Ganga and Saraswati, working separately can now a field in 8 and 12 hours respectively. If they work in stretches of one hour alternately, Ganga beginning at 9 a.m., when will the moving be completed ?

Options

A. 6 p.m .
B. 6.30 a.m.
C. 5 p.m.
D. 5.30 p.m.

Answer

To solve the problem, we need to determine when Ganga and Saraswati will complete the work if they alternate working on it for one hour each, starting with Ganga at 9 a.m.

### Step-by-Step Solution:

1. **Determine the Work Rates:**
- Ganga can complete the field in 8 hours. Therefore, her work rate is:
\[ \text{Work Rate of Ganga} = \frac{1 \text{ field}}{8 \text{ hours}} = \frac{1}{8} \text{ fields per hour} \]
- Saraswati can complete the field in 12 hours. Therefore, her work rate is:
\[ \text{Work Rate of Saraswati} = \frac{1 \text{ field}}{12 \text{ hours}} = \frac{1}{12} \text{ fields per hour} \]

2. **Calculate the Amount of Work Done in 2 Hours:**
- In the first hour (9 a.m. to 10 a.m.), Ganga works and completes:
\[ \text{Work done by Ganga} = \frac{1}{8} \text{ fields} \]
- In the second hour (10 a.m. to 11 a.m.), Saraswati works and completes:
\[ \text{Work done by Saraswati} = \frac{1}{12} \text{ fields} \]
- Therefore, the total work done in 2 hours is:
\[ \text{Total Work in 2 hours} = \frac{1}{8} + \frac{1}{12} \]
- To add these fractions, find a common denominator (which is 24):
\[ \frac{1}{8} = \frac{3}{24}, \quad \frac{1}{12} = \frac{2}{24} \]
\[ \text{Total Work in 2 hours} = \frac{3}{24} + \frac{2}{24} = \frac{5}{24} \text{ fields} \]

3. **Determine How Many Cycles Are Needed to Complete the Work:**
- The total work required is 1 field. The number of 2-hour cycles needed to complete the work is:
\[ \text{Number of Cycles} = \frac{1 \text{ field}}{\frac{5}{24} \text{ fields per 2 hours}} = \frac{24}{5} \text{ cycles} \approx 4.8 \text{ cycles} \]
- This means they will complete 4 full cycles (8 hours) and then have some work left.

4. **Calculate Work Done in 4 Full Cycles (8 Hours):**
- In 4 cycles (8 hours), the total work done is:
\[ \text{Work done in 8 hours} = 4 \times \frac{5}{24} = \frac{20}{24} = \frac{5}{6} \text{ fields} \]

5. **Calculate Remaining Work:**
- After 8 hours, the remaining work is:
\[ \text{Remaining Work} = 1 - \frac{5}{6} = \frac{1}{6} \text{ fields} \]

6. **Determine Who Works Next:**
- After 8 hours (which ends at 5 p.m.), Ganga will work for the next hour (from 5 p.m. to 6 p.m.):
- In this hour, Ganga will complete:
\[ \text{Work done by Ganga} = \frac{1}{8} \text{ fields} \]
- Since \(\frac{1}{8} > \frac{1}{6}\), she will finish the remaining work in less than an hour.

7. **Calculate Time to Complete Remaining Work:**
- To find the exact time taken to complete the remaining \(\frac{1}{6}\) fields:
\[ \text{Time} = \frac{\text{Remaining Work}}{\text{Work Rate of Ganga}} = \frac{\frac{1}{6}}{\frac{1}{8}} = \frac{8}{6} = \frac{4}{3} \text{ hours} \approx 1 \text{ hour and } 20 \text{ minutes} \]

8. **Final Completion Time:**
- Starting from 5 p.m., if Ganga works for \(\frac{4}{3}\) hours, she will finish at:
\[ 5 \text{ p.m.} + 1 \text{ hour and } 20 \text{ minutes} = 6:20 \text{ p.m.} \]

### Conclusion:
The work will be completed at **6:20 p.m.**.