The difference between S.I & C.I on a certain sum of money for 3 years at 10 % is Rs. 31. Find the sum of money.

Answer

To solve the problem, we need to find the principal amount (P) given that the difference between Compound Interest (C.I) and Simple Interest (S.I) for 3 years at a rate of 10% is Rs. 31.

### Step-by-step Solution:

1. **Understand the relationship between C.I and S.I:**
The difference between Compound Interest (C.I) and Simple Interest (S.I) for 3 years can be calculated using the formula:
\[ C.I - S.I = \frac{P \cdot r^2 \cdot (300 + r)}{100^3} \]
where:
- \( P \) = Principal amount
- \( r \) = Rate of interest (in percentage)

2. **Substitute the known values:**
In this case, we know:
- The difference \( C.I - S.I = 31 \)
- The rate \( r = 10 \% \)

Substituting these values into the formula gives:
\[ 31 = \frac{P \cdot 10^2 \cdot (300 + 10)}{100^3} \]

3. **Calculate \( r^2 \) and \( 300 + r \):**
- \( r^2 = 10^2 = 100 \)
- \( 300 + r = 300 + 10 = 310 \)

Therefore, the equation becomes:
\[ 31 = \frac{P \cdot 100 \cdot 310}{100^3} \]

4. **Simplify the equation:**
The denominator \( 100^3 = 100 \cdot 100 \cdot 100 = 1000000 \). Thus, we can rewrite the equation as:
\[ 31 = \frac{P \cdot 31000}{1000000} \]

5. **Cross-multiply to solve for \( P \):**
\[ 31 \cdot 1000000 = P \cdot 31000 \]
\[ 31000000 = P \cdot 31000 \]

6. **Isolate \( P \):**
\[ P = \frac{31000000}{31000} \]

7. **Perform the division:**
\[ P = 1000 \]

Thus, the principal amount is Rs. 1000.

### Final Answer:
The sum of money (Principal) is Rs. 1000.