On what sum does the difference between the compound interest and the simple interest for 3 years at 10% is Rs. 31 ?

Options

A. Rs. 1500
B. Rs. 1200
C. Rs. 1100
D. Rs. 1000

Answer

To solve the problem of finding the sum on which the difference between compound interest (CI) and simple interest (SI) for 3 years at a rate of 10% is Rs. 31, we can follow these steps:

### Step 1: Define the Variables
Let the principal amount (sum) be Rs. P.

### Step 2: Calculate Simple Interest (SI)
The formula for Simple Interest is:
\[ SI = \frac{P \times R \times T}{100} \]
Where:
- \( R = 10\% \)
- \( T = 3 \) years

Substituting the values:
\[ SI = \frac{P \times 10 \times 3}{100} = \frac{30P}{100} = \frac{3P}{10} \]

### Step 3: Calculate Compound Interest (CI)
The formula for Compound Interest for 3 years at the same rate is:
\[ CI = P \left(1 + \frac{R}{100}\right)^T - P \]
Substituting the values:
\[ CI = P \left(1 + \frac{10}{100}\right)^3 - P = P \left(1.1\right)^3 - P \]

Calculating \( (1.1)^3 \):
\[ (1.1)^3 = 1.1 \times 1.1 \times 1.1 = 1.331 \]
Thus,
\[ CI = P \times 1.331 - P = 0.331P \]

### Step 4: Find the Difference Between CI and SI
The difference between Compound Interest and Simple Interest is:
\[ CI - SI = 0.331P - \frac{3P}{10} \]

To combine these, convert \( \frac{3P}{10} \) to a decimal:
\[ \frac{3P}{10} = 0.3P \]
Now, substituting this back into the difference:
\[ CI - SI = 0.331P - 0.3P = 0.031P \]

### Step 5: Set the Difference Equal to Rs. 31
According to the problem, this difference equals Rs. 31:
\[ 0.031P = 31 \]

### Step 6: Solve for P
To find P, divide both sides by 0.031:
\[ P = \frac{31}{0.031} = 1000 \]

### Conclusion
The sum (principal amount) on which the difference between compound interest and simple interest for 3 years at 10% is Rs. 31 is Rs. 1000.