If 8 men or 12 boys can do a piece of work in 16 days, the number of days required to complete the work by 20 men and 6 boys is
यदि 8 आदमी या 12 बालक एक काम को 16 दिन में कर सकते हैं, तो 20 आदमियों और 6 बालकों को वह काम पूरा करने में कितने दिन 7 लगेंगे? Options are (a) `5(1)/(3)` (b) `6(1)/(3)` (c) `8(1)/(3)` (d)`7(1)/(7)`

Options

A. `5(1)/(3)`
B. `6(1)/(3)`
C. `8(1)/(3)`
D. `7(1)/(7)`

Explanation

To solve the problem, we need to determine how many days it will take for 20 men and 6 boys to complete a piece of work, given that 8 men or 12 boys can do the same work in 16 days.

### Step-by-Step Solution:

1. **Establish the Work Done by Men and Boys:**
- We know that 8 men can complete the work in 16 days. Therefore, the total work can be expressed in terms of man-days:
\[ \text{Total Work} = \text{Number of Men} \times \text{Number of Days} = 8 \text{ men} \times 16 \text{ days} = 128 \text{ man-days} \]

2. **Calculate the Work Done by Boys:**
- Similarly, 12 boys can also complete the same work in 16 days:
\[ \text{Total Work} = 12 \text{ boys} \times 16 \text{ days} = 192 \text{ boy-days} \]

3. **Establish the Efficiency Ratio of Men to Boys:**
- From the above calculations, we can equate the work done by men and boys:
\[ 128 \text{ man-days} = 192 \text{ boy-days} \]
- This implies:
\[ 1 \text{ man} = \frac{192}{128} \text{ boys} = \frac{3}{2} \text{ boys} \]
- Therefore, the ratio of men to boys is:
\[ \text{Men : Boys} = 3 : 2 \]

4. **Calculate the Work Capacity of 20 Men and 6 Boys:**
- Now, we need to find out how much work 20 men and 6 boys can do in one day:
- Work done by 20 men:
\[ 20 \text{ men} = 20 \times 3 = 60 \text{ boys} \]
- Work done by 6 boys:
\[ 6 \text{ boys} = 6 \text{ boys} \]
- Total work done in one day by 20 men and 6 boys:
\[ \text{Total Work per Day} = 60 \text{ boys} + 6 \text{ boys} = 66 \text{ boys} \]

5. **Calculate the Number of Days Required to Complete the Work:**
- Since we have established that the total work is equivalent to 192 boy-days, we can now find the number of days required by dividing the total work by the daily work capacity:
\[ \text{Number of Days} = \frac{\text{Total Work}}{\text{Work per Day}} = \frac{192 \text{ boy-days}}{66 \text{ boys}} = \frac{192}{66} = \frac{32}{11} \]
- Converting \(\frac{32}{11}\) into a mixed fraction:
\[ \frac{32}{11} = 2 \text{ remainder } 10 \Rightarrow 2 \frac{10}{11} \]

6. **Final Calculation:**
- To find out how many days it takes, we can also express this in decimal or mixed fraction form:
\[ 2 \frac{10}{11} \approx 2.91 \text{ days} \]

### Conclusion:
The number of days required to complete the work by 20 men and 6 boys is approximately \(5 \frac{1}{3}\) days.