8 women and 8 girls can finish a piece of work in 6 days, whereas 4 women and 10 girls can finish it in 8 days. In how many days will one girl finish it working alone?

Options

A. 120
B. 144
C. 72
D. 84

Answer

To solve the problem, we need to find the work done by women and girls in terms of their individual contributions. Let's denote the work done by one woman in one day as \( W \) and the work done by one girl in one day as \( G \).

### Step 1: Set up the equations based on the given information.

1. **From the first scenario**:
- 8 women and 8 girls can finish the work in 6 days.
- Therefore, the total work can be expressed as:
\[ \text{Total Work} = \text{(Work done per day)} \times \text{(Number of days)} = (8W + 8G) \times 6 \]

2. **From the second scenario**:
- 4 women and 10 girls can finish the work in 8 days.
- Thus, the total work can also be expressed as:
\[ \text{Total Work} = (4W + 10G) \times 8 \]

### Step 2: Equate the total work from both scenarios.

Since both expressions represent the same total work, we can set them equal to each other:
\[ (8W + 8G) \times 6 = (4W + 10G) \times 8 \]

### Step 3: Simplify the equation.

Expanding both sides:
\[ 48W + 48G = 32W + 80G \]

### Step 4: Rearrange the equation to isolate terms.

Moving all terms involving \( W \) and \( G \) to one side:
\[ 48W - 32W = 80G - 48G \]
\[ 16W = 32G \]

### Step 5: Simplify to find the relationship between \( W \) and \( G \).

Dividing both sides by 16:
\[ W = 2G \]

### Step 6: Substitute \( W \) back into one of the original equations to find \( G \).

Using the first scenario:
\[ \text{Total Work} = (8W + 8G) \times 6 \]
Substituting \( W = 2G \):
\[ \text{Total Work} = (8(2G) + 8G) \times 6 = (16G + 8G) \times 6 = 24G \times 6 = 144G \]

### Step 7: Find the total work in terms of \( G \).

Now we know the total work is \( 144G \).

### Step 8: Determine how many days one girl takes to finish the work alone.

If one girl does \( G \) work in one day, then the number of days \( D \) it takes for one girl to finish the total work is:
\[ D = \frac{\text{Total Work}}{\text{Work done by one girl in one day}} = \frac{144G}{G} = 144 \]

### Conclusion

Thus, one girl will finish the work alone in **144 days**.

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