Question

A and B can complete a task in 12 days, B and C in 15 days, A and C in 20 days. How many days will A alone take?

• A. 20
• B. 30
• C. 40
• D. 60

Hint:

Sum pairwise rates, divide by 2 to find total rate, and subtract to isolate individual rates.

Answer 1

The Correct Option is B

We need to find the number of days A takes to complete the task alone.
- Step 1: Define work rates. Let A’s rate be \( a \), B’s rate be \( b \), and C’s rate be \( c \) (work per day). The task is 1 unit of work.
- Step 2: Set up equations based on given dat(a)
- A and B together: \( a + b = \frac{1}{12} \) (complete in 12 days).
- B and C together: \( b + c = \frac{1}{15} \).
- A and C together: \( a + c = \frac{1}{20} \).
- Step 3: Solve for the sum of rates. Add all equations:
\[ (a + b) + (b + c) + (a + c) = \frac{1}{12} + \frac{1}{15} + \frac{1}{20} \] \[ 2a + 2b + 2c = \frac{5 + 4 + 3}{60} = \frac{12}{60} = \frac{1}{5} \] \[ a + b + c = \frac{1}{10} \] - Step 4: Isolate A’s rate. Subtract \( b + c = \frac{1}{15} \) from \( a + b + c = \frac{1}{10} \):
\[ a = \frac{1}{10} - \frac{1}{15} = \frac{3 - 2}{30} = \frac{1}{30} \] - Step 5: Calculate A’s time. Time = \( \frac{1}{\text{rate}} = \frac{1}{\frac{1}{30}} = 30 \) days.
- Step 6: Verify with other equations.
- Find \( b \): \( a + b = \frac{1}{12} \Rightarrow \frac{1}{30} + b = \frac{1}{12} \Rightarrow b = \frac{5 - 2}{60} = \frac{3}{60} = \frac{1}{20} \).
- Find \( c \): \( a + c = \frac{1}{20} \Rightarrow \frac{1}{30} + c = \frac{1}{20} \Rightarrow c = \frac{3 - 2}{60} = \frac{1}{60} \).
- Check: \( b + c = \frac{1}{20} + \frac{1}{60} = \frac{3 + 1}{60} = \frac{1}{15} \). Correct.
- Step 7: Check options.
- (a) 20: Incorrect.
- (b) 30: Correct.
- (c) 40: Incorrect.
- (d) 60: Incorrect.
Thus, the answer is b.