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Difficulty: MediumRatios, Rates, and Proportions

A logistics facility utilizes two automated sorting lines, Line A and Line B, to process incoming shipments. Line A processes shipments at a constant rate that is 20 percent greater than the constant rate of Line B. When both lines operate simultaneously, they process a combined total of 3,300 shipments in 3 hours. Working alone at its constant rate, how many hours will it take Line B to process 2,250 shipments?

Answer: 4.5 hours

Answer

4.5 hours
Let rr represent the rate of Line B in shipments per hour. Because Line A processes 20% faster than Line B, Line A's rate is 1.20r1.20r. Together, their combined processing rate is r+1.20r=2.20rr + 1.20r = 2.20r shipments per hour. Operating for 3 hours, the total shipments processed is 3×2.20r=6.60r=3,3003 \times 2.20r = 6.60r = 3,300. Solving for rr gives r=500r = 500 shipments per hour. To process 2,250 shipments alone, Line B requires 2,250500=4.5\frac{2,250}{500} = 4.5 hours.

Step-by-Step Solution

1
Define the relationship between the individual processing rates.
If Line B processes at rate rr shipments per hour, Line A processes at 1.20r1.20r shipments per hour.
Line A's rate is 20 percent greater than Line B's rate.
2
Find the combined processing rate.
Combined rate = r+1.20r=2.20rr + 1.20r = 2.20r shipments per hour.
When working together, individual rates add up.
3
Determine Line B's rate (rr).
3×2.20r=3,300  ⟹  6.60r=3,300  ⟹  r=5003 \times 2.20r = 3,300 \implies 6.60r = 3,300 \implies r = 500 shipments per hour.
Total work equals combined rate multiplied by time.
4
Calculate the time required for Line B to complete 2,250 shipments.
Time=2,250500=4.5\text{Time} = \frac{2,250}{500} = 4.5 hours.
Time taken equals total work divided by the individual rate.

Key Concept

Combined Rates and Ratio Relationships
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Topics

Ratios, Rates, and Proportions

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