A and B together can do a piece of work in 36 days, B and C together can do it in 24 days, and A and C together can do it in 18 days. If all three work together, in how many days will they finish the work?

Introduction / Context:
This is a classic three worker time and work problem where the pairwise times for different pairs of workers are given. The goal is to determine the time required when all three workers collaborate. It involves solving a small system of equations based on work rates rather than times.

Given Data / Assumptions:

• A and B together complete the work in 36 days.
• B and C together complete the work in 24 days.
• A and C together complete the work in 18 days.
• Each worker maintains a constant rate of work.
• We must find the time taken if A, B and C all work together.

Concept / Approach:
Let the individual daily work rates of A, B and C be a, b and c units per day respectively. Then the combined rates for the given pairs are a + b, b + c and a + c. These three expressions correspond to 1 / 36, 1 / 24 and 1 / 18 respectively. By adding these three equations and simplifying, we can find the total rate a + b + c, and then compute the time taken by all three together as the reciprocal of this total rate.

Step-by-Step Solution:
Step 1: Let total work be 1 unit. Let daily work rates of A, B and C be a, b and c respectively.Step 2: From the data, we have three equations based on pairwise work:(i) a + b = 1 / 36(ii) b + c = 1 / 24(iii) a + c = 1 / 18Step 3: Add all three equations: (a + b) + (b + c) + (a + c) = 1 / 36 + 1 / 24 + 1 / 18.Step 4: On the left side, this becomes 2(a + b + c). So 2(a + b + c) = 1 / 36 + 1 / 24 + 1 / 18.Step 5: Find the sum on the right. The common denominator of 36, 24 and 18 is 72 or 360, but we can compute directly. For convenience, use 72: 1 / 36 = 2 / 72, 1 / 24 = 3 / 72, and 1 / 18 = 4 / 72.Step 6: Sum = 2 / 72 + 3 / 72 + 4 / 72 = 9 / 72 = 1 / 8.Step 7: Therefore 2(a + b + c) = 1 / 8, giving a + b + c = 1 / 16.Step 8: Thus the combined daily rate of A, B and C together is 1 / 16 of the work per day.Step 9: Time taken when A, B and C work together = 1 divided by 1 / 16 = 16 days.

Verification / Alternative check:
We can quickly verify the arithmetic using another denominator like 360 to ensure the fraction sum is correct. Using 360, 1 / 36 = 10 / 360, 1 / 24 = 15 / 360, 1 / 18 = 20 / 360. Their sum is 45 / 360 = 1 / 8, which matches our earlier calculation. Therefore a + b + c = 1 / 16 is reliable, and 16 days is the correct combined time.

Why Other Options Are Wrong:

• 8 days: This corresponds to a rate of 1 / 8 per day, which is double the actual combined rate and inconsistent with the pairwise times.
• 30 days: This implies a very small combined rate, which would contradict the relatively short pairwise completion times.
• 32 days: This is also too long and not supported by the combined rate deduced from the three pairwise values.

Common Pitfalls:
Students may try to average the given times instead of working with rates, or they may incorrectly subtract equations, leading to errors in the total rate. Another common issue is adding fractions carelessly without a correct common denominator. Always use rates, not times, and handle fractions with care to avoid arithmetic mistakes.

Final Answer:
All three working together will finish the work in 16 days.