Formulate the following problems as a pair of equations, and hence find their solutions:
2 women and 5 men can together finish an embroidery work in 4 days, while 3 women and 6 men can finish it in 3 days. Find the time taken by 1 women alone to finish the work, and also that taken by 1 man alone.

Answer

To solve the problem, we need to formulate it as a pair of linear equations based on the information given. Let's denote:

- \( W \) = the amount of work done by one woman in one day
- \( M \) = the amount of work done by one man in one day

### Step 1: Formulate the equations

From the problem statement, we have two scenarios:

1. **Scenario 1**: 2 women and 5 men can finish the work in 4 days.
- The total work can be expressed as:
\[ \text{Total Work} = \text{(Work done in 1 day)} \times \text{(Number of days)} \]
\[ \text{Total Work} = (2W + 5M) \times 4 \]

2. **Scenario 2**: 3 women and 6 men can finish the work in 3 days.
- Similarly, the total work can be expressed as:
\[ \text{Total Work} = (3W + 6M) \times 3 \]

Since both expressions represent the same total work, we can set them equal to each other:
\[ (2W + 5M) \times 4 = (3W + 6M) \times 3 \]

### Step 2: Simplify the equation

Expanding both sides:
\[ 8W + 20M = 9W + 18M \]

Now, rearranging the equation to isolate \( W \) and \( M \):
\[ 8W - 9W + 20M - 18M = 0 \]
\[ -W + 2M = 0 \]
\[ W = 2M \quad \text{(Equation 1)} \]

### Step 3: Substitute \( W \) in one of the original equations

Now we can substitute \( W = 2M \) into one of the original equations to find \( M \). Let's use the first scenario:
\[ 2(2M) + 5M = \frac{1}{4} \quad \text{(since total work = 1)} \]
\[ 4M + 5M = \frac{1}{4} \]
\[ 9M = \frac{1}{4} \]
\[ M = \frac{1}{36} \quad \text{(Equation 2)} \]

### Step 4: Find \( W \)

Now substituting \( M = \frac{1}{36} \) back into Equation 1 to find \( W \):
\[ W = 2M = 2 \times \frac{1}{36} = \frac{1}{18} \]

### Step 5: Interpret the results

- The time taken by one man to finish the work alone:
\[ \text{Time for 1 man} = \frac{1}{M} = 36 \text{ days} \]

- The time taken by one woman to finish the work alone:
\[ \text{Time for 1 woman} = \frac{1}{W} = 18 \text{ days} \]

### Final Answer
- Time taken by one woman alone: **18 days**
- Time taken by one man alone: **36 days**