Directions: Each of the following questions consists of a question followed by three statements I, II and III. You have to study the question and the statements and decide which of the statement(s) is/are necessary to answer the question. An item costing ₹ 3000 is sold at a certain discount. Find the rate of discount offered. I. The profit earned after discount is 5%. II. Had the discount rate been doubled, the seller incurs a loss of 15%. III. The item is marked at a price 25% above the cost price.

### Concept & Logic We need to find the discount rate ($d\%$), where $\text{Selling Price (SP)} = \text{Marked Price (MP)} \times (1 - d)$. We are given the Cost Price ($\text{CP} = ₹ 3000$) in the question prompt. We need to evaluate pairs of statements to see if they yield the discount rate. ### Step-by-Step Solution * **From Question:** $\text{CP} = ₹ 3000$. * **Evaluating I and III:** * Statement I gives a $5\%$ profit, so $\text{SP} = 1.05 \times 3000 = ₹ 3150$. * Statement III gives $\text{MP} = 1.25 \times 3000 = ₹ 3750$. * Discount = $\text{MP} - \text{SP} = 3750 - 3150 = ₹ 600$. * Discount \% = $(600 / 3750) \times 100 = 16\%$. (I and III are sufficient). * **Evaluating II and III:** * Statement III gives $\text{MP} = ₹ 3750$. * Statement II says double discount ($2d$) yields $15\%$ loss, so new $\text{SP} = 0.85 \times 3000 = ₹ 2550$. * $\text{MP} \times (1 - 2d) = 2550 \Rightarrow 3750 \times (1 - 2d) = 2550 \Rightarrow 1 - 2d = 0.68 \Rightarrow 2d = 0.32 \Rightarrow d = 16\%$. (II and III are sufficient). * **Evaluating I and II:** * Let MP be $M$ and discount rate be $d$. * From I: $M \times (1 - d) = 1.05 \times 3000 = 3150 \Rightarrow M - Md = 3150$. * From II: $M \times (1 - 2d) = 0.85 \times 3000 = 2550 \Rightarrow M - 2Md = 2550$. * Subtracting the second equation from the first: $Md = 600$. * Substituting $Md = 600$ back into the first equation: $M - 600 = 3150 \Rightarrow M = 3750$. * Since $Md = 600$ and $M = 3750$, then $3750 \times d = 600 \Rightarrow d = 16\%$. (I and II are sufficient). * Since any pair of statements allows us to find the discount rate, any two statements are sufficient. ### Exam Strategy & Shortcut Recognize the algebraic structure. You have two unknowns: Marked Price (MP) and Discount rate ($d$). You need exactly two distinct linear equations involving these unknowns to solve for them. Statements I, II, and III each provide one distinct linear equation relating MP and $d$ to the known CP. Therefore, any two statements will form a solvable system of equations. ### Common Pitfall A common pitfall is giving up on checking "I and II" after seeing that "I and III" and "II and III" are much easier to calculate. Always check all pairs before concluding, especially when algebraic systems are involved. ### Final Answer Therefore, the correct answer is **Any two of the three**.