To determine the rate at which the housekeeper is cleaning, we first need to find out how much area (in terms of the first floor) she cleans per hour.
Given that she has cleaned \( \frac{1}{3} \) of the first floor in \( \frac{1}{5} \) of an hour, we can find the cleaning rate by calculating how much she would clean in one hour.
The rate of cleaning can be calculated as follows:
- Find the cleaning done in one hour by scaling the amount cleaned in \( \frac{1}{5} \) of an hour.
- The cleaning rate can be calculated as:
\[ \text{Cleaning Rate} = \left(\frac{\text{Amount cleaned}}{\text{Time taken}}\right) \times \text{Time in one hour} \]
The amount cleaned per hour is:
\[ \text{Cleaning Rate} = \frac{1/3}{1/5} = \frac{1}{3} \times \frac{5}{1} = \frac{5}{3} \]
So, the housekeeper cleans \( \frac{5}{3} \) of the first floor in one hour.
Therefore, the cleaning rate is:
\[ \frac{5}{3} \text{ floors per hour} \]
Thus, the housekeeper is cleaning at a rate of \( \frac{5}{3} \) of the first floor per hour, which is approximately 1.67 times the first floor in one hour.