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 :

Concept & Logic When a question provides the average of reciprocals along with multiple-choice options, it is often significantly faster to test the options rather than solving the algebraic equation, provided the relations are simple. Step-by-Step Solution * Let the third number be $x$. * The second number is $2x$. * The first number is $2(2x) = 4x$. * The numbers are $4x$, $2x$, and $x$. * Their reciprocals are $\frac{1}{4x}$, $\frac{1}{2x}$, and $\frac{1}{x}$. * Find the sum of these reciprocals. The common denominator is $4x$: $$\text{Sum} = \frac{1}{4x} + \frac{2}{4x} + \frac{4}{4x} = \frac{7}{4x}$$ * The average of the 3 reciprocals is: $$\text{Average} = \frac{\frac{7}{4x}}{3} = \frac{7}{12x}$$ * We are given that the average is $\frac{7}{72}$. Equate them: $$\frac{7}{12x} = \frac{7}{72}$$ $$12x = 72 \implies x = 6$$ * Substitute $x = 6$ back into our expressions for the numbers: Third = $6$ Second = $2(6) = 12$ First = $4(6) = 24$ * The numbers are 24, 12, 6. Exam Strategy & Shortcut **Option Elimination:** Notice that all given options already follow the "first is twice the second, second is twice the third" rule (4:2:1 ratio). Just test the reciprocals of Option (c) 24, 12, 6. Reciprocals: $\frac{1}{24}, \frac{1}{12}, \frac{1}{6}$. Sum = $\frac{1 + 2 + 4}{24} = \frac{7}{24}$. Average = $\frac{7}{24} \div 3 = \frac{7}{72}$. It matches perfectly. You can do this mental arithmetic in 10 seconds. Common Pitfall A classic error is forgetting to divide the sum of the reciprocals by 3 to find the average, mistakenly equating the sum $\frac{7}{4x}$ directly to $\frac{7}{72}$, yielding $x = 18$, which leads to the incorrect option (d) 72, 36, 18 (or 36, 18, 9 if further miscalculated). Final Answer Therefore, the correct answer is 24, 12, 6.