Question

If P, Q and R work together, they take 12 days to complete the work. However, if only P and R work, they take 20 days to complete the work. A schedule is made such that P and R work every day, while Q joins them only on alternate days i.e. on 1^st, 3^rd, 5^th, 7^th,… days. On which day will the work be completed?

• A. 15^th
• B. 14^th
• C. 13^th
• D. 17^th

Answer 1

The Correct Option is A

Work Calculation with P, Q, and R 

- If P, Q, and R work together, they complete the work in 12 days. Their combined rate of work is:

\[ \text{Combined rate of P, Q, and R} = \frac{1}{12} \text{ work per day} \]

- If only P and R work together, they complete the work in 20 days. Their combined rate of work is:

\[ \text{Combined rate of P and R} = \frac{1}{20} \text{ work per day} \]

- Since P and R complete \(\frac{1}{20}\) of the work each day, we can find Q's rate of work as follows:

\[ \text{Q's rate} = \frac{1}{12} - \frac{1}{20} \]

Finding the LCM of 12 and 20:

\[ \text{Q's rate} = \frac{5}{60} - \frac{3}{60} = \frac{2}{60} = \frac{1}{30} \]

- On alternate days, Q works with P and R, so the rate of work when all three work together is:

\[ \frac{1}{12} \text{ work per day} \]

- On the other days, P and R complete:

\[ \frac{1}{20} \text{ work per day} \]

- The total work done in two days is:

\[ \frac{1}{12} + \frac{1}{20} = \frac{5}{60} + \frac{3}{60} = \frac{8}{60} = \frac{2}{15} \]

- Therefore, in every two-day period, \(\frac{2}{15}\) of the work is completed. To complete the entire work:

\[ \frac{1}{\frac{2}{15}} = 7.5 \text{ two-day periods} \]

Since 7.5 periods correspond to 15 days, the work will be completed on the 15th day.

Conclusion: The correct answer is (a) 15th day.