Directions: Each of the questions given below consists of a question followed by three statements. You have to study the question and the statements and decide which of the statement(s) is/are necessary to answer the question. In how many years will a sum of money put at simple interest treble itself ? I. The interest earned in 4 years is half the sum. II. The rate of interest is $12 \frac{1}{2}$%. III. The sum doubles itself in 8 years at simple interest.

### Concept & Time for Multipliers at Simple Interest When a sum of money "trebles" (becomes 3 times itself), the final Amount is $3P$, which means the Simple Interest earned is exactly $2P$. To find the required time ($T$) for this to happen, we strictly need the Rate of Interest ($R$). $$T = \frac{SI \times 100}{P \times R}$$ ### Step-by-Step Solution * **From the Question:** Target $SI = 2P$. We need to find $T$, so we must find $R$. * **Analyze Statement I:** In $T = 4$ years, $SI = 0.5P$. * Using the formula: $0.5P = \frac{P \times R \times 4}{100} \Rightarrow 50 = 4R \Rightarrow R = 12.5\%$. * Since we can find $R$, we can solve the main question. (Statement I is sufficient). * **Analyze Statement II:** The rate of interest is directly given as $12.5\%$. * This gives us exactly what we need to solve the main question. (Statement II is sufficient). * **Analyze Statement III:** The sum doubles ($SI = P$) in $T = 8$ years. * Using the formula: $P = \frac{P \times R \times 8}{100} \Rightarrow 100 = 8R \Rightarrow R = 12.5\%$. * Since we can find $R$, we can solve the main question. (Statement III is sufficient). * Because each statement independently provides the required interest rate, any one of them alone is enough. ### Exam Strategy & Shortcut Recognize that questions asking about doubling, trebling, or fractional interest in relation to a principal are purely asking for the interest rate. Read the three statements: I provides a fractional interest over time (yields rate), II directly provides the rate, and III provides a doubling scenario over time (yields rate). All three provide the exact same missing variable. ### Common Pitfall A common mistake is doing all the calculations to find the final answer for $T$ (which is 16 years). Data sufficiency requires only verifying that the required variables *can* be found, saving you precious calculation time. ### Final Answer Therefore, the correct answer is **Any one of the three**.