If a : 2b = 5 : 6 and 3a : 2c = 5 : 6, then a : b : c is equal to:

When two given ratios share a common variable (here, a appears in both a : 2b and 3a : 2c), each ratio can be rewritten using its own scaling parameter, and equating the two expressions obtained for the shared variable links every quantity to one common parameter. The combined ratio a : b : c is then read directly from that single parameter.

1. From a : 2b = 5 : 6, write a = 5k and 2b = 6k, so b = 3k.

2. From 3a : 2c = 5 : 6, write 3a = 5m and 2c = 6m, so a = 5m/3 and c = 3m.

3. Equate the two expressions obtained for a: 5k = 5m/3, which gives k = m/3.

4. Substitute back: a = 5k = 5m/3, b = 3k = m, c = 3m.

5. Multiply every term by 3 to clear the fraction: a : b : c = 5m : 3m : 9m = 5 : 3 : 9.

Substituting a = 5, b = 3, c = 9 back into the original relations confirms consistency: a : 2b = 5 : 6, matching the first given relation, and 3a : 2c = 15 : 18 = 5 : 6, matching the second.

Therefore, a : b : c = 5 : 3 : 9.