At which sum, the simple interest at the rate of 3¼ per annum will be Rs. 210 in 2½ years.

Options

A. (a) 2311.25
B. (b) 2355.30
C. (c) 2584.60
D. (d) 2716.90

Answer

To find the principal amount (P) for which the simple interest (SI) will be Rs. 210 at a rate of 3¼% per annum over a period of 2½ years, we can follow these steps:

### Step 1: Convert the rate of interest to a fraction
The rate of interest given is 3¼%. We can convert this to an improper fraction:
\[ 3¼ = \frac{13}{4} \text{ percent} \]

### Step 2: Convert the time period to a fraction
The time period given is 2½ years. We can convert this to an improper fraction:
\[ 2½ = \frac{5}{2} \text{ years} \]

### Step 3: Use the formula for Simple Interest
The formula for Simple Interest (SI) is given by:
\[ SI = \frac{P \times R \times T}{100} \]
Where:
- SI = Simple Interest
- P = Principal amount
- R = Rate of interest
- T = Time period

### Step 4: Substitute the known values into the formula
We know that:
- SI = 210
- R = \frac{13}{4}
- T = \frac{5}{2}

Substituting these values into the formula, we have:
\[ 210 = \frac{P \times \frac{13}{4} \times \frac{5}{2}}{100} \]

### Step 5: Simplify the equation
To eliminate the fraction, we can multiply both sides of the equation by 100:
\[ 210 \times 100 = P \times \frac{13}{4} \times \frac{5}{2} \]
This simplifies to:
\[ 21000 = P \times \frac{13 \times 5}{4 \times 2} \]

### Step 6: Calculate the denominator
Calculating the denominator:
\[ 4 \times 2 = 8 \]
Thus, the equation becomes:
\[ 21000 = P \times \frac{65}{8} \]

### Step 7: Solve for P
To find P, we rearrange the equation:
\[ P = \frac{21000 \times 8}{65} \]

### Step 8: Calculate the value of P
Now, we calculate:
\[ P = \frac{168000}{65} \]
Calculating this gives:
\[ P \approx 2584.615 \]

### Conclusion
Thus, the principal amount (P) is approximately Rs. 2584.62.

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