$A$ runs twice as fast as $B$ and $B$ runs thrice as fast as $C$. The distance covered by $C$ in 72 minutes, will be covered by $A$ in (C.P.O., 2007; R.R.B., 2006)

### Concept & Speed-Time Inversely Proportional Relation Speed and Time are inversely proportional when the distance is constant. If the speed of an object is $n$ times that of another, the time taken by it will be $1/n$ of the time taken by the other to cover the same distance. $$ \text{Speed Ratio} = a : b \implies \text{Time Ratio} = \frac{1}{a} : \frac{1}{b} $$ ### Step-by-Step Solution 1. Let the speed of $C$ be $v_c$. 2. Since $B$ runs thrice as fast as $C$, speed of $B$ ($v_b$) = $3v_c$. 3. Since $A$ runs twice as fast as $B$, speed of $A$ ($v_a$) = $2 \times v_b = 2 \times 3v_c = 6v_c$. 4. The speed of $A$ is 6 times the speed of $C$. 5. Because time is inversely proportional to speed for a fixed distance, $A$ will take $\frac{1}{6}$th of the time $C$ takes. 6. Time taken by $C$ = 72 minutes. 7. Time taken by $A$ = $\frac{72}{6} = 12$ minutes. ### Exam Strategy & Shortcut Write down the speed ratios directly: $A : B : C = (2 \times 3) : 3 : 1 = 6 : 3 : 1$. The time ratio for the same distance is the inverse of the speed ratio: $T_a : T_b : T_c = \frac{1}{6} : \frac{1}{3} : \frac{1}{1} = 1 : 2 : 6$. If 6 units of time corresponds to 72 minutes, 1 unit of time corresponds to $\frac{72}{6} = 12$ minutes. ### Common Pitfall A common error is confusing speed and time relationships, accidentally multiplying the time by 6 (resulting in $72 \times 6$) instead of dividing it. ### Final Answer Therefore, the correct answer is **12 minutes**.