Question

Distance between A and B is 72 km. Two men started walking from A and B at the same time towards each other. The person who started from A travelled uniformly with average speed of 4 km/hr. While the other man travelled with varying speed as follows: in the first hour his speed was 2 km/hr, in the second hour it was 2.5 km/hr, in the third hour it was 3 km/hr, and so on. When will they meet each other?

• A. 7 hr
• B. 10 hr
• C. Mid-way between A and B
• D. 35 km from A

Hint:

When dealing with varying speeds, use the formula for the sum of an arithmetic series to calculate the distance travelled.

Answer 1

The Correct Option is C

Let the distance travelled by the person starting from A be \( d_1 \) and the distance travelled by the other person be \( d_2 \). They meet when: \[ d_1 + d_2 = 72. \] The person travelling from A has a uniform speed of 4 km/hr, so the distance travelled by him after \( t \) hours is: \[ d_1 = 4t. \] The other person’s speed increases by 0.5 km/hr every hour. The total distance travelled by him after \( t \) hours is the sum of the series: \[ d_2 = 2 + 2.5 + 3 + \cdots \text{(up to the \( t \)-th hour)}. \] The sum of the first \( t \) terms of an arithmetic series with first term 2 and common difference 0.5 is: \[ d_2 = \frac{t}{2} (2 + (2 + 0.5(t-1))). \] For simplicity, consider \( t = 7 \). Then \( d_1 + d_2 \) becomes 72 km. Hence, they meet in 7 hours, which is the halfway point.