Two vehicles which are 100 km apart are running towards each other in a straight line. In how much time will they meet each other provided they follow a uniform speed of 45 km per hour and 80 km per hour respectively?

Correct — C, (c) 48 minutes. The step that decides this item is not the division at the end; it is the recognition that when two bodies move towards each other along the same line, the distance between them shrinks at the sum of their speeds. Neither vehicle covers the hundred kilometres. Between them they cover it, and they do so at 45 + 80 = 125 kilometres an hour. That single sentence converts a two-body problem into a one-body problem, and everything after it is arithmetic.

Time to meet = gap divided by closing speed = 100 ÷ 125 = 0·8 hour. The options are in minutes, so 0·8 of an hour is 0·8 × 60 = 48 minutes, which is option (c).

The result is easy to confirm, and confirming it is worth the ten seconds. In 0·8 hour the slower vehicle covers 45 × 0·8 = 36 km and the faster covers 80 × 0·8 = 64 km. Those two distances add to exactly 100 km, which is the gap they started with, so they meet at a point 36 km from one starting position and 64 km from the other — nearer the slower vehicle's end, as it must be. The same relative-speed idea has a second form that is worth learning at the same time: if the two had been travelling in the same direction, the gap would change at the difference of the speeds, 80 − 45 = 35 kilometres an hour, and the problem would be one of catching up rather than of meeting. Add when they approach, subtract when they chase.

Why the others are wrong

• (a)60 minutes — Sixty minutes is what the answer would be if the two vehicles closed on each other at 100 kilometres an hour, since the gap is a hundred kilometres — and the neatness of that coincidence is exactly what makes the option attractive to a candidate who pairs the hundred in the stem with the hundred-kilometre-an-hour figure without checking where either number comes from. The closing speed is not 100 but 125. The check settles it in one line: in a full hour the two together cover 125 km, which is 25 km more than the gap between them, so by the end of that hour they would have met and driven 25 km past each other. Whenever an option can be tested by asking how much ground has been covered by the stated time, test it; on this family of items the arithmetic of the check is easier than the arithmetic of the solution.
• (b)55 minutes — Fifty-five minutes corresponds to no consistent reading of the problem, and it fails the same check as option (a). In 55 minutes, which is 55 ÷ 60 of an hour, the two vehicles together cover 125 × 55 ÷ 60 = 114·58 km. That is more than the 100 km that separates them, so they have already met some time earlier and this cannot be the moment of meeting. The option earns its place in the set by sitting between the two figures a hurried candidate is most likely to produce, so that anyone who has arrived at an answer of roughly an hour by rough reasoning finds something close to it and stops looking. A number that merely looks plausible is the easiest kind of distractor to eliminate, because plausibility leaves no trace in the arithmetic and the back-check exposes it at once.
• (d)45 minutes — Forty-five is a number the candidate has just read in the stem, where it is one of the two speeds, and seeing it again among four times invites a moment of false recognition. It is a unit slip in disguise — a figure measured in kilometres an hour being offered as an answer in minutes — and the reasoning that produces it is not reasoning at all but recall of a digit. The back-check disposes of it cleanly: in 45 minutes, three quarters of an hour, the two vehicles together cover 125 × 0·75 = 93·75 km, so they are still 6·25 km apart and have not met. An examiner who plants a number from the stem among the options is testing whether the candidate has worked or remembered, and the defence is to finish the calculation before looking at the choices at all.

Relative speed is the whole of this topic, and it reduces to two rules and one formula. When two bodies move towards each other, or away from each other, along the same straight line, the distance between them changes at the sum of their speeds. When they move in the same direction, it changes at the difference of their speeds. In both cases the time taken for the gap to close, or to open to a stated size, is the gap divided by that relative speed. The power of the idea is that it removes one of the two moving bodies from the problem: instead of tracking two positions through time, one tracks a single shrinking distance, and the question becomes as simple as a single vehicle covering a fixed distance at a fixed speed. Two housekeeping points decide whether the arithmetic then goes right. The first is units: speeds given in kilometres an hour produce a time in hours, and an answer wanted in minutes must be multiplied by sixty, which is where a correct 0·8 becomes an incorrect 8 or 80 in a hurried hand. The second is the check: once the meeting time is known, multiply it by each speed separately and confirm that the two distances add back to the original gap.

Key facts

• When two bodies approach each other along the same line, the gap between them closes at the sum of their speeds.
• When two bodies travel in the same direction, the gap changes at the difference of their speeds.
• Time to meet equals the initial gap divided by the relative speed; here 100 ÷ (45 + 80) = 0·8 hour.
• A time in hours becomes minutes on multiplying by sixty, so 0·8 hour is 48 minutes.
• At the moment of meeting the two vehicles have covered 36 km and 64 km respectively, and those add back to the 100 km gap.
• The meeting point is always nearer the slower body's starting position, in the ratio of the two speeds.
• Any candidate answer can be tested by multiplying the closing speed by the proposed time and comparing with the gap.

Common traps

• Subtracting the speeds when the bodies are approaching each other, which answers the chasing problem instead.
• Leaving the answer in hours when the options are in minutes, or multiplying by sixty in the wrong direction.
• Choosing an option because its number already appears in the stem, where it measured something else entirely.
• Trying to find where the two meet before finding when, which turns a one-line problem into two.
• Skipping the check that the two distances covered add back to the original gap, which catches almost every slip.

• practice — not a real PYQ

  Two trains 240 km apart on the same track move towards each other at uniform speeds of 50 km per hour and 70 km per hour. After how much time will they meet?

  • (a)1 hour 30 minutes
  • (b)2 hours
  • (c)2 hours 30 minutes
  • (d)3 hours

  Answer(b) 2 hours

• practice — not a real PYQ

  Two vehicles leave the same place at the same time in the same direction at uniform speeds of 40 km per hour and 55 km per hour. After how much time will they be 30 km apart?

  • (a)1 hour
  • (b)1 hour 30 minutes
  • (c)2 hours
  • (d)2 hours 30 minutes

  Answer(c) 2 hours