Understanding the Problem
To determine the time taken by P, Q, and R to complete the work together, we first need to find their individual work rates based on the given information.
Work Rates Calculation
- P and Q together complete the work in 18 days.
- Their combined rate is \( \frac{1}{18} \) (work/day).
- Q and R together complete the work in 24 days.
- Their combined rate is \( \frac{1}{24} \) (work/day).
- P and R together complete the work in 36 days.
- Their combined rate is \( \frac{1}{36} \) (work/day).
Setting Up the Equations
Let:
- Rate of P = \( p \)
- Rate of Q = \( q \)
- Rate of R = \( r \)
From the information given, we can set up the following equations:
- \( p + q = \frac{1}{18} \) (1)
- \( q + r = \frac{1}{24} \) (2)
- \( p + r = \frac{1}{36} \) (3)
Solving the Equations
To find \( p + q + r \), we can add all three equations:
\[
(p + q) + (q + r) + (p + r) = \frac{1}{18} + \frac{1}{24} + \frac{1}{36}
\]
Simplifying the right side, we find a common denominator (72):
- \( \frac{4}{72} + \frac{3}{72} + \frac{2}{72} = \frac{9}{72} = \frac{1}{8} \)
Thus,
\[
2(p + q + r) = \frac{1}{8}
\]
So,
\[
p + q + r = \frac{1}{16}
\]
Final Calculation
The combined work rate of P, Q, and R is \( \frac{1}{16} \) work per day. Therefore, the time taken to complete the entire work together is:
\[
\text{Time} = \frac{1}{\frac{1}{16}} = 16 \text{ days}
\]
Conclusion
P, Q, and R can complete the work together in 16 days.