Two boys $A$ and $B$ start at the same time to ride from Delhi to Meerut, 60 km away. $A$ travels 4 km an hour slower than $B$, $B$ reaches Meerut and at once turns back meeting $A$, 12 km from Meerut. A's rate was (M.B.A., 2011)

### Concept & Proportionality of Distance and Speed When two objects travel for the exact same amount of time, the ratio of the distances they cover is equal to the ratio of their respective speeds. $$\frac{Speed_A}{Speed_B} = \frac{Distance_A}{Distance_B}$$ ### Step-by-Step Solution 1. The total distance between Delhi and Meerut is $60$ km. 2. Boy $B$ rides to Meerut and turns back, meeting $A$ at a point $12$ km away from Meerut. 3. Calculate the distance covered by $B$: $B$ travelled all the way to Meerut ($60$ km) and then $12$ km back. Distance covered by $B = 60 + 12 = 72$ km. 4. Calculate the distance covered by $A$: $A$ travelled from Delhi but hasn't reached Meerut yet. He is $12$ km short of it. Distance covered by $A = 60 - 12 = 48$ km. 5. Since they started at the same time and meet at the same time, their travel durations are identical. Thus, the ratio of their speeds equals the ratio of distances covered: $$\frac{Speed_A}{Speed_B} = \frac{48}{72} = \frac{2}{3}$$ 6. Let $A$'s speed be $2x$ and $B$'s speed be $3x$. 7. The problem states $A$ travels $4$ km/hr slower than $B$: $Speed_B - Speed_A = 4$ $3x - 2x = 4 \Rightarrow x = 4$ 8. We need $A$'s rate (speed): $A$'s speed $= 2x = 2 \times 4 = 8$ km/hr. ### Exam Strategy & Shortcut Visualizing the distances is the key. In the time it takes the meeting to happen, they have collectively covered twice the distance between the cities ($60 \times 2 = 120$ km). $B$ covers $60 + 12 = 72$ km. $A$ covers $48$ km. Ratio is $48 : 72$, or $2 : 3$. Difference in ratio is $1$ unit, which corresponds to the $4$ km/hr speed difference. Therefore, $A$'s speed ($2$ units) is $8$ km/hr. No equations needed! ### Common Pitfall A common mistake is incorrectly calculating $A$'s distance as $60$ or $12$. Careful reading is required: $B$ reaches Meerut and turns back, meeting $A$ *who is still on his way*. ### Final Answer Therefore, the correct answer is **8 km/hr**.