4 boys and 5 girls can do a piece of work in 10 days. 6 boys and 6 girls can do the same work in 7 days In how many days can 2 boys and 7 girls complete the same work, working together?

Options

A. 15 days
B. 14 days
C. 21 days
D. 18 days

Answer

To solve the problem step by step, let's break it down:

### Step 1: Determine the work done by boys and girls
We know that:
- 4 boys and 5 girls can complete the work in 10 days.
- 6 boys and 6 girls can complete the same work in 7 days.

Let the work done by 1 boy in 1 day be \( B \) and the work done by 1 girl in 1 day be \( G \).

From the first scenario:
\[ (4B + 5G) \times 10 = 1 \text{ (Total work)} \]
This simplifies to:
\[ 4B + 5G = \frac{1}{10} \quad \text{(Equation 1)} \]

From the second scenario:
\[ (6B + 6G) \times 7 = 1 \text{ (Total work)} \]
This simplifies to:
\[ 6B + 6G = \frac{1}{7} \quad \text{(Equation 2)} \]

### Step 2: Simplify the equations
Now, we can simplify both equations:
- From Equation 1:
\[ 4B + 5G = \frac{1}{10} \]
- From Equation 2:
\[ 6B + 6G = \frac{1}{7} \implies B + G = \frac{1}{42} \quad \text{(Divide by 6)} \]

### Step 3: Solve for \( B \) and \( G \)
Now we have two equations:
1. \( 4B + 5G = \frac{1}{10} \)
2. \( B + G = \frac{1}{42} \)

From the second equation, we can express \( G \) in terms of \( B \):
\[ G = \frac{1}{42} - B \]

Substituting \( G \) in Equation 1:
\[ 4B + 5\left(\frac{1}{42} - B\right) = \frac{1}{10} \]
This simplifies to:
\[ 4B + \frac{5}{42} - 5B = \frac{1}{10} \]
\[ -B + \frac{5}{42} = \frac{1}{10} \]
\[ -B = \frac{1}{10} - \frac{5}{42} \]
Finding a common denominator (210):
\[ -B = \frac{21}{210} - \frac{25}{210} = -\frac{4}{210} = -\frac{2}{105} \]
Thus,
\[ B = \frac{2}{105} \]

Now substituting \( B \) back to find \( G \):
\[ G = \frac{1}{42} - \frac{2}{105} \]
Finding a common denominator (210):
\[ G = \frac{5}{210} - \frac{4}{210} = \frac{1}{210} \]

### Step 4: Calculate total work
Now, we can calculate the total work in terms of units:
\[ \text{Total Work} = 1 \text{ unit} = 210 \text{ units} \]

### Step 5: Calculate work done by 2 boys and 7 girls
Now we need to find out how many days 2 boys and 7 girls can complete the work:
\[ \text{Work done by 2 boys and 7 girls in one day} = 2B + 7G \]
Substituting the values of \( B \) and \( G \):
\[ = 2\left(\frac{2}{105}\right) + 7\left(\frac{1}{210}\right) \]
Calculating:
\[ = \frac{4}{105} + \frac{7}{210} \]
Finding a common denominator (210):
\[ = \frac{4 \times 2}{210} + \frac{7}{210} = \frac{8 + 7}{210} = \frac{15}{210} = \frac{1}{14} \]

### Step 6: Calculate the number of days to complete the work
If 2 boys and 7 girls can do \( \frac{1}{14} \) of the work in one day, then the total number of days \( D \) to complete the work is:
\[ D = \frac{1 \text{ (Total Work)}}{\frac{1}{14}} = 14 \text{ days} \]

### Final Answer
Thus, 2 boys and 7 girls can complete the work in **14 days**.

---