A watch which gains 5 seconds in 3 minutes was set right at 7 a.m. In the afternoon of the same day, when the watch indicated quarter past 4 o'clock, the true time is

### Concept & Faulty Clocks Proportions Establish a ratio between the "incorrect time" and "true time". Use this multiplier on the total elapsed incorrect hours to find the true elapsed hours. $$ \text{True Time Elapsed} = \text{Incorrect Elapsed} \times \left( \frac{\text{True Interval}}{\text{Incorrect Interval}} \right) $$ ### Step-by-Step Solution * Given: Watch gains 5 seconds every 3 minutes. Set correct at 7 a.m. Current false time = 4:15 p.m. * Calculate the ratio of incorrect time to correct time: 3 minutes of correct time = $3 \text{ min } 5 \text{ sec}$ of incorrect time. * Convert incorrect interval to minutes: $3 + \frac{5}{60} = 3 \frac{1}{12} = \frac{37}{12}$ minutes. * Total incorrect time elapsed from 7 a.m. to 4:15 p.m. is 9 hours and 15 minutes = $9 \frac{1}{4}$ hours = $\frac{37}{4}$ hours. * Set up ratio to find true elapsed hours: $\text{True Hours} = \frac{37}{4} \times \frac{3}{\frac{37}{12}} = \frac{37}{4} \times \frac{36}{37} $. * Simplify the calculation: $\text{True Hours} = \frac{36}{4} = 9$ hours exactly. * True time = 7 a.m. + 9 hours = 4 p.m. ### Exam Strategy & Shortcut Notice the clean fractions. $\frac{37}{12}$ mins incorrect = 3 mins true. The total incorrect time is $\frac{37}{4}$ hours. The 37s cancel out perfectly when multiplying by the inverse ratio $\frac{36}{37}$, immediately yielding 9 true hours. 7 a.m. + 9 = 4 p.m. ### Common Pitfall Trying to subtract the accumulated 5-second intervals linearly based on the incorrect time elapsed, rather than setting up an exact proportional ratio. ### Final Answer Therefore, the correct answer is **4 p.m.**.