Question : 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 $\frac{7}{12}$. The numbers are:
Option 1: 20, 10 and 5
Option 2: 4, 2 and 1
Option 3: 36, 18 and 9
Option 4: 16, 8 and 4
Correct Answer: 4, 2 and 1
Solution :
Let us assume the third number is $x$.
Second number = 2$x$
First number = 2(2$x$) = 4$x$
Reciprocals of the three numbers = $\frac{1}{4x}, \frac{1}{2x}, \frac{1}{x}$
Average of the reciprocals = $\frac{7}{12}$
Sum of the reciprocals = $\frac{1}{4x}+\frac{1}{2x}+\frac{1}{x}=\frac{7}{4x}$
Average of reciprocals = $\frac{7}{4x} × \frac{1}{3} = \frac{7}{12x}$
So, $\frac{7}{12x} = \frac{7}{12}$
⇒ $x$ = 1
Hence, the numbers are 4, 2 and 1.
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