Question

A and B can complete a work in 12 days, B and C in 15 days, and A and C in 20 days. How long will A alone take to complete the work?

• A. 20 days
• B. 24 days
• C. 30 days
• D. 36 days

Hint:

For work-rate problems with pairs, sum all pair rates and divide by 2 to find individual rates, then isolate the required person's rate.

Answer 1

The Correct Option is C

- Step 1: Let A's rate = $\dfrac{1}{A}$, B's rate = $\dfrac{1}{B}$, C's rate = $\dfrac{1}{C}$.
- Step 2: Given: A and B: $\dfrac{1}{A} + \dfrac{1}{B} = \dfrac{1}{12}$; B and C: $\dfrac{1}{B} + \dfrac{1}{C} = \dfrac{1}{15}$; A and C: $\dfrac{1}{A} + \dfrac{1}{C} = \dfrac{1}{20}$.
- Step 3: Add all: $2\left(\dfrac{1}{A} + \dfrac{1}{B} + \dfrac{1}{C}\right) = \dfrac{1}{12} + \dfrac{1}{15} + \dfrac{1}{20}$. LCM = 60, so $\dfrac{5 + 4 + 3}{60} = \dfrac{12}{60} = \dfrac{1}{5}$. Thus, $\dfrac{1}{A} + \dfrac{1}{B} + \dfrac{1}{C} = \dfrac{1}{10}$.
- Step 4: Find A's rate: $\dfrac{1}{A} = \left(\dfrac{1}{A} + \dfrac{1}{B} + \dfrac{1}{C}\right) - \left(\dfrac{1}{B} + \dfrac{1}{C}\right) = \dfrac{1}{10} - \dfrac{1}{15} = \dfrac{3 - 2}{30} = \dfrac{1}{30}$.
- Step 5: A's time = 30 days. Verify: Use other pairs, same result.
- Step 6: Check options: Option (c) is 30 days, which matches.