A and B can complete a work in 10 days; B and C in 12 days; C and A in 15…

Concept

When people work together, their daily work rates add. If a person finishes one whole job in d days, that person’s rate is 1/d of the job per day. Pairwise rates can be combined algebraically to isolate an individual rate.

Application

1. Let the daily rates of A, B, and C be a, b, and c. From the three pair statements: a + b = 1/10, b + c = 1/12, and c + a = 1/15.

2. Add the three equations: 2(a + b + c) = 1/10 + 1/12 + 1/15 = (6 + 5 + 4)/60 = 15/60 = 1/4.

3. Divide by 2: a + b + c = 1/8.

4. Subtract b + c = 1/12 from the total: a = 1/8 - 1/12 = (3 - 2)/24 = 1/24.

5. Since A completes 1/24 of the work per day, A alone takes 24 days.

Cross-check

1. Using a = 1/24, the equation a + b = 1/10 gives b = 1/10 - 1/24 = 7/120.

2. Then b + c = 1/12 gives c = 1/12 - 7/120 = 1/40.

3. Now c + a = 1/40 + 1/24 = (3 + 5)/120 = 8/120 = 1/15, which agrees with the third pair rate.

Therefore, A alone can complete the work in 24 days.