Inferring individual rates from joint output: A can copy 75 pages in 25 hours. Working together, A and B can copy 135 pages in 27 hours. Assuming constant rates, how long will B alone take to copy 42 pages?

Introduction / Context:
This question requires deducing the individual rates of two workers from given production data. Once A’s solo rate and the A+B combined rate are known, B’s solo rate is the difference. Finally, time for B to complete a specified number of pages is pages divided by B’s rate.

Given Data / Assumptions:

• A copies 75 pages in 25 hours ⇒ rate_A = 75/25 = 3 pages/hour.
• A and B together copy 135 pages in 27 hours ⇒ rate_{A+B} = 135/27 = 5 pages/hour.
• Rates are constant; no setup or idle time.

Concept / Approach:
Since the combined rate equals the sum of individual rates, rate_B = rate_{A+B} − rate_A. Then compute B’s time for a 42-page job using T = pages / rate_B.

Step-by-Step Solution:
rate_A = 3 pages/hourrate_{A+B} = 5 pages/hourrate_B = 5 − 3 = 2 pages/hourTime for B to copy 42 pages = 42 / 2 = 21 hours

Verification / Alternative check:
Check with proportions: In 21 hours, B would copy 42 pages, and A would copy 63 pages in the same time at 3 pages/hour; together they would produce 105 pages in 21 hours, which scales to 135 pages in 27 hours at 5 pages/hour, consistent with the given joint data.

Why Other Options Are Wrong:

• 18 hours: implies B’s rate 42/18 = 2.333… pages/hour, which contradicts the deduced rate 2 pages/hour.
• 24 hours: implies B’s rate 1.75 pages/hour, too slow.
• 15 hours: implies B’s rate 2.8 pages/hour, too fast given the combined rate constraint.

Common Pitfalls:

• Dividing 135 by 27 incorrectly; ensure rate_{A+B} = 5 pages/hour.
• Assuming the difference of times instead of the difference of rates.

Final Answer:
21 hours