A group of women working together at the same rate can build a wall in $45$ hours. When the work started, all the women did not start working together. They joined the worked over a period of time, one by one, at equal intervals. Once at work, each one stayed till the work was complete. If the first woman worked 5 times as many hours as the last woman, for how many hours did the first woman work?
1. Let the work done in 1 hour by 1 woman be \( k \), and let the number of women be \( y \).
2. Let the interval after each woman arrives be \( t \), and let the time for which all of them work together be \( \eta \).
3. The total work \( W \) can be expressed as the sum of the work done by each woman. The first woman works for \((y-1)t + \eta\) hours, the second woman works for \((y-2)t + \eta\) hours, and so on, until the last woman who works for \(\eta\) hours.
4. Therefore, the total work \( W \) is given by:
   \[
   W = k \left( (y-1)t + \eta \right) + k \left( (y-2)t + \eta \right) + \cdots + k \eta
   \]
5. This can be simplified using the sum of an arithmetic series:
   \[
   W = k \left( \sum_{i=0}^{y-1} ((y-1-i)t + \eta) \right)
   \]
6. The sum inside the parentheses is:
   \[
   \sum_{i=0}^{y-1} ((y-1-i)t + \eta) = \sum_{i=0}^{y-1} (y-1-i)t + \sum_{i=0}^{y-1} \eta
   \]
7. The first sum is the sum of the first \( y \) terms of an arithmetic series with the first term \( (y-1)t \) and the last term \( 0 \):
   \[
   \sum_{i=0}^{y-1} (y-1-i)t = t \sum_{i=0}^{y-1} (y-1-i) = t \sum_{i=0}^{y-1} i = t \cdot \frac{(y-1)y}{2}
   \]
8. The second sum is simply \( y \eta \):
   \[
   \sum_{i=0}^{y-1} \eta = y \eta
   \]
9. Therefore, the total work \( W \) is:
   \[
   W = k \left( t \cdot \frac{(y-1)y}{2} + y \eta \right)
   \]
10. We know that \( ky = \frac{W}{45} \), so:
    \[
    W = 45ky
    \]
11. Equating the two expressions for \( W \):
    \[
    k \left( t \cdot \frac{(y-1)y}{2} + y \eta \right) = 45ky
    \]
12. Dividing both sides by \( ky \):
    \[
    t \cdot \frac{(y-1)}{2} + \eta = 45
    \]
13. We also know that the first woman worked 5 times as many hours as the last woman, so:
    \[
    (y-1)t + \eta = 5\eta
    \]
14. Simplifying this, we get:
    \[
    (y-1)t = 4\eta
    \]
15. Substituting \( (y-1)t = 4\eta \) into \( t \cdot \frac{(y-1)}{2} + \eta = 45 \):
    \[
    4\eta \cdot \frac{1}{2} + \eta = 45
    \]
16. Simplifying, we get:
    \[
    2\eta + \eta = 45
    \]
    \[
    3\eta = 45
    \]
    \[
    \eta = 15
    \]
17. Therefore, the first woman worked for:
    \[
    (y-1)t + \eta = 4\eta + \eta = 5\eta = 5 \times 15 = 75 \text{ hours}
    \]

The final answer is \(\boxed{75}\) hours.