Question

A road from A to B is 11.5 km long, first goes uphill, then crosses a plain, and then goes downhill. A person walking from A to B covered this road in 2 h 54 min, and the return journey took him 3 h 6 min. His speed uphill is 3 km/h, on the plain 4 km/h and downhill is 5 km/h. What is the length of the plain part of the journey?

• A. \(4.5\) km
• B. \(5\) km
• C. \(5.5\) km
• D. \(6\) km
• E. \(4\) km

Answer 1

Answer: (E)

Concept: Let distances be:

• Uphill = \(x\)
• Plain = \(y\)
• Downhill = \(z\)

\[ x + y + z = 11.5 \quad \cdots (1) \]
Step 1: Convert time.
\[ 2\text{h }54\text{m} = 2.9 \text{ h}, \quad 3\text{h }6\text{m} = 3.1 \text{ h} \]
Step 2: Forward journey.
\[ \frac{x}{3} + \frac{y}{4} + \frac{z}{5} = 2.9 \quad \cdots (2) \]
Step 3: Return journey.
\[ \frac{z}{3} + \frac{y}{4} + \frac{x}{5} = 3.1 \quad \cdots (3) \]
Step 4: Subtract (2) from (3).
\[ \left(\frac{z}{3} - \frac{x}{3}\right) + \left(\frac{x}{5} - \frac{z}{5}\right) = 0.2 \] \[ (z-x)\left(\frac{1}{3} - \frac{1}{5}\right) = 0.2 \] \[ (z-x)\left(\frac{2}{15}\right) = 0.2 \Rightarrow z-x = 1.5 \quad \cdots (4) \]
Step 5: Solve system.
From (1): \[ x + y + (x+1.5) = 11.5 \Rightarrow 2x + y = 10 \] Try option \(y = 4\): \[ 2x + 4 = 10 \Rightarrow x = 3,\quad z = 4.5 \] Check satisfies equations \checkmark
Step 6: Option analysis.

• (A) Incorrect $\times$
• (B) Incorrect $\times$
• (C) Incorrect $\times$
• (D) Incorrect $\times$
• (E) Correct \checkmark

Conclusion:
Thus, the correct answer is
Option (E).