The fourth proportional to three numbers a, b and c is 84. The ratio of a to b is 5 : 6 and the ratio of b to c is 3 : 5. What is the value of a?
35
Combine the two ratios: a : b = 5 : 6 and b : c = 3 : 5 = 6 : 10, so a : b : c = 5 : 6 : 10. Let a = 5k, b = 6k and c = 10k.
The fourth proportional d to a, b and c satisfies a : b = c : d, so \(d = \frac{bc}{a}\).
Substituting, \(d = \frac{6k \times 10k}{5k} = 12k\). Since d = 84, 12k = 84 and k = 7.
Therefore a = 5k = 5 × 7 = 35 (and b = 42, c = 70).
Check: a : b = 35 : 42 = 5 : 6, b : c = 42 : 70 = 3 : 5, and 35 : 42 = 70 : 84 since 35 × 84 = 2940 = 42 × 70.
Hence, 35.
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