Question

Given below are three statements. Study the question and statements to identify which options are necessary to answer the question: Question: What is the principal sum? I. The interest amount after 30 months is half the interest amount after five years.
II. The sum amounts to ₹750 in five years at simple interest.
III. The rate of interest is $8%$ p.a.

• A. One and three only
• B. Two and three only
• C. One and two only
• D. One and three only or two and three only

Hint:

In simple interest problems, to find the principal you must know both the interest rate and either the interest amount or the total amount over a given time period.

Answer 1

The Correct Option is B

Step 1: Understanding the problem.
We need to find the principal $P$ for a simple interest situation. The standard formula is: \[ SI = \frac{P \times R \times T}{100} \] To find $P$, we must know $R$ (rate) and $SI$ (simple interest over a known period).
Step 2: Evaluating statement I.
Statement I says: "The interest after 30 months is half the interest after 5 years."
In simple interest, $SI$ is directly proportional to time. Since $30$ months = $2.5$ years, and $2.5$ years is indeed half of $5$ years, this statement is always true and gives no new numerical information. It does not help in finding $P$.
Step 3: Evaluating statement II.
Statement II: "The sum amounts to ₹750 in five years."
The "sum" here is the total amount $A = P + SI$. Over five years, this total is known.
Step 4: Evaluating statement III.
Statement III: "The rate of interest is $8%$ p.a."
This gives the rate $R$.
Step 5: Combining statements.
- Using II and III:
From $A = P + SI$ and $SI = \frac{P \times 8 \times 5}{100} = 0.4P$, we have: \[ 750 = P + 0.4P = 1.4P ⇒ P = \frac{750}{1.4} = 535.71 \] Hence, II and III together are sufficient to find $P$.
- Using I and III:
Rate is known, and proportionality of time is known, but no amount ($A$ or $SI$) is given, so $P$ cannot be found.
- Using I and II:
We know the total amount but not the rate $R$, so we cannot find $P$ uniquely.
Thus, only statements II and III together are sufficient. \[ \boxed{\text{Two and three only}} \]