Question

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

• A. 24 days
• B. 30 days
• C. 36 days
• D. 40 days

Hint:

For combined work problems, express in rates, sum them, and isolate the desired worker's rate.

Answer 1

The Correct Option is B

To solve the problem, we need to find the number of days A alone will take to complete the task. We know:

• A and B together complete the task in 12 days.
• B and C together complete the task in 15 days.
• A and C together complete the task in 20 days.

We can solve this using equations by letting the work done by each person individually be represented as:

• A's work in one day = \( \frac{1}{a} \)
• B's work in one day = \( \frac{1}{b} \)
• C's work in one day = \( \frac{1}{c} \)

From the given data, we can establish the following equations:

• \( \frac{1}{a} + \frac{1}{b} = \frac{1}{12} \) (Equation 1)
• \( \frac{1}{b} + \frac{1}{c} = \frac{1}{15} \) (Equation 2)
• \( \frac{1}{a} + \frac{1}{c} = \frac{1}{20} \) (Equation 3)

By adding all three equations, we get:

\(\frac{1}{a} + \frac{1}{b} + \frac{1}{b} + \frac{1}{c} + \frac{1}{a} + \frac{1}{c} = \frac{1}{12} + \frac{1}{15} + \frac{1}{20}\)

Simplifying gives:

\(2(\frac{1}{a} + \frac{1}{b} + \frac{1}{c}) = \frac{1}{12} + \frac{1}{15} + \frac{1}{20}\)

To find a common denominator for the right-hand side:

\(\frac{1}{12} = \frac{5}{60}\), \(\frac{1}{15} = \frac{4}{60}\), \(\frac{1}{20} = \frac{3}{60}\)

Add them: \( \frac{5 + 4 + 3}{60} = \frac{12}{60} = \frac{1}{5} \)

Therefore:

\(2(\frac{1}{a} + \frac{1}{b} + \frac{1}{c}) = \frac{1}{5}\)

Divide both sides by 2:

\(\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = \frac{1}{10}\)

Now use Equation 1: \(\frac{1}{a} + \frac{1}{b} = \frac{1}{12}\).

Subtract from \(\frac{1}{10}\):

\(\frac{1}{c} = \frac{1}{10} - \frac{1}{12}\)

Calculate \(\frac{1}{c}\) by finding a common denominator (60):

\(\frac{1}{10} = \frac{6}{60}\), \(\frac{1}{12} = \frac{5}{60}\)

\(\frac{1}{c} = \frac{6}{60} - \frac{5}{60} = \frac{1}{60}\)

So, \(C\) completes in 60 days.

Use Equation 3: \(\frac{1}{a} + \frac{1}{c} = \frac{1}{20}\)

\(\frac{1}{a} + \frac{1}{60} = \frac{1}{20}\)

\(\frac{1}{a} = \frac{1}{20} - \frac{1}{60}\)

Find a common denominator (60):

\(\frac{1}{20} = \frac{3}{60}\), \(\frac{1}{a} = \frac{3}{60} - \frac{1}{60} = \frac{2}{60} = \frac{1}{30}\)

Thus, A alone completes the work in:

30 days