The S.I. occured on a sum of money at the rate of interest 5% per annum for two years. Is 410 Rs. The compound interest is the same as S.I as occured on other sum of money at the same rate of interest and the same time. Find the difference between the two sum of money (principle) ?

Options

A. Rs. 80
B. Rs. 90
C. Rs. 150
D. None of these

Answer

To solve the problem step by step, we will first calculate the principal amount based on the simple interest (S.I.) given, and then we will find the principal amount for the compound interest (C.I.) before determining the difference between the two principal amounts.

### Step 1: Understand the Simple Interest Formula
The formula for Simple Interest (S.I.) is:
\[ \text{S.I.} = \frac{P \times R \times T}{100} \]
Where:
- \( P \) = Principal amount
- \( R \) = Rate of interest per annum
- \( T \) = Time in years

### Step 2: Substitute the Known Values for S.I.
From the question, we know:
- S.I. = 410 Rs.
- R = 5% per annum
- T = 2 years

Substituting these values into the S.I. formula gives:
\[ 410 = \frac{P \times 5 \times 2}{100} \]

### Step 3: Solve for Principal (P)
Rearranging the equation to find \( P \):
\[ 410 = \frac{10P}{100} \]
\[ 410 = \frac{P}{10} \]
\[ P = 410 \times 10 = 4100 \text{ Rs.} \]

### Step 4: Calculate the Compound Interest
For Compound Interest (C.I.), the formula is:
\[ \text{C.I.} = P \left(1 + \frac{R}{100}\right)^T - P \]
Using the same principal amount \( P \) as calculated for S.I.:
\[ \text{C.I.} = P \left(1 + \frac{5}{100}\right)^2 - P \]
\[ = P \left(1 + 0.05\right)^2 - P \]
\[ = P \left(1.05\right)^2 - P \]
\[ = P \times 1.1025 - P \]
\[ = P \times (1.1025 - 1) \]
\[ = P \times 0.1025 \]

### Step 5: Set C.I. Equal to S.I.
Since the C.I. is equal to the S.I. of another sum of money, we set:
\[ P \times 0.1025 = 410 \]
Substituting \( P = 4100 \):
\[ 4100 \times 0.1025 = 410 \]

### Step 6: Find the Principal for C.I.
Let’s denote the principal for C.I. as \( P' \):
\[ P' \times 0.1025 = 410 \]
\[ P' = \frac{410}{0.1025} = 4000 \text{ Rs.} \]

### Step 7: Calculate the Difference Between the Two Principals
Now, we find the difference between the two principal amounts:
\[ \text{Difference} = P - P' = 4100 - 4000 = 100 \text{ Rs.} \]

### Final Answer
The difference between the two sums of money (principals) is **100 Rs.**.