Two clocks are set right at 6:10 p.m. on Saturday. The first clock loses 2 minutes per hour whereas the second clock gains twice as much as the first one loses per hour. What is the difference between the times shown by the two clocks at 12:00 noon the next day? (in minutes)

We are given two clocks that are set correctly at 6:10 p.m. on Saturday. The first clock loses 2 minutes per hour, while the second clock gains 4 minutes per hour (twice the rate the first clock loses).

### Step 1: Calculate the total elapsed time.
The time between 6:10 p.m. on Saturday and 12:00 noon on Sunday is the duration we need to consider.

- From 6:10 p.m. to midnight is \(5\) hours and \(50\) minutes.
- From midnight to 12:00 noon on Sunday is \(12\) hours.

Thus, the total time elapsed is:

\[
5\ \text{hours}\ 50\ \text{minutes} + 12\ \text{hours} = 17\ \text{hours}\ 50\ \text{minutes} = 17 + \frac{50}{60} = 17.8333\ \text{hours}.
\]

### Step 2: Calculate the total time lost by the first clock.
The first clock loses 2 minutes per hour. Over \(17.8333\) hours, the total time lost by the first clock is:

\[
2 \times 17.8333 = 35.6666\ \text{minutes}.
\]

### Step 3: Calculate the total time gained by the second clock.
The second clock gains 4 minutes per hour. Over \(17.8333\) hours, the total time gained by the second clock is:

\[
4 \times 17.8333 = 71.3333\ \text{minutes}.
\]

### Step 4: Calculate the difference in time between the two clocks.
The difference in time between the two clocks is the total time gained by the second clock minus the total time lost by the first clock:

\[
71.3333 - 35.6666 = 35.6667\ \text{minutes}.
\]

Thus, the difference between the times shown by the two clocks at 12:00 noon on Sunday is approximately **35.67 minutes**.

Would you like more details on any step or have any questions?

Here are five related questions you can explore:
1. How would the problem change if the first clock lost 1 minute per hour instead of 2 minutes?
2. What if both clocks lost time at different rates instead of one gaining time?
3. How long would it take for the two clocks to show a 1-hour difference?
4. What if the clocks started at 12:00 noon instead of 6:10 p.m.?
5. How does the time difference scale if we let more days pass before checking the clocks?

**Tip:** When dealing with time, it's helpful to express fractional hours as decimals to avoid confusion.

Solve the problem of two clocks set correctly at 6:10 p.m. on Saturday, where one loses 2 minutes per hour and the other gains 4 minutes per hour. Learn how to calculate the time difference by noon the next day with step-by-step instructions, covering key time calculation concepts. Suitable for Grades 8-10.

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