To determine what part of all the garbage bins the first garbage truck will empty when both crews work together, we first need to find their respective rates of working.
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First Truck Rate: The first truck empties all the bins in 5 hours. Therefore, its rate of work is: \[ \text{Rate}_{\text{First Truck}} = \frac{1 \text{ (all bins)}}{5 \text{ (hours)}} = \frac{1}{5} \text{ bins/hour} \]
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Second Truck Rate: The second truck empties all the bins in 8 hours. Thus, its rate is: \[ \text{Rate}_{\text{Second Truck}} = \frac{1 \text{ (all bins)}}{8 \text{ (hours)}} = \frac{1}{8} \text{ bins/hour} \]
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Combined Rate: When both trucks are working together, their combined rate is the sum of their individual rates: \[ \text{Combined Rate} = \frac{1}{5} + \frac{1}{8} \] To sum these fractions, we need a common denominator. The least common multiple of 5 and 8 is 40. We can rewrite the rates as: \[ \frac{1}{5} = \frac{8}{40} \quad \text{and} \quad \frac{1}{8} = \frac{5}{40} \] Now summing the two rates: \[ \text{Combined Rate} = \frac{8}{40} + \frac{5}{40} = \frac{13}{40} \text{ bins/hour} \]
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Time Taken Together: Let \( t \) be the total time they work together in hours to empty all the bins. Since they can empty all bins together in the combined rate, it takes: \[ 1 = \left(\frac{13}{40}\right) t \implies t = \frac{40}{13} \text{ hours} \]
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Part of Bins Emptied by the First Truck in \( t \): The first truck will empty: \[ \text{Part}{\text{First Truck}} = \text{Rate}{\text{First Truck}} \times t = \left(\frac{1}{5}\right) \times \left(\frac{40}{13}\right) \] Simplifying: \[ \text{Part}_{\text{First Truck}} = \frac{40}{65} \implies \frac{8}{13} \]
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Calculating the Equivalent Decimal Value: To find the approximate decimal value of \( \frac{8}{13} \): \[ \frac{8}{13} \approx 0.615 \]
Based on the options given (0.6, 0.5, 0.7, 0.4), the closest approximation is:
0.6.
Thus, the part of all the garbage bins that the first garbage truck will empty is approximately 0.6.
3 answers