If a : (b + c) = 1 : 3 and c : (a + b) = 5 : 7, then b : (a + c) is equal to –

Concept — A ratio statement p : q = m : n is nothing but the linear equation n·p = m·q written in ratio form. When two such statements tie three unknowns together, the standard method is to express every unknown in terms of one reference quantity. Any ratio that is then asked for becomes a division in which the reference quantity cancels, so the answer is fixed by the given relations alone and does not depend on the actual sizes of a, b and c.

Application

1. Turn the first ratio into an equation: a : (b + c) = 1 : 3 gives 3a = b + c, hence b = 3a − c.

2. Turn the second ratio into an equation: c : (a + b) = 5 : 7 gives 7c = 5(a + b).

3. Substitute b = 3a − c into the second equation: 7c = 5(a + 3a − c) = 5(4a − c) = 20a − 5c.

4. Collect like terms: 7c + 5c = 20a, so 12c = 20a, which simplifies to 3c = 5a.

5. Introduce a single reference quantity k. From 3c = 5a, taking a = 3k gives c = 5k.

6. Back-substitute into b = 3a − c: b = 9k − 5k = 4k. Hence a : b : c = 3 : 4 : 5.

7. Form the required ratio: b : (a + c) = 4k : (3k + 5k) = 4k : 8k = 1 : 2, and k cancels as expected.

Cross-check — put k = 1, that is a = 3, b = 4, c = 5, and test the two given conditions as well as the required one.

Both given conditions are satisfied by a : b : c = 3 : 4 : 5, so b : (a + c) = 1 : 2.