If (a-b) : (b-c) : (c-d) = 1 : 2 : 3, then what is the value of (a+d) : c?

Concept

When consecutive differences are in the ratio m:n:p, represent them as mk, nk, and pk using one common scale k. Reconstruct the variables by adding these differences from one endpoint.

A ratio is unchanged by a common nonzero scale, so an expression involving the variables can often be reduced without choosing particular values.

Application

1. Let a - b = k, b - c = 2k, and c - d = 3k.

2. From c - d = 3k, obtain c = d + 3k.

3. Then b = c + 2k = d + 5k.

4. Next, a = b + k = d + 6k.

5. Therefore, a + d = 2d + 6k = 2(d + 3k) = 2c, so (a + d) : c = 2 : 1.

Cross-check

Choose d = 1 and k = 1. Then (a, b, c, d) = (7, 6, 4, 1); the consecutive differences are 1, 2, and 3, while (a + d) : c = 8 : 4 = 2 : 1.

Thus, (a + d) : c = 2 : 1.