Akshara and Siddharth travel from point P to Q, a distance of 72 km at a rate of 8 km/h and 10 km/h, respectively. Siddharth reaches Q first and returns immediately and meets Akshara at R. Find the distance from P to R.

Correct — B. Let t be the time from the start to the meeting at R.

Akshara covers 8t, so R is 8t km from P.
Siddharth covers P→Q (72 km) and then back from Q to R, a further (72 − 8t) km.
His distance is also 10t, so 10t = 72 + 72 − 8t.

18t = 144 → t = 8 hours
PR = 8 × 8 = 64 km → option (b).

Why the others are wrong

• (a)
  65 km — 65 km fails the timing check: Akshara needs 65⁄8 = 8.125 h to reach it, while Siddharth needs (144 − 65)⁄10 = 7.9 h. They must arrive at the same instant.
• (c)
  63 km — 63 km gives Akshara 63⁄8 = 7.875 h against Siddharth's (144 − 63)⁄10 = 8.1 h — two different meeting times, so R cannot be there.
• (d)
  66 km — 66 km gives Akshara 66⁄8 = 8.25 h against Siddharth's (144 − 66)⁄10 = 7.8 h. Only 64 km balances the pair at 8 hours each.

This is a meet-after-a-turn-around problem, and it has a one-line shortcut.

By the moment they meet, Akshara has walked P→R and Siddharth has walked P→Q→R. Laid end to end that is twice PQ — 144 km — whatever R turns out to be.

Both have been travelling for the same time, so add the speeds: 144 ÷ (8 + 10) = 8 hours. Akshara's own distance, 8 × 8 = 64 km, is what the question asks for.

Key facts

• When one traveller covers the full distance d and turns back to meet the other, the two together cover 2d by the meeting instant.
• Both travel for the same time here, so combined distance ÷ combined speed gives that time: 144 ÷ 18 = 8 hours.
• At the meeting Siddharth has covered 80 km and Akshara 64 km, and 80 + 64 = 144.

Common traps

• Using 72 km as the combined distance rather than 144 km, which halves the time.
• Answering with Siddharth's 80 km, or with the 8 km gap between R and Q.
• Adding the speeds without first checking that both travelled for the same length of time.

No directly related past PYQ was found.