A and B together can complete a job in 72 days, B and C together can complete it in 120 days, and A and C together can complete it in 90 days. In how many days can A alone complete the job?

Introduction / Context:
This is a classic time and work problem involving three workers A, B and C, where we know the times taken by different pairs of them to complete a job. Our task is to use this information to find the time taken by A alone. This type of question tests the ability to form and solve a system of equations based on combined work rates and then isolate one worker's rate.

Given Data / Assumptions:
- A and B together complete the job in 72 days. - B and C together complete the job in 120 days. - A and C together complete the job in 90 days. - All work at constant rates. - Total work is treated as one complete job.

Concept / Approach:
If someone finishes a job in T days, their daily work rate is 1 / T. Let the individual rates of A, B and C be a, b and c jobs per day respectively. We are given three equations in terms of these rates: a + b, b + c and a + c. By adding and subtracting these equations appropriately, we can solve for each rate. Once A's rate a is found, A's time to finish the job alone is simply 1 / a days. This approach uses algebraic manipulation of simultaneous linear equations.

Step-by-Step Solution:
Step 1: Let A's rate = a, B's rate = b, C's rate = c jobs per day. Step 2: From A and B together: a + b = 1 / 72. Step 3: From B and C together: b + c = 1 / 120. Step 4: From A and C together: a + c = 1 / 90. Step 5: Add equations for A + B and A + C: (a + b) + (a + c) = 1 / 72 + 1 / 90. Step 6: Left side becomes 2a + (b + c). Right side: use denominator 360. 1 / 72 = 5 / 360 and 1 / 90 = 4 / 360, so sum = 9 / 360 = 1 / 40. Step 7: We know b + c = 1 / 120 from Step 3. Step 8: Substitute b + c into the expression: 2a + 1 / 120 = 1 / 40. Step 9: Solve for a: 2a = 1 / 40 - 1 / 120. Step 10: Use denominator 120: 1 / 40 = 3 / 120, so 2a = (3 / 120) - (1 / 120) = 2 / 120 = 1 / 60. Step 11: Therefore, a = 1 / 120 job per day. Step 12: Time taken by A alone to complete one job = 1 / a = 120 days.

Verification / Alternative check:
We can confirm the correctness by reconstructing b and c and checking one of the original pair equations. Once we know a = 1 / 120, from a + b = 1 / 72, we get b = 1 / 72 - 1 / 120, which is positive and reasonable. Similarly, from a + c = 1 / 90 we get a valid c. Recalculating a + b, b + c and a + c returns the original 1 / 72, 1 / 120 and 1 / 90, validating our solution. So A taking 120 days alone is fully consistent.

Why Other Options Are Wrong:
130, 150 or 100 days: These do not match the rate derived from the simultaneous equations and would lead to inconsistencies with the given combined times.
90 days: This is the time for A and C together, not for A alone. It ignores the need to separate out B's and C's contributions.

Common Pitfalls:
A common mistake is to try to average the given times rather than use algebraic equations. Another error is mishandling fractions when adding and subtracting the combined rates. Careful use of a common denominator and systematic solving of the three linear equations is essential. Always remember that the combined rate of two workers is simply the sum of their individual rates, not the average of their times.

Final Answer:
A alone can complete the job in 120 days.