Directions: Read the information carefully and answer the questions. A, B, C and D together can do a work 'X' in 14 days. The ratio of efficiency of A to that of C is $3 : 2$. C is $50\%$ less efficient than that of B and B is $33\frac{1}{3}\%$ more efficient than D. A and B together can complete work 'X' in 'x' days, while A and D can do the same work together in 'y' days. Two persons P and Q together can complete another work 'Y' in $(x - 9)$ days, while P and R together can complete the same work in $(y - 4)$ days. If P, Q and R together can complete the work 'Y' in $\frac{3x}{0.5y - 8}$ days, then efficiency of 'R' is what percent more or less than the efficiency of 'Q'?

### Concept & Time and Work Efficiency This problem utilizes the concept of efficiency ratios to find total work and individual work rates. First, we must decode the efficiency relationship from the common directions to find the values of $x$ and $y$, then apply those variables to the specific problem. $$ \text{Total Work} = \text{Total Efficiency} \times \text{Number of Days} $$ ### Step-by-Step Solution 1. **Determine the efficiencies from the directions:** Let the efficiencies of A, B, C, and D be $E_A, E_B, E_C,$ and $E_D$. - $E_A : E_C = 3 : 2 \implies$ Let $E_A = 3k, E_C = 2k$. - C is $50\%$ less efficient than B: $E_C = 0.5 \times E_B \implies E_B = 2 \times E_C = 4k$. - B is $33\frac{1}{3}\%$ (which is $\frac{1}{3}$) more efficient than D: $E_B = E_D + \frac{1}{3}E_D = \frac{4}{3}E_D$. - $4k = \frac{4}{3}E_D \implies E_D = 3k$. Ratio of efficiencies (A : B : C : D) = $3 : 4 : 2 : 3$. 2. **Calculate 'x' and 'y':** - Total Efficiency = $3 + 4 + 2 + 3 = 12$ units/day. - Total Work 'X' = $12 \times 14 = 168$ units. - Efficiency of (A + B) = $3 + 4 = 7$ units/day. Time 'x' = $\frac{168}{7} = 24$ days. So, $x = 24$. - Efficiency of (A + D) = $3 + 3 = 6$ units/day. Time 'y' = $\frac{168}{6} = 28$ days. So, $y = 28$. 3. **Solve for Work 'Y' conditions:** - (P + Q) time = $x - 9 = 24 - 9 = 15$ days. - (P + R) time = $y - 4 = 28 - 4 = 24$ days. - (P + Q + R) time = $\frac{3x}{0.5y - 8} = \frac{3(24)}{0.5(28) - 8} = \frac{72}{14 - 8} = \frac{72}{6} = 12$ days. - Let Total Work 'Y' be the LCM of $(15, 24, 12)$, which is $120$ units. 4. **Calculate individual efficiencies of P, Q, and R:** - Efficiency of (P + Q) = $\frac{120}{15} = 8$ units/day. - Efficiency of (P + R) = $\frac{120}{24} = 5$ units/day. - Efficiency of (P + Q + R) = $\frac{120}{12} = 10$ units/day. - Efficiency of R = (P + Q + R) - (P + Q) = $10 - 8 = 2$ units/day. - Efficiency of Q = (P + Q + R) - (P + R) = $10 - 5 = 5$ units/day. 5. **Find the required percentage:** - R is less than Q by $(5 - 2) = 3$ units. - Percentage = $\left(\frac{3}{5}\right) \times 100 = 60\%$. ### Exam Strategy & Shortcut When dealing with multiple entities doing the same work, immediately set the total work to the Least Common Multiple (LCM) of their completion times. This converts complex fraction additions into simple integer arithmetic, drastically reducing calculation time. ### Common Pitfall A frequent error is misinterpreting "$33\frac{1}{3}\%$ more efficient than D". Students often write $B = \frac{1}{3}D$ instead of $B = D + \frac{1}{3}D = \frac{4}{3}D$. Always pay attention to the word "more". ### Final Answer Therefore, the correct answer is **60%**.