The sum of three numbers is 264. If the first number be twice the second and third number be one-third of the first, then the second number is:

## Concept & Logic When multiple variables are related to each other, express all of them in terms of a single common variable to create a simple linear equation based on their sum. ## Step-by-Step Solution * **Given:** Let the three numbers be $A$, $B$, and $C$. Sum $= A + B + C = 264$. * **Calculation / Deduction:** * First condition: The first number is twice the second. * $A = 2B \Rightarrow B = \frac{A}{2}$ * Second condition: The third number is one-third of the first. * $C = \frac{A}{3}$ * Substitute $B$ and $C$ in terms of $A$ into the sum equation: * $A + \frac{A}{2} + \frac{A}{3} = 264$ * Find a common denominator ($6$): * $$ \frac{6A + 3A + 2A}{6} = 264 $$ * $$ \frac{11A}{6} = 264 $$ * $11A = 264 \times 6 = 1584$ * $A = \frac{1584}{11} = 144$ * The question asks for the second number ($B$): * $B = \frac{A}{2} = \frac{144}{2} = 72$ ## Exam Strategy & Shortcut **Ratio Method:** Establish a direct ratio. $A:B = 2:1$ (which is $6:3$) $A:C = 3:1$ (which is $6:2$) Combined ratio $A:B:C = 6:3:2$. Total parts $= 6 + 3 + 2 = 11$ parts. $11 \text{ parts} = 264 \Rightarrow 1 \text{ part} = 24$. Second number is $3$ parts $= 3 \times 24 = 72$. This is much faster than fractions. ## Common Pitfall Students often get bogged down by fractions when using algebraic substitution. Switching to LCM-based ratios as shown in the shortcut completely bypasses the risk of fraction arithmetic errors. ## Final Answer **Therefore, the correct answer is 72.**