Question: The Q1 is the tallest residential building in Australia, with $78$ floors.
The window cleaner spends $8$ hours a day, $5$ days a week, suspended in a basket on the outside of the Q1. There are $32$ windows on each floor, and he is allowed $21$ minutes per window before moving to the next one. He was complaining that by the time he has finished all the windows, he has to start back at the beginning with dirty windows all over again.
How long does it actually take him to clean all the windows on Q1?
Give your answer in weeks and days.
1. **State the problem:** We need to find how long it takes the window cleaner to clean all the windows on the Q1 building, which has $78$ floors and $32$ windows per floor, with $21$ minutes spent per window.
2. **Calculate total number of windows:**
$$\text{Total windows} = 78 \times 32 = 2496$$
3. **Calculate total cleaning time in minutes:**
$$\text{Total time (minutes)} = 2496 \times 21 = 52416$$
4. **Convert total time to hours:**
$$\text{Total time (hours)} = \frac{52416}{60} = 873.6$$
5. **Calculate how many days it takes cleaning $8$ hours per day:**
$$\text{Days} = \frac{873.6}{8} = 109.2$$
6. **Calculate how many weeks and days this corresponds to, given $5$ working days per week:**
$$\text{Weeks} = \left\lfloor \frac{109.2}{5} \right\rfloor = 21$$
$$\text{Remaining days} = 109.2 - 21 \times 5 = 109.2 - 105 = 4.2$$
7. **Interpretation:** It takes $21$ full weeks and about $4.2$ days to clean all the windows.
**Final answer:**
The window cleaner takes approximately **21 weeks and 4 days** to clean all the windows on Q1.