The simple interest on a certain sum for 2 years is 50 and the compound interest is 55. Find the rate of interest per annum and the sum.

Options

A. 6% P.a. Rs. 200
B. 15% P.a. ,Rs. 150
C. 20% P.a. ,Rs. 125
D. 18% P.a. , Rs. 175

Answer

To solve the problem, we need to find the rate of interest per annum and the sum (principal amount) given the simple interest (SI) and compound interest (CI) for 2 years.

### Step 1: Understand the relationship between SI and CI
The formula for Simple Interest (SI) is:
\[ \text{SI} = \frac{P \times R \times T}{100} \]
Where:
- \( P \) = Principal amount (sum)
- \( R \) = Rate of interest per annum
- \( T \) = Time in years

The formula for Compound Interest (CI) is:
\[ \text{CI} = P \left(1 + \frac{R}{100}\right)^T - P \]
For 2 years, this can be simplified to:
\[ \text{CI} = P \left(1 + \frac{R}{100}\right)^2 - P \]

### Step 2: Set up equations based on the given information
From the problem, we know:
- SI for 2 years = 50
- CI for 2 years = 55

Using the SI formula:
\[ 50 = \frac{P \times R \times 2}{100} \]
This simplifies to:
\[ P \times R = 2500 \quad \text{(Equation 1)} \]

Using the CI formula:
\[ 55 = P \left( \left(1 + \frac{R}{100}\right)^2 - 1 \right) \]
This can be expanded to:
\[ 55 = P \left( \frac{R^2}{10000} + \frac{2R}{100} \right) \]
Multiplying through by 10000 to eliminate the fraction gives:
\[ 550000 = P (R^2 + 200R) \quad \text{(Equation 2)} \]

### Step 3: Substitute Equation 1 into Equation 2
From Equation 1, we can express \( P \) in terms of \( R \):
\[ P = \frac{2500}{R} \]

Substituting this into Equation 2:
\[ 550000 = \left(\frac{2500}{R}\right)(R^2 + 200R) \]
This simplifies to:
\[ 550000R = 2500(R^2 + 200R) \]
Dividing both sides by 2500 gives:
\[ 220R = R^2 + 200R \]
Rearranging this gives:
\[ R^2 - 20R = 0 \]
Factoring out \( R \):
\[ R(R - 20) = 0 \]

### Step 4: Solve for R
This gives us two solutions:
1. \( R = 0 \) (not possible as it represents no interest)
2. \( R = 20 \)

Thus, the rate of interest per annum is \( R = 20\% \).

### Step 5: Find the principal amount (P)
Now, substituting \( R = 20 \) back into Equation 1:
\[ P \times 20 = 2500 \]
\[ P = \frac{2500}{20} = 125 \]

### Final Answer
The rate of interest per annum is \( 20\% \) and the sum (principal amount) is \( 125 \).

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