A conveyor belt delivers baggage at the rate of 3 tons in 5 minutes and a second conveyor belt delivers baggage at the rate of 1 ton in 2 minutes. How much time will it take to get 33 tons of baggage delivered using both the conveyor belts together?

### Concept & Additive Rates When multiple machines work together simultaneously, their output rates (work per unit time) are additive. $$ \text{Total Time} = \frac{\text{Total Target Work}}{\text{Combined Output Rate}} $$ ### Step-by-Step Solution * **Calculate Rate of Belt 1:** Belt 1 delivers $3$ tons in $5$ minutes. Rate of Belt 1 = $\frac{3}{5}$ tons / minute. * **Calculate Rate of Belt 2:** Belt 2 delivers $1$ ton in $2$ minutes. Rate of Belt 2 = $\frac{1}{2}$ tons / minute. * **Calculate Combined Rate:** Combined rate = $\frac{3}{5} + \frac{1}{2}$ Take LCM of $5$ and $2$, which is $10$. Combined rate = $\frac{6}{10} + \frac{5}{10} = \frac{11}{10}$ tons / minute. * **Calculate Total Time for 33 Tons:** Total work needed = $33$ tons. Time taken = $\frac{\text{Total Work}}{\text{Combined Rate}}$ Time = $\frac{33}{\frac{11}{10}} = 33 \times \frac{10}{11} = 3 \times 10 = 30$ minutes. ### Exam Strategy & Shortcut Instead of fractions, you can equalize the time to find combined output in a fixed interval. In $10$ minutes (LCM of $5$ and $2$): Belt 1 delivers $3 \times 2 = 6$ tons. Belt 2 delivers $1 \times 5 = 5$ tons. Together they deliver $6 + 5 = 11$ tons in $10$ minutes. To deliver $33$ tons ($3$ times the amount), it will take $3 \times 10 = 30$ minutes. ### Common Pitfall A common mistake is trying to add the times directly or averaging the rates rather than strictly summing the capacities (tons per minute). ### Final Answer Therefore, the correct answer is **30 min**.