G and H can do a work in 30 days .If H alone can do it in 50 days ,then G alone can do it in ___ days .

Options

A. 75
B. 70
C. 60
D. 65

Answer

To solve the problem, we need to determine how many days G alone can complete the work, given that G and H together can complete it in 30 days, and H alone can complete it in 50 days.

### Step-by-Step Solution:

1. **Understand the Work Done Together**:
- G and H together can complete the work in 30 days. This means their combined work rate (efficiency) is:
\[ \text{Efficiency of (G + H)} = \frac{1 \text{ work}}{30 \text{ days}} = \frac{1}{30} \]

2. **Calculate H's Efficiency**:
- H alone can complete the work in 50 days. Therefore, H's efficiency is:
\[ \text{Efficiency of H} = \frac{1 \text{ work}}{50 \text{ days}} = \frac{1}{50} \]

3. **Calculate G's Efficiency**:
- Since we know the combined efficiency of G and H, we can find G's efficiency by subtracting H's efficiency from their combined efficiency:
\[ \text{Efficiency of G} = \text{Efficiency of (G + H)} - \text{Efficiency of H} \]
\[ \text{Efficiency of G} = \frac{1}{30} - \frac{1}{50} \]

4. **Finding a Common Denominator**:
- The least common multiple of 30 and 50 is 150. We can convert the fractions:
\[ \frac{1}{30} = \frac{5}{150}, \quad \frac{1}{50} = \frac{3}{150} \]
- Now, substituting back:
\[ \text{Efficiency of G} = \frac{5}{150} - \frac{3}{150} = \frac{2}{150} = \frac{1}{75} \]

5. **Calculate the Time Taken by G Alone**:
- Now that we have G's efficiency, we can find out how many days G would take to complete the work alone:
\[ \text{Time taken by G} = \frac{1 \text{ work}}{\text{Efficiency of G}} = \frac{1}{\frac{1}{75}} = 75 \text{ days} \]

### Final Answer:
G alone can complete the work in **75 days**.