A sum of money amounts to Rs 868 in 2 years at a simple interest. If the rate of interest increased by 25%, then the sum amounts to Rs 910 during the same period. Find the sum?

Options

A. A)Rs 650
B. B)Rs 750
C. C)Rs 850
D. D)Rs 700

Answer

To solve the problem step by step, we will use the information provided about the amounts and the changes in the rate of interest.

### Step 1: Understand the given information
We have two amounts:
- Amount after 2 years at the original interest rate: \( A_1 = 868 \) Rs
- Amount after 2 years at the increased interest rate: \( A_2 = 910 \) Rs

### Step 2: Define the variables
Let:
- \( P \) = Principal amount (the sum we need to find)
- \( R \) = Original rate of interest (in percentage)
- \( K \) = A constant that relates to the interest earned per unit rate

### Step 3: Set up the equations based on the amounts
From the information given, we can express the amounts in terms of the principal and interest:
1. For the first amount:
\[ A_1 = P + \text{SI}_1 = P + \left( \frac{P \times R \times 2}{100} \right) = 868 \]
This can be rewritten as:
\[ P + 2 \times K = 868 \quad \text{(where \( K = \frac{P \times R}{100} \))} \]

2. For the second amount (with the increased rate of interest):
\[ A_2 = P + \text{SI}_2 = P + \left( \frac{P \times (R + 0.25R) \times 2}{100} \right) = 910 \]
This simplifies to:
\[ P + 2 \times (1.25K) = 910 \]
Which can be rewritten as:
\[ P + 2.5K = 910 \]

### Step 4: Set up the system of equations
Now we have two equations:
1. \( P + 2K = 868 \) (Equation 1)
2. \( P + 2.5K = 910 \) (Equation 2)

### Step 5: Solve the system of equations
We can subtract Equation 1 from Equation 2 to eliminate \( P \):
\[ (P + 2.5K) - (P + 2K) = 910 - 868 \]
This simplifies to:
\[ 0.5K = 42 \]
Now, solving for \( K \):
\[ K = 42 \times 2 = 84 \]

### Step 6: Substitute \( K \) back to find \( P \)
Now we can substitute the value of \( K \) back into one of the equations to find \( P \). Using Equation 1:
\[ P + 2 \times 42 = 868 \]
This simplifies to:
\[ P + 84 = 868 \]
Now, solving for \( P \):
\[ P = 868 - 84 = 784 \]

### Step 7: Conclusion
The principal amount \( P \) is Rs 784.