A watch that gains uniformly, is two minutes slow at noon on Monday and is 4 minutes 48 seconds fast at 2 pm on the following Monday. When was the watch showing the correct time?

The watch is 2 minutes slow at noon on Monday and 4 minutes 48 seconds fast at 2 pm on the following Monday. The watch gains uniformly (runs faster than actual time at a constant rate).

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From noon Monday to 2 pm the following Monday:

Noon Monday to noon next Monday = 7 days = $7 \times 24 = 168$ hours

Noon next Monday to 2 pm next Monday = 2 hours

Total time = $168 + 2 = 170$ hours

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The watch went from 2 minutes slow to 4 minutes 48 seconds fast.

Total gain by the watch = $2$ min $+ 4$ min $48$ sec

Converting to seconds: $2$ min $= 120$ seconds, $4$ min $48$ sec $= 288$ seconds

Total gain = $120 + 288 = 408$ seconds

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The watch needs to gain 2 minutes (120 seconds) from the starting point to show correct time.

If 408 seconds are gained in 170 hours, then 120 seconds are gained in:

Time needed $= \dfrac{120}{408} \times 170$ hours

$= \dfrac{5}{17} \times 170$

$= 50$ hours

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Starting from noon Monday, add 50 hours:

24 hours later → Noon Tuesday

48 hours later → Noon Wednesday

50 hours later → 2 pm Wednesday

Therefore, the watch shows the correct time at 2 pm on Wednesday.