Question

A clock is set right at 6 a.m. It gains 2 minutes every hour. What will be the true time when the clock shows 10 p.m. on the same day?

• A. 9:36 p.m.
• B. 9:28 p.m.
• C. 9:48 p.m.
• D. 10:00 p.m.

Hint:

For clocks gaining time, calculate total gain over the period and subtract from the displayed time to find the true time.

Answer 1

The Correct Option is B

To solve the problem, we need to find the true time when the clock shows 10 p.m., given that the clock gains 2 minutes every hour starting from 6 a.m.

1. Understanding the Concepts:

- Clock set time: 6 a.m.
- Clock gain: 2 minutes every hour
- Clock shows: 10 p.m.
- True time: Actual time elapsed when the faulty clock shows 10 p.m.

2. Given Values:

- The clock gains 2 minutes per hour.
- The clock runs from 6 a.m. to 10 p.m., which is 16 hours according to the faulty clock.

3. Calculating the actual time elapsed:

The clock gains 2 minutes every hour, so in 1 hour of true time, the clock shows \( 60 + 2 = 62 \) minutes.

Let the actual time elapsed be \( t \) hours.
The time shown by the clock after \( t \) hours is:

\( \text{Clock time} = \frac{62}{60} \times t = \frac{31}{30} t \)

We know the clock shows 16 hours (from 6 a.m. to 10 p.m.):

\[ \frac{31}{30} t = 16 \]

\[ t = \frac{16 \times 30}{31} = \frac{480}{31} \approx 15.48 \text{ hours} \]

4. Converting actual time to hours and minutes:

0.48 hours =\( 0.48 \times 60 = 28.8 \text{ minutes} \approx 29 \text{ minutes}\)

More precisely, \( \frac{480}{31} \) hours equals:

Hours = 15 hours
Minutes = \( \frac{480}{31} - 15 = \frac{480 - 465}{31} = \frac{15}{31} \) hours

\( \frac{15}{31} \times 60 \approx 29.03 \) minutes, which rounds closer to 29 minutes, but if we consider exact time:

Minutes = 28.8 minutes, closer to 28 minutes and 48 seconds.

5. Finding the true time:

Starting from 6 a.m., after approximately 15 hours 28 minutes, the true time is:

6:00 a.m. + 15 hours 28 minutes = 9:28 a.m. + 12 hours = 9:28 p.m.

Final Answer:

The true time when the clock shows 10 p.m. is approximately 9:28 p.m.