Two men at points $R$ and $S$, 76 km apart, set out at the same time to walk towards each other. The man at $R$ walks uniformly at the rate of $4 \frac{1}{2}$ km/hr ; the man at $S$ walks at the constant rate of $3 \frac{1}{4}$ km/hr for the first hour, at $3 \frac{3}{4}$ km/hr for the second hour, and so on, in arithmetic progression. If the men meet $x$ km nearer $R$ than $S$ in an integral number of hours, then $x$ is

### Concept & Arithmetic Progression (AP) in Distance When speed changes every hour by a constant amount, the distance covered per hour forms an Arithmetic Progression. The total distance is the sum of the distances covered by both individuals. Sum of AP: $$S_n = \frac{n}{2} [2a + (n-1)d]$$ ### Step-by-Step Solution 1. **Distance covered by man at R:** He walks at a constant speed of $4.5$ km/hr. In $n$ hours, distance $= 4.5n$. 2. **Distance covered by man at S:** His speeds form an AP: $a = 3.25$ ($\frac{13}{4}$), common difference $d = 3.75 - 3.25 = 0.5$ ($\frac{1}{2}$). Sum of distance in $n$ hours $= \frac{n}{2} \left[2\left(\frac{13}{4}\right) + (n-1)\left(\frac{1}{2}\right)\right]$ $= \frac{n}{2} \left[\frac{13}{2} + \frac{n-1}{2}\right] = \frac{n}{2} \left[\frac{12+n}{2}\right] = \frac{n(12+n)}{4}$. 3. **Equate to total distance:** Total distance is 76 km. $$4.5n + \frac{n(12+n)}{4} = 76$$ Multiply by 4 to clear the denominator: $$18n + 12n + n^2 = 304$$ $$n^2 + 30n - 304 = 0$$ 4. **Solve the quadratic equation:** Factorizing: $(n + 38)(n - 8) = 0$. Since time must be positive, $n = 8$ hours. 5. **Calculate meeting point distances:** Distance from R $= 4.5 \times 8 = 36$ km. Distance from S $= 76 - 36 = 40$ km. 6. **Find $x$:** The man meets 36 km from R and 40 km from S. Difference $= 40 - 36 = 4$ km. Thus, he meets 4 km nearer to R than S. ### Exam Strategy & Shortcut Since $n$ must be an integer, you can quickly estimate. Average speed of man at R = 4.5. Average speed of man at S starts at 3.25 and increases. Combined speed is roughly around 8-9 km/hr. $76 / \approx 9 = 8$ hours roughly. Plug in $n=8$ to check: Man R distance in 8 hours = 36. Man S distance = sum of 8 terms of AP: $3.25 + 3.75 + 4.25 + 4.75 + 5.25 + 5.75 + 6.25 + 6.75 = 40$. $36 + 40 = 76$. It matches perfectly. Difference is $40 - 36 = 4$. ### Common Pitfall Misinterpreting "meet $x$ km nearer R than S". This phrasing means calculating the difference between the distance from S and the distance from R ($|D_S - D_R|$), not just finding the distance from R. ### Final Answer Therefore, the correct answer is **4**.