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Difficulty: MediumRatios, Rates, and Proportions

An older printing press can produce 420420 brochures every 1515 minutes. A newer printing press can produce brochures at a rate that is 1.51.5 times the rate of the older press. If both presses run simultaneously at their respective constant rates, how many minutes will it take them to produce a total of 2,1002,100 brochures?

  1. A
    35
  2. B
    50
  3. 30Answer
  4. D
    75

Answer

The correct answer is 3030 minutes.
To find the time needed, first calculate the individual rates in brochures per minute. The older press prints 420420 brochures in 1515 minutes, which is 42015=28\frac{420}{15} = 28 brochures per minute. The newer press prints at 1.51.5 times this rate, which is 1.5×28=421.5 \times 28 = 42 brochures per minute. Working together, their combined rate is 28+42=7028 + 42 = 70 brochures per minute. Dividing the total target of 2,1002,100 brochures by the combined rate of 7070 brochures per minute yields 2,10070=30\frac{2,100}{70} = 30 minutes. Thus, the correct option is 3030.

Step-by-Step Solution

1
Calculate the rate of the older printing press in brochures per minute.
The older press prints at a rate of 42015=28\frac{420}{15} = 28 brochures per minute.
Finding the unit rate allows us to easily combine and compare the performance of both presses.
2
Calculate the rate of the newer printing press.
The newer press prints at a rate of 28×1.5=4228 \times 1.5 = 42 brochures per minute.
The problem states the newer press operates at 1.51.5 times the rate of the older press.
3
Determine the combined rate of both presses working together.
The combined rate is 28+42=7028 + 42 = 70 brochures per minute.
When both presses run simultaneously, their rates add up to form the total production rate.
4
Calculate the time required to produce 2,1002,100 brochures.
The required time is 2,10070=30\frac{2,100}{70} = 30 minutes.
Dividing the total work by the combined rate gives the total time needed to complete the task.

Key Concept

Solving rate problems by finding individual unit rates and combining them to find the total time required for a joint task.
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Topics

Ratios, Rates, and Proportions

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