A clock which gains uniformly was 10 minutes slow at 3 PM and 20 minutes fast…

Concept: A clock that gains or loses time uniformly changes its reading error at a constant rate. Given the error at two known moments, the rate equals the total change in error divided by the elapsed real time between them, and the moment the error becomes zero can be found by projecting that rate from either reading.

Application:

1. At 3 PM the clock is 10 minutes slow, so its error is -10 minutes; at 6 PM it is 20 minutes fast, so its error is +20 minutes.

2. The elapsed real time between the two readings is 3 hours.

3. The total change in error over this period is 20 - (-10) = 30 minutes, giving a gain rate of 30 / 3 = 10 minutes per hour.

4. For the clock to show the right time, its error must rise from -10 minutes to 0, a gain of exactly 10 minutes.

5. At 10 minutes gained per hour, gaining 10 minutes takes 10 / 10 = 1 hour.

6. One hour after 3 PM is 4:00 PM, so the clock displays the correct time at 4:00 PM.

Cross-check: From 4:00 PM to 6 PM is 2 more hours, during which the clock gains 10 x 2 = 20 minutes, taking its error from 0 to +20 minutes -- exactly the fast reading given for 6 PM, confirming the result.