P and Q are two points on a highway, 300 km apart. Two bikers A and B start at the same time from P towards Q. The speed of B is 12 km/h less than that of A. On reaching point Q, biker A immediately returns towards P. He meets B on the road at a place 60 km away from Q. What is the speed of A?

Correct — C, 36 km/h. Both riders start together and are still riding when they meet, so the two of them have been on the road for exactly the same length of time. Count the distances at that moment. A has run the full 300 km to Q and then turned back 60 km, so A has covered 360 km. B is still short of Q by 60 km, so B has covered 240 km. Equal times means 360 divided by A's speed equals 240 divided by B's speed, and B's speed is 12 less than A's: 360/v = 240/(v − 12). Cross-multiplying, 360v − 4320 = 240v, so 120v = 4320 and v = 36 km/h. B rides at 24 km/h; A takes 10 hours for 360 km and B takes 10 hours for 240 km, which confirms it.

• (a)38 km/h — Would make B's speed 26 km/h, and the ratio of distances would have to be 38 : 26, that is about 1.46. The actual ratio 360 : 240 is exactly 1.5, which the pair 36 and 24 satisfies.
• (b)30 km/h — Gives B 18 km/h and a distance ratio of 30 : 18, or 1.67 — too large. A would cover 360 km in 12 hours while B covered only 216 km.
• (d)35 km/h — Gives B 23 km/h. In the 360/35 hours A needs, B would cover about 236.6 km, so B would be about 63.4 km short of Q rather than 60.

In every meeting problem the shared quantity is time. Write each traveller's distance at the moment of meeting, divide by that traveller's speed, and set the two expressions equal — one equation, one unknown, once the speeds are tied together. The only real work is counting the distances correctly when one of them has turned around.

The turn at Q is what the item is testing. A does not stop at Q; A goes 300 km out and 60 km back, so A's distance is 300 + 60. B is caught 60 km before Q, so B's distance is 300 − 60. Together they have covered 600 km, which is twice the road — a useful check, because in any out-and-back meeting of this kind the combined distance is exactly twice the separation. The ratio 360 : 240 reduces to 3 : 2, so the speeds are in the ratio 3 : 2 as well, and a difference of 12 km/h in a 3 : 2 ratio means one part is 12 and the speeds are 36 and 24.

• Both riders start at the same instant, so their travel times to the meeting point are equal.
• A covers 300 + 60 = 360 km; B covers 300 − 60 = 240 km.
• 360 : 240 reduces to 3 : 2, so the speeds are in the ratio 3 : 2.
• A difference of 12 km/h across a 3 : 2 ratio makes one part 12, so the speeds are 36 and 24 km/h.
• Both take 10 hours, and the combined distance of 600 km is twice the 300 km between P and Q.

• Giving A a distance of 300 − 60 instead of 300 + 60.
• Trying to find the meeting time first, when the time cancels out.
• Mixing up which rider is the slower one when the difference is given.

A meeting item whose only difficulty is counting one rider's distance after a turnaround.

• practice — not a real PYQ

  Two towns are 240 km apart. Two cars start together from one town towards the other; the faster reaches the far town and turns back, meeting the slower car 40 km from that town. What is the ratio of their speeds?

  • (a)3 : 2
  • (b)7 : 5
  • (c)5 : 3
  • (d)4 : 3

  Answer(b) 7 : 5 — the faster car covers 280 km and the slower 200 km in the same time, and 280 : 200 is 7 : 5.

• practice — not a real PYQ

  If two cyclists cover distances in the ratio 5 : 3 in the same time and one rides 8 km/h faster than the other, the faster speed is

  • (a)15 km/h
  • (b)20 km/h
  • (c)24 km/h
  • (d)25 km/h

  Answer(b) 20 km/h — the speeds are in the ratio 5 : 3, so one part is 4 and the speeds are 20 and 12.