2 men or 5 women or 7 boys can finish a work in 469 days. then the number of days taken by 7 men, 5 women and 2 boys to finish the work is :

Options

A. 134
B. 106
C. 100
D. 98

Answer

To solve the problem step by step, we need to determine how many days it will take for 7 men, 5 women, and 2 boys to finish the work, given that 2 men, 5 women, or 7 boys can finish the work in 469 days.

### Step 1: Determine the work done by each group
Given:
- 2 men can finish the work in 469 days.
- 5 women can finish the work in 469 days.
- 7 boys can finish the work in 469 days.

This means:
- Work done by 2 men in 469 days = 1 work unit
- Work done by 5 women in 469 days = 1 work unit
- Work done by 7 boys in 469 days = 1 work unit

### Step 2: Calculate the efficiency of each group
The efficiency (work done per day) can be calculated as:
- Efficiency of 2 men = \( \frac{1}{469} \) work units per day
- Efficiency of 5 women = \( \frac{1}{469} \) work units per day
- Efficiency of 7 boys = \( \frac{1}{469} \) work units per day

Thus, the efficiency of each individual can be calculated as follows:
- Efficiency of 1 man = \( \frac{1}{2} \times \frac{1}{469} = \frac{1}{938} \) work units per day
- Efficiency of 1 woman = \( \frac{1}{5} \times \frac{1}{469} = \frac{1}{2345} \) work units per day
- Efficiency of 1 boy = \( \frac{1}{7} \times \frac{1}{469} = \frac{1}{3283} \) work units per day

### Step 3: Calculate the total efficiency of 7 men, 5 women, and 2 boys
Now, we calculate the total efficiency of 7 men, 5 women, and 2 boys:
- Total efficiency of 7 men = \( 7 \times \frac{1}{938} = \frac{7}{938} \)
- Total efficiency of 5 women = \( 5 \times \frac{1}{2345} = \frac{5}{2345} \)
- Total efficiency of 2 boys = \( 2 \times \frac{1}{3283} = \frac{2}{3283} \)

### Step 4: Find a common denominator and add the efficiencies
To add these fractions, we need a common denominator. The least common multiple (LCM) of 938, 2345, and 3283 can be used.

After calculating the LCM, we can express each efficiency with that common denominator and sum them up.

### Step 5: Calculate the total work done per day
Let’s denote the total efficiency as \( E \). The total work done per day by 7 men, 5 women, and 2 boys will be \( E \).

### Step 6: Calculate the number of days to finish the work
The total number of days taken to finish the work can be calculated using the formula:
\[ \text{Days} = \frac{1 \text{ work unit}}{E} \]

### Final Calculation
After performing the calculations, we find that the total efficiency \( E \) leads to:
\[ \text{Days} = 98 \text{ days} \]

Thus, the number of days taken by 7 men, 5 women, and 2 boys to finish the work is **98 days**.

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