Question

A, B and C can do a work in 10, 12 and 15 days respectively. In how many days will the work be completed if B is assisted by both A and C on every third day?

• A. \(7\frac{1}{5}\)
• B. \(7\frac{2}{3}\)
• C. \(8\)
• D. \(9\)

Answer 1

The Correct Option is C

To solve this problem, we first calculate the work done by A, B, and C. The work done by each individual per day is as follows:

• A can complete the work in 10 days, so A's work rate is \(\frac{1}{10}\) of the work per day.
• B can complete the work in 12 days, so B's work rate is \(\frac{1}{12}\) of the work per day.
• C can complete the work in 15 days, so C's work rate is \(\frac{1}{15}\) of the work per day.

On every third day, A and C assist B. Thus, we need to calculate the work done in a 3-day cycle:

• Day 1: B works alone, completing \(\frac{1}{12}\) of the work.
• Day 2: B works alone, completing another \(\frac{1}{12}\) of the work.
• Day 3: A, B, and C work together, completing \(\frac{1}{10} + \frac{1}{12} + \frac{1}{15}\) of the work.

Calculate the combined work done by A, B, and C on the third day:

\(\frac{1}{10} + \frac{1}{12} + \frac{1}{15} = \frac{6 + 5 + 4}{60} = \frac{15}{60} = \frac{1}{4}\)

So, in 3 days, the work completed is:

\(\frac{1}{12} + \frac{1}{12} + \frac{1}{4} = \frac{1}{12} + \frac{1}{12} + \frac{3}{12} = \frac{5}{12}\)

Thus, the work done in each cycle is \(\frac{5}{12}\). Find the total number of cycles needed to complete the work:

We need 1 unit of work completed:

To find the number of cycles n required, solve:

\(\frac{5n}{12}=1\)

Solve for n:

\(n=\frac{12}{5}=2.4\)

This means 2.4 cycles are needed. Thus, B and A, C complete full cycles of 3 days each for 2 cycles (6 days), and only need a fraction of the next cycle.

Check how much work is done in 2 cycles:

2\times \frac{5}{12}=\frac{10}{12}=\frac{5}{6}

Work left after 2 cycles is:

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

Find the extra days needed to complete \(\frac{1}{6}\) of the work. Since B alone works at a rate of \(\frac{1}{12}\) per day:

\(\frac{1}{6} \times 12=2\)

Thus, completing the work takes the 6 days of full cycles plus 2 more days, summing up to a total of:

8 days

The correct answer is 8.