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Difficulty: Very hardRatios, Rates, and Proportions

A manufacturing company uses two types of machines, Type P and Type Q, to complete a production run of custom components. The rate of production of a Type P machine is to that of a Type Q machine in the ratio 3:23:2. A team consisting of 44 Type P machines and 33 Type Q machines can complete the entire production run in 1010 hours when working together. The company starts the run with this team, but after 44 hours, 22 of the Type P machines break down and are removed from service. To complete the remainder of the production run on schedule, additional Type Q machines are added to the remaining machines. What is the total number of Type Q machines that must be added?

  1. A
    2
  2. 3Answer
  3. C
    5
  4. D
    6

Answer

3 additional Type Q machines must be added.
To complete the production run on schedule, we express the rate of Type P and Type Q machines as 3r3r and 2r2r respectively. The initial team of 44 Type P and 33 Type Q machines produces at a combined rate of 4(3r)+3(2r)=18r4(3r) + 3(2r) = 18r components per hour. The total work for the 1010-hour project is 18r×10=180r18r \times 10 = 180r components. In the first 44 hours, the team completes 18r×4=72r18r \times 4 = 72r components, leaving 180r−72r=108r180r - 72r = 108r components to be completed in the remaining 66 hours. This requires a combined rate of 108r/6=18r108r / 6 = 18r components per hour. After 22 Type P machines break down, the remaining machines (22 Type P and 33 Type Q) provide a rate of 2(3r)+3(2r)=12r2(3r) + 3(2r) = 12r components per hour. To reach the required rate of 18r18r, an additional rate of 18r−12r=6r18r - 12r = 6r is needed. Since each Type Q machine has a rate of 2r2r, the number of additional Type Q machines needed is 6r/2r=36r / 2r = 3.

Step-by-Step Solution

1
Define the individual production rates of Type P and Type Q machines using a common variable rr.
Let the rate of a Type P machine be 3r3r components per hour, and the rate of a Type Q machine be 2r2r components per hour.
This represents the given 3:23:2 ratio of their production rates.
2
Calculate the combined rate of the initial team and the total work required for the production run.
Combined initial rate =4(3r)+3(2r)=18r= 4(3r) + 3(2r) = 18r components per hour. Total work W=18r×10 hours=180rW = 18r \times 10 \text{ hours} = 180r components.
The team consists of 44 Type P and 33 Type Q machines working for 1010 hours.
3
Determine the work completed in the first 44 hours and the remaining work.
Completed work =18r×4=72r= 18r \times 4 = 72r components. Remaining work =180r−72r=108r= 180r - 72r = 108r components.
The team worked for 44 hours before the breakdown occurred.
4
Find the required rate to complete the remaining work in the remaining time.
Remaining time =10−4=6= 10 - 4 = 6 hours. Required rate =108r/6=18r= 108r / 6 = 18r components per hour.
The production run must be completed on the original schedule of 1010 total hours.
5
Calculate the rate of the remaining machines and the rate deficit.
Current rate of remaining machines (22 Type P and 33 Type Q) =2(3r)+3(2r)=12r= 2(3r) + 3(2r) = 12r components per hour. Rate deficit =18r−12r=6r= 18r - 12r = 6r components per hour.
Two Type P machines broke down, leaving 22 Type P and the original 33 Type Q machines.
6
Find the number of additional Type Q machines needed to cover the rate deficit.
Number of additional Type Q machines =6r/2r=3= 6r / 2r = 3.
Each Type Q machine contributes a rate of 2r2r components per hour.

Key Concept

Work-rate problems with varying machine efficiencies and changing team compositions over time.
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Topics

Ratios, Rates, and Proportions

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