To find the ratio of efficiencies of P, Q, and R, we can use the information given about the time taken by pairs of workers to complete the work.
1. **Understanding the problem**:
- P and Q together can complete the work in 18 days.
- Q and R together can complete the work in 24 days.
- P and R together can complete the work in 36 days.
2. **Calculating the work done**:
- Let's denote the total work as W. The work done by P and Q in one day is \( \frac{W}{18} \).
- The work done by Q and R in one day is \( \frac{W}{24} \).
- The work done by P and R in one day is \( \frac{W}{36} \).
3. **Setting up equations for efficiencies**:
- Let the efficiencies of P, Q, and R be denoted as \( p, q, \) and \( r \) respectively.
- From the information given, we can write the following equations based on the work done in one day:
- \( p + q = \frac{W}{18} \) (1)
- \( q + r = \frac{W}{24} \) (2)
- \( r + p = \frac{W}{36} \) (3)
4. **Eliminating W**:
- To eliminate W, we can multiply each equation by the least common multiple (LCM) of the denominators:
- LCM of 18, 24, and 36 is 72.
- Multiplying the equations by 72 gives:
- \( 72(p + q) = 4W \) (1)
- \( 72(q + r) = 3W \) (2)
- \( 72(r + p) = 2W \) (3)
5. **Rearranging the equations**:
- From (1): \( 72p + 72q = 4W \)
- From (2): \( 72q + 72r = 3W \)
- From (3): \( 72r + 72p = 2W \)
6. **Expressing in terms of efficiencies**:
- Now we can express the equations in terms of \( p, q, r \):
- \( 72p + 72q = 4W \) → \( 72p + 72q = 4 \)
- \( 72q + 72r = 3W \) → \( 72q + 72r = 3 \)
- \( 72r + 72p = 2W \) → \( 72r + 72p = 2 \)
7. **Solving the equations**:
- We can solve these equations simultaneously to find the values of \( p, q, \) and \( r \).
- By adding all three equations:
\[
(72p + 72q) + (72q + 72r) + (72r + 72p) = 4 + 3 + 2
\]
\[
144p + 144q + 144r = 9
\]
\[
p + q + r = \frac{9}{144} = \frac{1}{16}
\]
8. **Finding individual efficiencies**:
- Now we can express each efficiency in terms of a common variable.
- From the first equation, we can express \( p \) in terms of \( q \):
\[
p = \frac{4}{72} - q = \frac{1}{18} - q
\]
- Similarly, we can express \( q \) and \( r \) from the other equations.
9. **Calculating the ratio**:
- After solving for \( p, q, \) and \( r \), we can find the ratio of their efficiencies:
\[
p : q : r = 4 : 3 : 2
\]
Thus, the ratio of efficiencies of P, Q, and R is **4 : 3 : 2**.
