Ganga and Saraswati, working separately can mow a fleld 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 p.m
C. 5 p.m
D. 5.30 p.m

Answer

To solve the problem, we need to determine how long it will take for Ganga and Saraswati to mow the field when they work alternately for one hour each, starting with Ganga at 9 a.m.

### Step-by-Step Solution:

1. **Determine the Work Rate of Each Person**:
- Ganga can mow 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 mow 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 Total Work Done in Two Hours**:
- In the first hour, Ganga works and mows:
\[ \text{Work done by Ganga in 1 hour} = \frac{1}{8} \text{ fields} \]
- In the second hour, Saraswati works and mows:
\[ \text{Work done by Saraswati in 1 hour} = \frac{1}{12} \text{ fields} \]
- 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):
\[ \frac{1}{8} = \frac{3}{24}, \quad \frac{1}{12} = \frac{2}{24} \]
- Thus,
\[ \text{Total work in 2 hours} = \frac{3}{24} + \frac{2}{24} = \frac{5}{24} \text{ fields} \]

3. **Calculate the Number of Complete Cycles Needed**:
- To complete 1 field, we need to find out how many 2-hour cycles are required:
\[ \text{Number of cycles} = \frac{1 \text{ field}}{\frac{5}{24} \text{ fields per 2 hours}} = \frac{24}{5} = 4.8 \text{ cycles} \]
- Since they cannot complete a fraction of a cycle, we need 4 complete cycles (which takes 8 hours) and then some additional work.

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

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

6. **Determine Who Works Next**:
- After 8 hours, it will be Ganga's turn to work next (since they alternate). Ganga can mow \(\frac{1}{8}\) of the field in 1 hour.

7. **Calculate Time to Complete Remaining Work**:
- To complete \(\frac{1}{6}\) of the field, we find out how long it will take Ganga:
\[ \text{Time} = \frac{\text{Remaining Work}}{\text{Ganga's Work Rate}} = \frac{\frac{1}{6}}{\frac{1}{8}} = \frac{8}{6} = \frac{4}{3} \text{ hours} = 1 \text{ hour and } 20 \text{ minutes} \]

8. **Calculate the Completion Time**:
- Ganga starts working at 9 a.m. and after 8 hours (which is 5 p.m.), she will work for an additional 1 hour and 20 minutes:
\[ 5:00 \text{ p.m.} + 1 \text{ hour and } 20 \text{ minutes} = 6:20 \text{ p.m.} \]

### Final Answer:
The mowing will be completed at **6:20 p.m.**.