1. **State the problem:** Máiréad can clean a window in 9 minutes, Diarmuid in 12 minutes. (i) They start cleaning at the same time and finish a window simultaneously. Find after how many minutes this first happens. 2. **Formula and concept:** To find when they finish together, find the least common multiple (LCM) of their cleaning times, because they finish windows at multiples of their individual times. 3. **Calculate LCM:** - Máiréad's times: 9, 18, 27, 36, ... - Diarmuid's times: 12, 24, 36, 48, ... The first common time is 36 minutes. 4. **Answer for (i):** They will both finish a window together after **36 minutes**. --- 5. **Problem (ii):** A building has 35 windows. How long will it take Máiréad and Diarmuid to clean all 35 windows working together? 6. **Formula:** When working together, their combined rate is the sum of their individual rates. - Máiréad's rate: $\frac{1}{9}$ windows per minute - Diarmuid's rate: $\frac{1}{12}$ windows per minute Combined rate: $$ \frac{1}{9} + \frac{1}{12} = \frac{4}{36} + \frac{3}{36} = \frac{7}{36} \text{ windows per minute} $$ 7. **Calculate time to clean 35 windows:** $$ \text{Time} = \frac{\text{Total windows}}{\text{Combined rate}} = \frac{35}{\frac{7}{36}} = 35 \times \frac{36}{7} $$ 8. **Simplify:** $$ 35 \times \frac{36}{7} = \cancel{35}^{5} \times \frac{36}{\cancel{7}^{1}} = 5 \times 36 = 180 \text{ minutes} $$ 9. **Answer for (ii):** It will take them **180 minutes** to clean all 35 windows working together.