The minute hand of a clock overtakes the hour hand at intervals of 70min of the correct time how much in a day does the clock gains or loss

Options

A. `7200/77` min gain
B. `7200/77` min loss
C. `7300/77` min loss
D. `7300/78` min gain

Answer

To solve the problem of how much time a clock gains or loses in a day when the minute hand overtakes the hour hand at intervals of 70 minutes, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
The minute hand overtakes the hour hand every 70 minutes. We need to determine how much time the clock gains or loses in a day (24 hours).

2. **Using the Formula**:
We use the formula:
\[ \text{Time gained or lost} = \left( \text{Correct Time} - \text{Given Minutes} \right) \times \frac{60 \times 24}{\text{Given Minutes}} \]
Here, the correct time is 60 minutes (1 hour), and the given minutes is 70 minutes.

3. **Substituting Values**:
Substitute the values into the formula:
\[ \text{Time gained or lost} = \left( 60 - 70 \right) \times \frac{60 \times 24}{70} \]

4. **Calculating the Difference**:
Calculate \(60 - 70\):
\[ 60 - 70 = -10 \]

5. **Calculating the Multiplication**:
Now, calculate \(60 \times 24\):
\[ 60 \times 24 = 1440 \]

6. **Dividing by Given Minutes**:
Now divide by the given minutes (70):
\[ \frac{1440}{70} = 20.57 \text{ (approximately)} \]

7. **Final Calculation**:
Now multiply the difference by the result:
\[ \text{Time gained or lost} = -10 \times 20.57 \approx -205.7 \]

8. **Interpreting the Result**:
Since the result is negative, it indicates that the clock is losing time. Thus, the clock loses approximately 205.7 minutes in a day.

9. **Converting Minutes to Hours and Minutes**:
To convert 205.7 minutes into hours and minutes:
- 205.7 minutes = 3 hours and 25.7 minutes (approximately 26 minutes).

### Conclusion:
The clock loses approximately 3 hours and 26 minutes in a day.

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