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Repeated Identity

Joining two ratios that share a common quantity into a single three-way ratio.

⏱ 5 min · 🎯 4 things to master

Sometimes a problem gives you two ratios that share a person in the middle — "Alice to Bob is 3 to 4" and "Bob to Charlie is 2 to 5". Bob appears in both, but with different numbers. The repeated-identity method joins the two ratios into one by making that shared quantity match. Then you can compare all three at once.

Parents: let your child find the shared person before revealing. The "Method tip" boxes name the move a marker rewards — making the repeated quantity the same in both ratios.

By the end you'll be able to join two ratios into a single three-way ratio and solve. Let's link them.

Find the shared quantity

The is the person named in both ratios. In "Alice : Bob = 3 : 4" and "Bob : Charlie = 2 : 5", that is Bob. The trouble is Bob is 4 in the first ratio and 2 in the second — the same Bob, two different numbers.

🤔 Predict first: Alice : Bob = 3 : 4 and Bob : Charlie = 2 : 5. Who is the repeated identity?

Make Bob match

Make Bob the same number in both ratios using the lowest common multiple of 4 and 2, which is 4. The first ratio already has Bob as 4. Double the second ratio so Bob becomes 4: Bob : Charlie = 2 : 5 becomes 4 : 10. Now both agree that Bob is 4, so you can write the whole thing: Alice : Bob : Charlie = 3 : 4 : 10.

Try the find-one-unit step. If the three of them have 51 items altogether, step one unit until the three bars total 51.

Join the ratios, then find one unit

Predict first: After making Bob match, what is Bob : Charlie?

The total is 3 + 4 + 10 = 17 units = 51, so 1 unit = 3. Charlie has 10 × 3 = 30.

Watch out — these are easily mixed up

Quick recap

🎯 Mastery check

Answer all 6 — your progress is saved on this device.

  1. In "P : Q = 2 : 3" and "Q : R = 4 : 5", who is the repeated identity?

  2. Q is 3 in the first ratio and 4 in the second. Match Q at…

  3. To change Bob : Charlie from 2 : 5 to 4 : 10, you…

  4. Alice : Bob : Charlie = 3 : 4 : 10 and the total is 34. What is one unit?

  5. With Alice : Bob : Charlie = 3 : 4 : 10 and 1 unit = 2, how much does Charlie have?

  6. Why must the shared quantity be made equal before combining the ratios?