In how many minimum number of complete years, the interest on 212.50 P at 3% per annum will be in exact number of rupees?

Options

A. 6
B. 8
C. 9
D. 7

Answer

To solve the problem of finding the minimum number of complete years in which the interest on ₹212.50 at a rate of 3% per annum will be an exact number of rupees, we can follow these steps:

### Step 1: Understand the Formula for Simple Interest
The formula for calculating Simple Interest (SI) is given by:
\[ SI = \frac{P \times R \times T}{100} \]
where:
- \( P \) = Principal amount (initial investment)
- \( R \) = Rate of interest per annum
- \( T \) = Time in years

### Step 2: Substitute the Given Values
Here, the principal \( P \) is ₹212.50, the rate \( R \) is 3%, and we need to find \( T \) such that the interest is an exact number of rupees.

Substituting the values into the formula:
\[ SI = \frac{212.50 \times 3 \times T}{100} \]

### Step 3: Simplify the Expression
To eliminate the decimal, we can multiply the principal by 100 to make calculations easier:
\[ SI = \frac{21250 \times 3 \times T}{10000} \]
This simplifies to:
\[ SI = \frac{63750 \times T}{10000} \]
Further simplifying gives:
\[ SI = 6.375 \times T \]

### Step 4: Determine When the Interest is an Exact Rupee
For the interest \( SI \) to be an exact number of rupees, \( 6.375 \times T \) must be an integer. This means \( T \) must be such that \( 6.375 \times T \) results in a whole number.

### Step 5: Find the Minimum Value of \( T \)
The decimal \( 6.375 \) can be expressed as a fraction:
\[ 6.375 = \frac{6375}{1000} = \frac{51}{8} \]
Thus, we have:
\[ SI = \frac{51}{8} \times T \]
For \( SI \) to be an integer, \( T \) must be a multiple of 8 (since the denominator is 8).

### Step 6: Conclusion
The minimum complete number of years \( T \) that satisfies this condition is:
\[ T = 8 \]

Thus, the minimum number of complete years for the interest to be an exact number of rupees is **8 years**.

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