Workers P, Q and R together can make one table in 40 minutes, while P and Q together can make the same table in 60 minutes. How many minutes will R alone take to make one table?

Introduction / Context:
This question involves three workers making one table, where the combined time for all three is known and the combined time for only two of them is also known. The aim is to isolate the contribution of the third worker by working with rates of work in tables per minute.

Given Data / Assumptions:

• P, Q and R together can make one table in 40 minutes.
• P and Q together can make the table in 60 minutes.
• Each worker works at a constant rate.
• We need to find the time taken by R alone to finish one table.

Concept / Approach:
The idea is similar to standard work rate problems. A worker who completes one table in T minutes has a rate of 1 / T table per minute. The combined rate of P, Q and R is known, as well as the combined rate of only P and Q. Subtracting these rates will give the rate of R alone. The reciprocal of R's rate provides the required time in minutes.

Step-by-Step Solution:
Step 1: Let the total work be 1 table.Step 2: Time taken by P, Q and R together = 40 minutes, so their combined rate = 1 / 40 table per minute.Step 3: Time taken by P and Q together = 60 minutes, so their combined rate = 1 / 60 table per minute.Step 4: Rate of R alone = rate of (P + Q + R) minus rate of (P + Q).Step 5: Compute rate of R = 1 / 40 - 1 / 60.Step 6: Use common denominator 120. Then 1 / 40 = 3 / 120 and 1 / 60 = 2 / 120.Step 7: Rate of R = 3 / 120 - 2 / 120 = 1 / 120 table per minute.Step 8: Time taken by R alone = 1 divided by (1 / 120) = 120 minutes.

Verification / Alternative check:
We can confirm by computing the total rate again. If R's rate is 1 / 120, then rate of P + Q + R should be 1 / 60 + 1 / 120 = 2 / 120 + 1 / 120 = 3 / 120 = 1 / 40 table per minute. This matches the given time of 40 minutes for all three together, confirming that 120 minutes for R alone is correct.

Why Other Options Are Wrong:

• 100 minutes: This corresponds to a rate of 1 / 100, which would make the combined time of all three less than 40 minutes.
• 90 minutes: This rate is 1 / 90, also larger than 1 / 120, giving a faster combined completion than specified.
• 150 minutes: This is a slower rate (1 / 150) and would give a different combined rate that does not match the 40 minute requirement.

Common Pitfalls:
Some students accidentally add the times instead of the rates or take an average of the times 40 and 60 directly, which is incorrect. Others forget to use a common denominator when subtracting fractions. Always remember that in work problems, it is the rates (work per unit time) that add or subtract, not the times themselves.

Final Answer:
R alone will take 120 minutes to make one table.