Question

A car travels from A to B at 60 km/h and returns at 40 km/h. What is the average speed for the round trip?

• A. 48 km/h
• B. 50 km/h
• C. 52 km/h
• D. 45 km/h

Hint:

For equal distances, use \( \frac{2 \cdot s_1 \cdot s_2}{s_1 + s_2} \) to find average speed quickly.

Answer 1

The Correct Option is A

We need to calculate the average speed for the round trip.
- Step 1: Understand average spee(d) Average speed is total distance divided by total time:
\[ \text{Average speed} = \frac{\text{Total distance}}{\text{Total time}} \] - Step 2: Define distances and speeds. Let the distance from A to B be \( d \) km. Total distance (A to B and back) = \( 2d \). Speed from A to B = 60 km/h, from B to A = 40 km/h.
- Step 3: Calculate time for each leg. Time for A to B:
\[ t_1 = \frac{d}{60} \] Time for B to A:
\[ t_2 = \frac{d}{40} \] Total time:
\[ t_1 + t_2 = \frac{d}{60} + \frac{d}{40} \] Find a common denominator (LCM of 60 and 40 is 120):
\[ \frac{d}{60} = \frac{2d}{120}, \quad \frac{d}{40} = \frac{3d}{120} \] \[ \text{Total time} = \frac{2d + 3d}{120} = \frac{5d}{120} = \frac{d}{24} \] - Step 4: Compute average spee(d)
\[ \text{Average speed} = \frac{2d}{\frac{d}{24}} = 2d \times \frac{24}{d} = 48 \, \text{km/h} \] - Step 5: Alternative metho(d) Use the formula for average speed over equal distances:
\[ \text{Average speed} = \frac{2 \cdot s_1 \cdot s_2}{s_1 + s_2} = \frac{2 \cdot 60 \cdot 40}{60 + 40} = \frac{4800}{100} = 48 \, \text{km/h} \] - Step 6: Verify with options.
- (a) 48: Matches exactly.
- (b) 50: Incorrect.
- (c) 52: Incorrect.
- (d) 45: Incorrect.
- Step 7: Double-check. The harmonic mean confirms the calculation, ensuring no arithmetic errors.
Thus, the answer is a.