share the same real quantity, make that quantity look the same in units before combining anything. Identity first, scaling second, merging third.

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