Alekh and Alia together can complete a task in 5 days, while Alia and Aman together can finish the same task in 15 days. Alekh works on the task for 4 days, Alia works for 5 days, and then Aman completes the remaining work alone in 4 days. How many days will Alia take to finish the entire work by herself?

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**Solution:**

1. Let the daily work done by Alekh, Alia, and Aman be *x*, *y*, and *z* respectively.

2. According to the problem:
   - Alekh and Alia together can finish the work in 5 days, so their combined daily work is:
     $$x + y = \frac{1}{5}$$

- Alia and Aman together can finish the work in 15 days, so their combined daily work is:
     $$y + z = \frac{1}{15}$$

3. Alekh works for 4 days, Alia for 5 days, and Aman completes the remaining work alone in 4 days. The total work done is 1, so:
   $$4x + 5y + 4z = 1$$

4. Using the first equation, express *x* as:
   $$x = \frac{1}{5} - y$$

5. Substitute *x* into the total work equation:
   $$4\left(\frac{1}{5} - yight) + 5y + 4z = 1$$

Simplify:
   $$\frac{4}{5} - 4y + 5y + 4z = 1$$

$$\frac{4}{5} + y + 4z = 1$$

6. From the second equation, express *z* as:
   $$z = \frac{1}{15} - y$$

7. Substitute *z* into the above equation:
   $$\frac{4}{5} + y + 4\left(\frac{1}{15} - yight) = 1$$

Simplify:
   $$\frac{4}{5} + y + \frac{4}{15} - 4y = 1$$

$$\frac{4}{5} + \frac{4}{15} - 3y = 1$$

8. Convert fractions to a common denominator (15):
   $$\frac{12}{15} + \frac{4}{15} - 3y = 1$$

$$\frac{16}{15} - 3y = 1$$

9. Rearranging:
   $$-3y = 1 - \frac{16}{15} = \frac{15}{15} - \frac{16}{15} = -\frac{1}{15}$$

$$3y = \frac{1}{15}$$

$$y = \frac{1}{45}$$

10. Since *y* is the daily work done by Alia, the number of days Alia takes to complete the work alone is:
    $$	ext{Days} = \frac{1}{y} = 45$$

**Answer:** 45 days

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