1. **Problem statement:** A hotel has 320 rooms and 32 cleaners. With all cleaners working, it takes 2 hours to clean all rooms. On Monday, 20% of cleaners were not available. Will cleaning take more than 2 hours? 2. **Formula and concept:** The time taken to complete a job is inversely proportional to the number of workers if the work rate per cleaner is constant. This means: $$\text{Time} \times \text{Number of cleaners} = \text{Constant}$$ 3. **Calculate available cleaners on Monday:** $$20\% \text{ of } 32 = 0.20 \times 32 = 6.4 \approx 6 \text{ cleaners not available}$$ So, cleaners available = $$32 - 6 = 26$$ 4. **Calculate new time taken:** Using inverse proportionality: $$T_1 \times N_1 = T_2 \times N_2$$ Where: - $$T_1 = 2$$ hours (original time) - $$N_1 = 32$$ cleaners (original number) - $$N_2 = 26$$ cleaners (new number) - $$T_2 = ?$$ new time Rearranging: $$T_2 = \frac{T_1 \times N_1}{N_2} = \frac{2 \times 32}{26} = \frac{64}{26} \approx 2.46 \text{ hours}$$ 5. **Conclusion:** Since $$2.46 > 2$$, cleaning will take more than 2 hours on Monday. 6. **Regarding the proportionality statements:** - Ishan says cleaners are directly proportional to time taken. - Mira says cleaners are inversely proportional to time taken. Since more cleaners reduce the time needed, the relationship is inverse proportionality. **Therefore, Mira is correct.** **Reason:** If the number of cleaners increases, the time taken decreases, and vice versa, which defines inverse proportionality.