Assume 'A' does 500 J of work in 'x' minutes and 'B' does 1000 J of work in 20 minutes. If the power delivered by 'A' is P₁ and 'B' is P₂ and P₁ = 2P₂, then 'x', in minutes is:

Correct — B, 5. Power is the rate of doing work, work divided by time. B does 1000 J in 20 minutes, so P₂ works out at 50 J per minute. A is to deliver twice that, so P₁ is 100 J per minute. A has only 500 J of work to do, and at 100 J per minute that takes 500 ÷ 100 = 5 minutes. Set out as a ratio the arithmetic is even shorter: P₁/P₂ = (500/x) ÷ (1000/20) = 10/x, and setting that equal to 2 gives x = 5 at once. Two things are worth noticing about the units. The strict SI unit of power is the watt, one joule per second, so a purist would convert 20 minutes to 1200 seconds and get P₂ = 0.833 W and P₁ = 1.667 W — and then x = 300 s, which is the same 5 minutes. The conversion cancels because the question fixes only the ratio of the two powers, and a ratio does not care what unit of time you keep the clock in. The sense of the answer is easy to sanity-check as well: A has half as much work to get through and is working twice as fast, so it should need a quarter of B's time, and a quarter of 20 minutes is 5.

• (a)10 — The commonest slip — halving 20 once instead of twice. It follows from using only one of the two facts, either that A's work is half or that A's power is double, when both apply and their effects multiply.
• (c)20 — This is B's time, and it would make P₁ = 500/20 = 25 J per minute, which is half of P₂ rather than double it.
• (d)25 — Larger than B's time, so A would be working more slowly than B, not twice as fast. The relation has been inverted somewhere in the working.

Work measures energy transferred, power measures how fast it is transferred. The SI unit of work and energy is the joule; the SI unit of power is the watt, defined as one joule per second, with the kilowatt and the horsepower as the practical units used for machines. Two agents can do identical work and differ entirely in power: a man and a crane can each lift the same load to the same height, doing the same work, but the crane does it in a fraction of the time and so delivers far more power.

This is a two-step problem dressed as a one-step one, and the mistake it is built to catch is using half the information. A has less work AND more power, and both effects shorten its time, so the time falls by a factor of four rather than two. Working in a ratio is the safest route because it makes both factors visible at once and removes any anxiety about units. A general form worth carrying: t₁/t₂ = (W₁/W₂) × (P₂/P₁), which here gives (1/2) × (1/2) = 1/4, and a quarter of 20 minutes is 5. The related idea to keep alongside is that energy equals power multiplied by time, which is why household electricity is billed in kilowatt-hours — a unit of energy, not of power, despite the way it is usually spoken about.

• Power is work divided by time; its SI unit is the watt, equal to one joule per second.
• In this problem P₂ = 1000 J ÷ 20 min = 50 J per minute and P₁ = 100 J per minute, so x = 500 ÷ 100 = 5 minutes.
• Because only the ratio of the two powers is fixed, working in minutes or in seconds gives the same answer.
• Half the work at twice the power takes a quarter of the time, since the two factors multiply.
• Energy equals power multiplied by time, which is why the kilowatt-hour is a unit of energy and not of power.

• Applying only one of the two conditions and halving the time once instead of twice.
• Converting minutes to seconds under the impression that the answer depends on it; a fixed power ratio makes the unit irrelevant.
• Reading the kilowatt-hour as a unit of power; it is energy, being power multiplied by time.

As a small algebraic comparison of two rates like this, or as a direct computation of power from work and time, or as an electricity-bill calculation.

• practice — not a real PYQ

  A machine does 3000 J of work in 1 minute. Its power output is:

  • (a)50 W
  • (b)300 W
  • (c)3000 W
  • (d)180000 W

  Answer(a) 50 W — power is work divided by time in seconds, so 3000 J ÷ 60 s = 50 watts.

• practice — not a real PYQ

  Two workers do the same amount of work, one in 10 minutes and the other in 40 minutes. The ratio of the power delivered by the first to that by the second is:

  • (a)1 : 4
  • (b)4 : 1
  • (c)1 : 2
  • (d)2 : 1

  Answer(b) 4 : 1 — for equal work, power is inversely proportional to the time taken, so a quarter of the time means four times the power.