1. **Problem Statement:** An aeroplane covers a certain distance at a speed of 840 km/hr in 6 hours. We need to find the speed required to cover the same distance in $1 \frac{2}{3}$ hours. 2. **Formula Used:** The relation between speed, distance, and time is given by: $$\text{Speed} = \frac{\text{Distance}}{\text{Time}}$$ 3. **Step 1: Calculate the distance covered in the first scenario.** $$\text{Distance} = \text{Speed} \times \text{Time} = 840 \times 6 = 5040 \text{ km}$$ 4. **Step 2: Convert $1 \frac{2}{3}$ hours to an improper fraction or decimal.** $$1 \frac{2}{3} = \frac{5}{3} \text{ hours} \approx 1.6667 \text{ hours}$$ 5. **Step 3: Calculate the required speed to cover 5040 km in $\frac{5}{3}$ hours.** $$\text{Speed} = \frac{5040}{\frac{5}{3}} = 5040 \times \frac{3}{5} = 3024 \text{ km/hr}$$ 6. **Explanation:** The speed must increase because the time to cover the same distance decreases. AI tools helped verify the calculations and understand the inverse relationship between speed and time when distance is constant. --- 7. **Problem Statement:** If a box of sweets is divided among 24 children, each gets 5 sweets. How many sweets would each get if the number of children is reduced by 4? 8. **Step 1: Calculate total sweets.** $$\text{Total sweets} = 24 \times 5 = 120$$ 9. **Step 2: Calculate sweets per child when children are reduced by 4.** $$\text{New number of children} = 24 - 4 = 20$$ $$\text{Sweets per child} = \frac{120}{20} = 6$$ 10. **Explanation:** Reducing the number of children increases sweets per child. AI tools helped confirm the division and reasoning. --- 11. **Problem Statement:** Harry and Joe together can mop a warehouse in 8 hours. Harry alone takes 12 hours. Find how long Joe alone would take. 12. **Step 1: Calculate work rates.** $$\text{Harry's rate} = \frac{1}{12} \text{ of work per hour}$$ $$\text{Combined rate} = \frac{1}{8} \text{ of work per hour}$$ 13. **Step 2: Calculate Joe's rate.** $$\text{Joe's rate} = \text{Combined rate} - \text{Harry's rate} = \frac{1}{8} - \frac{1}{12} = \frac{3}{24} - \frac{2}{24} = \frac{1}{24}$$ 14. **Step 3: Calculate time Joe takes alone.** $$\text{Time} = \frac{1}{\text{Joe's rate}} = 24 \text{ hours}$$ 15. **Explanation:** Work rates add when working together. AI tools helped understand and verify the subtraction of rates and the final calculation. **AI Tools Used:** Calculations and verifications were done using ChatGPT and WolframAlpha to ensure accuracy and understanding of formulas and problem-solving steps.