4 boys and 7 girls can complete a work in 7 days, while 3 boys and 6 girls can complete it in 9 days. In how many days will 9 girls complete it?

Let's solve the problem step by step.

### Step 1: Define the variables
- Let the work done by one boy in one day be \( b \) (units of work).
- Let the work done by one girl in one day be \( g \) (units of work).

### Step 2: Set up the equations
Based on the problem:

- **Equation 1:** \( 4b + 7g \) is the work done by 4 boys and 7 girls in one day.
  
  Since they can complete the work in 7 days:
  \[
  7 \times (4b + 7g) = 1 \quad \text{(1 unit of work)}
  \]
  Simplifying:
  \[
  4b + 7g = \frac{1}{7} \quad \text{(Equation 1)}
  \]

- **Equation 2:** \( 3b + 6g \) is the work done by 3 boys and 6 girls in one day.
  
  Since they can complete the work in 9 days:
  \[
  9 \times (3b + 6g) = 1 \quad \text{(1 unit of work)}
  \]
  Simplifying:
  \[
  3b + 6g = \frac{1}{9} \quad \text{(Equation 2)}
  \]

### Step 3: Solve the system of equations
Now we solve the system of linear equations:

1. **Multiply Equation 1 by 3** to align coefficients of \( b \):
   \[
   12b + 21g = \frac{3}{7}
   \]

2. **Multiply Equation 2 by 4** to align coefficients of \( b \):
   \[
   12b + 24g = \frac{4}{9}
   \]

3. **Subtract the first equation from the second** to eliminate \( b \):
   \[
   (12b + 24g) - (12b + 21g) = \frac{4}{9} - \frac{3}{7}
   \]
   Simplifying:
   \[
   3g = \frac{4}{9} - \frac{3}{7}
   \]

4. **Find the common denominator** and solve for \( g \):
   \[
   3g = \frac{28 - 27}{63} = \frac{1}{63}
   \]
   \[
   g = \frac{1}{189} \quad \text{(units of work per day by one girl)}
   \]

### Step 4: Substitute \( g \) back to find \( b \)
Use \( g = \frac{1}{189} \) in Equation 1 or 2 to find \( b \).

Using Equation 1:
\[
4b + 7\left(\frac{1}{189}\right) = \frac{1}{7}
\]
\[
4b + \frac{7}{189} = \frac{1}{7}
\]
\[
4b + \frac{1}{27} = \frac{1}{7}
\]
\[
4b = \frac{1}{7} - \frac{1}{27}
\]
\[
4b = \frac{27 - 7}{189} = \frac{20}{189}
\]
\[
b = \frac{5}{189} \quad \text{(units of work per day by one boy)}
\]

### Step 5: Calculate the time taken by 9 girls to complete the work
Now, we know that the work done by 9 girls in one day is \( 9g \). So:
\[
9g = 9 \times \frac{1}{189} = \frac{9}{189} = \frac{1}{21}
\]
This means 9 girls will take 21 days to complete the work.

### Final Answer
9 girls will complete the work in **21 days**.

Would you like more details or have any questions?

### Follow-Up Questions:
1. How long would it take for 5 boys and 4 girls to complete the same work?
2. What if the work required 2 times more effort; how many days would 9 girls need?
3. How would the solution change if only 1 boy and 1 girl were working together?
4. What if instead of 9 girls, we had 12 girls? How many days would it take?
5. Can this problem be solved using matrix methods? If so, how?

### Tip:
Always check units and simplify the equations early when working with multiple variables in word problems.

Solve the work and time problem where 4 boys and 7 girls complete a task in 7 days, and 3 boys and 6 girls complete it in 9 days. Learn how to find the number of days 9 girls will take using algebra and systems of linear equations. Suitable for students in Grades 9-12.

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