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Difficulty: MediumAlgebraic Word Problems and Modeling

A commercial print shop uses two high-speed printing presses, Press Alpha and Press Beta. Press Alpha operates at a constant rate of 120120 pages per minute, while Press Beta operates at a constant rate of 180180 pages per minute. Press Alpha begins printing a job of 15,00015,000 pages at 9:00 AM. At 9:15 AM, Press Beta is turned on to assist Press Alpha, and both presses continue printing simultaneously at their respective constant rates until the job is completed. How many total minutes after 9:00 AM will the entire 15,00015,000-page job be finished?

Answer: 59 minutes

Answer

The entire 15,000-page job will be finished 59 minutes after 9:00 AM.
Press Alpha operates alone for the first 15 minutes, completing 15×120=1,80015 \times 120 = 1,800 pages. That leaves 15,000−1,800=13,20015,000 - 1,800 = 13,200 pages. Once Press Beta joins at 9:15 AM, the combined rate becomes 120+180=300120 + 180 = 300 pages per minute. The remaining pages require 13,200/300=4413,200 / 300 = 44 minutes. Summing the 15-minute initial period and the 44-minute joint period gives a total of 59 minutes after 9:00 AM.

Step-by-Step Solution

1
Find the work completed by Press Alpha during the 15-minute staggered start period.
1,800 pages completed.
Press Alpha ran alone for 15 minutes at 120 pages per minute.
2
Determine the remaining work to be done after 9:15 AM.
13,200 pages remaining.
Subtract the completed pages from the total batch size of 15,000 pages.
3
Calculate the combined work rate of Press Alpha and Press Beta.
300 pages per minute.
When working together, rates add linearly: 120 + 180 = 300.
4
Calculate time needed to complete the remaining pages.
44 minutes.
Divide remaining work (13,200 pages) by combined rate (300 pages/min).
5
Calculate total elapsed time from 9:00 AM.
59 minutes.
Combine the 15 initial minutes with the 44 subsequent minutes.

Key Concept

Linear work-rate equations with staggered initial start times
Estimated Time:1m 30s
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Topics

Algebraic Word Problems and Modeling

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