Multiple choice

Two cyclists, P and Q, start from two towns, M and N, and ride towards each other on a straight path at the same time. After meeting at a point along the way, P takes 5 hours more than Q to reach his destination. If P was cycling at a speed that is 3 km/hr less than the speed of Q, then which of the following can be the speed of Q and the time taken (in hours) to meet, respectively?

  1. 8 km/hr and 7 hours

  2. 9 km/hr and 6 hours

  3. 10 km/hr and 5 hours

  4. 12 km/hr and 4 hours
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let v be the speed of Q, so P's speed is v-3. Let t be the time to meet. Distance M to meeting point is (v-3)t, and N to meeting point is vt. After meeting, P covers the distance vt at speed v-3, taking vt/(v-3) hours. Q covers (v-3)t at speed v, taking (v-3)t/v hours. The difference is 5 hours. Testing option B: v=9, v-3=6. Meeting time t=6. P takes (9*6)/6 = 9 hours. Q takes (6*6)/9 = 4 hours. 9-4=5, which matches.

AI explanation

When two objects start simultaneously and meet, the square of the ratio of their speeds equals the inverse ratio of their additional times to reach their destinations, giving v1/v2 = sqrt(t2/t1). Let P's additional time be 5 + x and Q's be x; substituting the speeds gives (v-3)/v = sqrt(x/(x+5)). We check the given options to see which pair satisfies this relationship: testing the speed of 9 km/hr and time of 6 hours for Q gives a ratio of 6/9 = 2/3, which perfectly equals sqrt(4/9).

Keep practicing — related questions