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1. Of the three numbers, the first is twice the second and the second is twice the third. The average of the reciprocal of the numbers is $\frac{7}{72}$ . The numbers are:

A. 36, 18, 9

B. 24, 12, 6

C. 20, 10, 5

D. 16, 8, 4

Solution:

\begin{aligned} Let three numbers be x, y, z . \\ Given, \\ x = 2 y \\ \Rightarrow x = 4 z \\ \Rightarrow y = 2 z \\ \Rightarrow z = z \\ The average of reciprocal numbers is \frac{7}{72} \\ \frac{\frac{1}{x} + \frac{1}{y} + \frac{1}{z}}{3} = \frac{7}{72} \\ \Rightarrow \frac{y z + x z + x y}{3 x y z} = \frac{7}{72} \\ \Rightarrow \frac{2 z \times z + 4 z \times z + 4 z \times 2 z}{3 (4 z \times 2 z \times z)} = \frac{7}{72} \\ \Rightarrow \frac{2 z^{2} + 4 z^{2} + 8 z^{2}}{3 \times 8 z^{3}} = \frac{7}{72} \\ \Rightarrow \frac{14 z^{2}}{24 z^{3}} = \frac{7}{72} \\ \Rightarrow 504 = 84 z \\ z = 6 \\ So, x = 4 z = 4 \times 6 = 24, \\ \Rightarrow y = 2 z = 2 \times 6 = 12 \\ Thus the numbers are 24, 12, 6 \end{aligned}

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