Painter A can paint a house alone in 40 days and painter B can paint the same house alone in 60 days. With the additional help of painter C, they are able to finish painting the house in only 20 days when all three work together. In how many days can painter C alone complete painting the entire house?

Introduction / Context:
In this time and work aptitude question, three painters A, B, and C are working together to paint a single house. You are given the individual times of A and B, and the combined time of A, B, and C. The concept tested is work rate or efficiency and how to compute the time taken by one worker when the combined time of several workers is known.

Given Data / Assumptions:

• Painter A can complete the work alone in 40 days.
• Painter B can complete the same work alone in 60 days.
• Painters A, B, and C together complete the work in 20 days.
• The total work is assumed to be 1 unit (one complete house painted).
• All painters work at constant rates and their efficiencies do not change over time.

Concept / Approach:
The key idea is to convert time taken into daily work rates. If a worker finishes 1 unit of work in T days, then the worker's rate is 1/T units per day. When multiple workers operate together, their rates add algebraically. Once the combined rate of all three is known, we subtract the rates of A and B from the total to obtain the rate of C alone and then invert that rate to get C's time.

Step-by-Step Solution:
Step 1: Let the total work be 1 unit. Step 2: Rate of painter A = 1/40 units per day. Step 3: Rate of painter B = 1/60 units per day. Step 4: A, B, and C together finish the work in 20 days, so their combined rate = 1/20 units per day. Step 5: Let rate of painter C be Rc. Then 1/40 + 1/60 + Rc = 1/20. Step 6: Compute 1/40 + 1/60. Using the LCM of 120, we get 1/40 = 3/120 and 1/60 = 2/120, so 1/40 + 1/60 = 5/120 = 1/24. Step 7: Substitute into the equation: 1/24 + Rc = 1/20. Step 8: Therefore Rc = 1/20 - 1/24. Step 9: Using the LCM of 120 again, 1/20 = 6/120 and 1/24 = 5/120, so Rc = 1/120 units per day. Step 10: Time taken by C alone = 1 / Rc = 1 / (1/120) = 120 days.

Verification / Alternative check:
We can verify by checking combined work in 20 days. In 20 days, A does 20 * (1/40) = 1/2 of the work and B does 20 * (1/60) = 1/3 of the work. Together A and B complete 1/2 + 1/3 = 5/6 of the work. The remaining 1 - 5/6 = 1/6 of the work must be done by C in 20 days. C's daily work then is (1/6) / 20 = 1/120, which matches our calculated rate and confirms that C alone needs 120 days.

Why Other Options Are Wrong:

• 100 days: This would correspond to a higher rate than 1/120, which does not match the given combined time of 20 days.
• 90 days: Implies an even higher work rate for C, causing the combined team to finish in less than 20 days.
• 80 days: Also makes the combined rate too large and contradicts the given total time of 20 days.

Common Pitfalls:
Students may mistakenly add times directly instead of adding rates, or they may confuse the idea of “more efficient” with fewer days. Another frequent error is miscalculating fractions during subtraction of the rates. Always convert to a common denominator and carefully simplify the fractions to avoid arithmetic mistakes.

Final Answer:
Painter C alone will complete the entire painting job in 120 days.