Two persons P and Q together can complete a work 'Y' in $(x+12)$ days, while Q and R together can complete the same work in $(y - 2)$ days. If P, Q, and R together can complete the work 'Y' in $\frac{6x}{y - 7}$ days, find by what percentage the efficiency of Q is more or less than the efficiency of P.

Given from previous data:
- $x = 12$
- $y = 14$

Question

Two persons P and Q together can complete a work 'Y' in $(x+12)$ days, while Q and R together can complete the same work in $(y - 2)$ days. If P, Q, and R together can complete the work 'Y' in $\frac{6x}{y - 7}$ days, find by what percentage the efficiency of Q is more or less than the efficiency of P.

Given from previous data:
- $x = 12$
- $y = 14$

Accepted Answer

**Solution:**

1. Substitute the given values of $x$ and $y$:
   - $x = 12$
   - $y = 14$

2. Calculate the days taken by each pair and group:
   - P and Q together complete work 'Y' in $x + 12 = 12 + 12 = 24$ days.
   - Q and R together complete work 'Y' in $y - 2 = 14 - 2 = 12$ days.
   - P, Q, and R together complete work 'Y' in $\frac{6x}{y - 7} = \frac{6 	imes 12}{14 - 7} = \frac{72}{7}$ days.

3. Find the total work 'Y' in units:
   - LCM of 24, 12, and $\frac{72}{7}$ is 72 units.

4. Calculate efficiencies (work done per day):
   - Efficiency of P and Q together = $\frac{72}{24} = 3$ units/day
   - Efficiency of Q and R together = $\frac{72}{12} = 6$ units/day
   - Efficiency of P, Q, and R together = $\frac{72}{\frac{72}{7}} = 7$ units/day

5. Using these, find individual efficiencies:
   - Let efficiency of P = $p$, Q = $q$, and R = $r$.
   - From P + Q = 3, Q + R = 6, and P + Q + R = 7:
     - Subtract (P + Q) from (P + Q + R): $7 - 3 = r = 4$
     - Substitute $r = 4$ in Q + R = 6: $q + 4 = 6 \Rightarrow q = 2$
     - Substitute $q = 2$ in P + Q = 3: $p + 2 = 3 \Rightarrow p = 1$

6. Calculate the percentage difference between efficiencies of Q and P:
   $$	ext{Percentage difference} = \frac{q - p}{p} 	imes 100 = \frac{2 - 1}{1} 	imes 100 = 100\%$$

**Answer:** 100\%

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