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Difficulty: MediumRatios, Rates, and Percentages

A manufacturing facility operates two assembly lines, Line 1 and Line 2, producing high-precision components.

- Line 1 operates at a constant rate of R1R_1 components per hour, and Line 2 operates at a constant rate of R2R_2 components per hour.
- Operating Line 1 for 55 hours and Line 2 for 66 hours yields a total output of 1,2001,200 components.
- If Line 1's production rate is increased by 25%25\% and Line 2's production rate is decreased by 15%15\%, running both lines together for 44 hours yields a total output of 940940 components.

Which of the following represents the original production rate of Line 1 (R1R_1) and Line 2 (R2R_2), respectively, in components per hour?

  1. R1=120R_1 = 120 components/hour, R2=100R_2 = 100 components/hourAnswer
  2. B
    R1=100R_1 = 100 components/hour, R2=120R_2 = 120 components/hour
  3. C
    R1=150R_1 = 150 components/hour, R2=85R_2 = 85 components/hour
  4. D
    R1=140R_1 = 140 components/hour, R2=90R_2 = 90 components/hour
  5. E
    R1=110R_1 = 110 components/hour, R2=110R_2 = 110 components/hour

Answer

R1=120R_1 = 120 components/hour and R2=100R_2 = 100 components/hour
The correct answer accurately determines the original rates by translating the scenario into two simultaneous equations (5R1+6R2=12005 R_1 + 6 R_2 = 1200 and 4(1.25R1+0.85R2)=9404(1.25 R_1 + 0.85 R_2) = 940). Solving this system yields R1=120R_1 = 120 components per hour and R2=100R_2 = 100 components per hour.

Step-by-Step Solution

1
Formulate the first linear equation from the initial production scenario.
5R1+6R2=12005 R_1 + 6 R_2 = 1200
Line 1 runs for 55 hours at rate R1R_1 and Line 2 runs for 66 hours at rate R2R_2 to produce 1,2001,200 total components.
2
Express the modified rates after percentage changes.
Modified rate for Line 1 is 1.25R11.25 R_1; modified rate for Line 2 is 0.85R20.85 R_2.
Increasing R1R_1 by 25%25\% gives R1×1.25R_1 \times 1.25, and decreasing R2R_2 by 15%15\% gives R2×(1−0.15)=0.85R2R_2 \times (1 - 0.15) = 0.85 R_2.
3
Formulate and simplify the second equation from the modified production scenario.
5R1+3.4R2=9405 R_1 + 3.4 R_2 = 940
Running both modified lines together for 44 hours gives 4(1.25R1+0.85R2)=9404(1.25 R_1 + 0.85 R_2) = 940, which simplifies to 5R1+3.4R2=9405 R_1 + 3.4 R_2 = 940.
4
Solve the system of equations by subtracting the simplified second equation from the first equation.
2.6R2=260  ⟹  R2=1002.6 R_2 = 260 \implies R_2 = 100
(5R1+6R2)−(5R1+3.4R2)=1200−940  ⟹  2.6R2=260  ⟹  R2=100(5 R_1 + 6 R_2) - (5 R_1 + 3.4 R_2) = 1200 - 940 \implies 2.6 R_2 = 260 \implies R_2 = 100.
5
Substitute R2=100R_2 = 100 back into the first equation to solve for R1R_1.
R1=120R_1 = 120
5R1+6(100)=1200  ⟹  5R1=600  ⟹  R1=1205 R_1 + 6(100) = 1200 \implies 5 R_1 = 600 \implies R_1 = 120.

Key Concept

Solving simultaneous linear rate equations involving percentage increases and decreases
Estimated Time:2m 0s
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Topics

Ratios, Rates, and Percentages

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