If 3 men or 9 boys can finish a work in 21 days . In how many days, can 5 men and 6 boys together do the same work ?

Options

A. 12 days
B. 8 days
C. 14 days
D. None the above

Answer

To solve the problem step by step, we will first determine the work done by men and boys, and then find out how many days it will take for 5 men and 6 boys to complete the same work.

### Step 1: Determine the total work done by men and boys

Given that:
- 3 men can finish the work in 21 days.
- 9 boys can finish the same work in 21 days.

Let's denote:
- The work done by 1 man in 1 day as \( M \) (units of work).
- The work done by 1 boy in 1 day as \( B \) (units of work).

**Total work done by 3 men in 21 days:**
\[ \text{Total Work} = \text{Work per day} \times \text{Number of days} = 3M \times 21 = 63M \text{ units} \]

**Total work done by 9 boys in 21 days:**
\[ \text{Total Work} = \text{Work per day} \times \text{Number of days} = 9B \times 21 = 189B \text{ units} \]

Since both expressions represent the same total work, we can set them equal to each other:
\[ 63M = 189B \]

### Step 2: Simplify the equation

Dividing both sides by 63:
\[ M = \frac{189B}{63} = 3B \]

This tells us that the work done by one man in one day is equivalent to the work done by three boys in one day.

### Step 3: Calculate the total work in terms of \( k \)

Let’s express \( M \) and \( B \) in terms of a common variable \( k \):
- Let \( B = k \) (work done by one boy in one day).
- Then \( M = 3k \) (work done by one man in one day).

### Step 4: Find the total work in terms of \( k \)

Using either of the earlier equations, we can find the total work:
\[ \text{Total Work} = 63M = 63(3k) = 189k \text{ units} \]

### Step 5: Calculate the work done by 5 men and 6 boys in one day

Now, we need to find out how much work 5 men and 6 boys can do together in one day:
\[ \text{Work done by 5 men} = 5M = 5(3k) = 15k \]
\[ \text{Work done by 6 boys} = 6B = 6k \]
\[ \text{Total work done by 5 men and 6 boys in one day} = 15k + 6k = 21k \]

### Step 6: Calculate the number of days taken by 5 men and 6 boys to complete the work

Let \( X \) be the number of days taken to complete the work:
\[ \text{Total Work} = \text{Work done in one day} \times \text{Number of days} \]
\[ 189k = 21k \times X \]

Dividing both sides by \( 21k \):
\[ X = \frac{189k}{21k} = 9 \text{ days} \]

### Conclusion

Thus, 5 men and 6 boys together can complete the work in **9 days**.

---