Time taken by P, Q, and R together to complete the job is $37.5\%$ less than time taken by Q and R together to complete the job, while P and Q together can do the job in $Z$ days, and R alone can complete it in $Y$ days. Share of R is ₹$8000$ out of total share of P, Q, and R together (₹$16000$). If P, Q and R together can complete the job in $(Z - K)$ days, which of the following is/are definitely true. I. $\frac{Y}{2} = K$ II. If time taken by P, Q and R together to complete the job is 15 day less than that of P and Q together, then time taken by S to complete the job is 20 days, if R is half efficient that as that of S. III. $Z = Y$

### Concept & Ratio Allocation Use the relationship between share and work done to find efficiencies, and relate total work to time taken. $$ \text{Work Ratio} = \text{Share Ratio} $$ $$ \text{Total Work} = \text{Efficiency} \times \text{Time} $$ ### Step-by-Step Solution * **Given:** Share of R = ₹8000 out of ₹16000. This means R does exactly 50% of the work. * Therefore, Efficiency of R = Efficiency of (P + Q). * **Given:** Time taken by P, Q, R together is 37.5% (or $\frac{3}{8}$) less than Q and R together. * Time(P,Q,R) = $\left(1 - \frac{3}{8}\right) \times$ Time(Q,R) = $\frac{5}{8} \times$ Time(Q,R). * Efficiency is inversely proportional to time: Eff(P,Q,R) = $\frac{8}{5} \times$ Eff(Q,R). * Let Eff(P+Q+R) = $8x$ and Eff(Q+R) = $5x$. Then Eff(P) = $8x - 5x = 3x$. * Since R does 50% of work, Eff(R) = $4x$. Then Eff(Q) = Eff(Q+R) - Eff(R) = $5x - 4x = x$. * Check: Eff(P+Q) = $3x + x = 4x$ (which equals Eff(R), consistent with shares). * **Testing Statement III:** Time by P&Q = $Z$. Total work = $4x \times Z$. Time by R = $Y$. Total work = $4x \times Y$. Thus, $Z = Y$. (Statement III is true). * **Testing Statement I:** Time by P,Q,R = $\frac{\text{Total Work}}{\text{Eff(P+Q+R)}} = \frac{4x \times Z}{8x} = \frac{Z}{2}$. * We are given Time(P,Q,R) = $Z - K$. So, $\frac{Z}{2} = Z - K \implies K = \frac{Z}{2}$. Since $Z = Y$, $K = \frac{Y}{2}$. (Statement I is true). * **Testing Statement II:** Time(P,Q,R) = $\frac{Z}{2}$. Time(P,Q) = $Z$. Difference = $\frac{Z}{2} = 15 \implies Z = 30$. * Eff(S) = $2 \times \text{Eff(R)} = 8x$. Time by S = $\frac{\text{Total Work}}{\text{Eff(S)}} = \frac{4x \times 30}{8x} = 15$ days. * The statement claims S takes 20 days, which is false. (Statement II is false). * Therefore, only I and III are definitely true. ### Exam Strategy & Shortcut Focus directly on the share ratio to quickly establish that R is half the total workforce. This instantly yields $Y = Z$ without needing the 37.5% complex ratio, validating statement III and I simultaneously. ### Common Pitfall Misinterpreting "37.5% less than" as just 37.5%. Always subtract from 100% (or 1) to get the actual multiplier (62.5% or 5/8). ### Final Answer Therefore, the correct answer is **I and III only**.