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-by-step Solution:** - Calculate the distance covered in the first case: $$\text{Distance} = 840 \times 6 = 5040 \text{ km}$$ - Convert $1 \frac{2}{3}$ hours to an improper fraction or decimal: $$1 \frac{2}{3} = \frac{5}{3} \text{ hours} \approx 1.6667 \text{ hours}$$ - Calculate the required speed to cover 5040 km in $\frac{5}{3}$ hours: $$\text{Speed} = \frac{5040}{\frac{5}{3}} = 5040 \times \frac{3}{5} = 3024 \text{ km/hr}$$ 4. **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. 1. **Problem Statement:** A box of sweets is divided among 24 children, each getting 5 sweets. Find how many sweets each child would get if the number of children is reduced by 4. 2. **Formula Used:** Total sweets = Number of children $\times$ sweets per child. 3. **Step-by-step Solution:** - Calculate total sweets: $$24 \times 5 = 120 \text{ sweets}$$ - New number of children: $$24 - 4 = 20$$ - Sweets per child now: $$\frac{120}{20} = 6 \text{ sweets}$$ 4. **Explanation:** Reducing the number of children increases sweets per child. AI tools helped confirm the division and reasoning. 1. **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. 2. **Formula Used:** Work rate formula: $$\frac{1}{\text{Time together}} = \frac{1}{\text{Harry's time}} + \frac{1}{\text{Joe's time}}$$ 3. **Step-by-step Solution:** - Let Joe's time be $x$ hours. - Using the formula: $$\frac{1}{8} = \frac{1}{12} + \frac{1}{x}$$ - Solve for $\frac{1}{x}$: $$\frac{1}{x} = \frac{1}{8} - \frac{1}{12} = \frac{3}{24} - \frac{2}{24} = \frac{1}{24}$$ - Therefore: $$x = 24 \text{ hours}$$ 4. **Explanation:** Joe takes 24 hours alone. AI tools helped understand and solve the work-rate problem efficiently.