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Difficulty: MediumAlgebraic Word Problems and Modeling

A university archive uses two high-speed scanners, Machine A and Machine B, to digitize manuscript pages. Machine A operates at a constant rate of 6060 pages per hour, and Machine B operates at a constant rate of 8080 pages per hour. On Monday, Machine A starts scanning at 8:00 AM. At 9:30 AM, Machine B is brought online to work alongside Machine A. If both machines continue scanning together at their respective constant rates until a total of 510510 pages have been scanned, at what time will the scanning project be completed?

  1. A
    11:00 AM
  2. B
    11:38 AM
  3. 12:30 PMAnswer
  4. D
    2:45 PM
  5. E
    3:30 PM

Answer

12:30 PM
The option stating 12:30 PM is correct. Machine A operates alone from 8:00 AM to 9:30 AM (1.5 hours), completing 60×1.5=9060 \times 1.5 = 90 pages. This leaves 510−90=420510 - 90 = 420 pages to be scanned. From 9:30 AM onward, both machines work together at a combined rate of 60+80=14060 + 80 = 140 pages per hour. The remaining 420 pages require 420140=3\frac{420}{140} = 3 hours. Adding 3 hours to 9:30 AM gives a completion time of 12:30 PM.

Step-by-Step Solution

1
Calculate the number of pages scanned by Machine A during its head start
60 pages/hr×1.5 hours=90 pages60 \text{ pages/hr} \times 1.5 \text{ hours} = 90 \text{ pages}
Machine A operates alone from 8:00 AM to 9:30 AM, which is 1.51.5 hours.
2
Determine the remaining number of pages to be scanned
510−90=420 pages510 - 90 = 420 \text{ pages}
Subtract the pages already completed from the total requirement of 510 pages.
3
Calculate the combined scanning rate of both machines
60+80=140 pages/hr60 + 80 = 140 \text{ pages/hr}
When both machines operate simultaneously, their individual rates add together.
4
Calculate the time required to scan the remaining pages
420 pages140 pages/hr=3 hours\frac{420 \text{ pages}}{140 \text{ pages/hr}} = 3 \text{ hours}
Divide the remaining work by the combined rate.
5
Determine the final completion time
9:30 AM+3 hours=12:30 PM9:30 \text{ AM} + 3 \text{ hours} = 12:30 \text{ PM}
Add the combined work duration to the time when both machines began working together (9:30 AM).

Key Concept

Combined Work Rates with Staggered Start Times
Estimated Time:1m 30s
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Topics

Algebraic Word Problems and Modeling

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