If the C.I & S.I on a certain sum of money for 2 years are Rs. 65 & Rs. 60 respectively at certain rate of interest. Find the sum of money & rate of interest.

Answer

To solve the problem of finding the sum of money (P) and the rate of interest (R) given the compound interest (C.I) and simple interest (S.I) for 2 years, we can follow these steps:

### Step 1: Identify the given values
- C.I for 2 years = Rs. 65
- S.I for 2 years = Rs. 60

### Step 2: Use the formula for the difference between C.I and S.I
The difference between compound interest and simple interest for 2 years can be expressed as:
\[ C.I - S.I = \frac{P \cdot R^2}{100^2} \]
Where:
- P = Principal amount (sum of money)
- R = Rate of interest

### Step 3: Calculate the difference
From the given values:
\[ C.I - S.I = 65 - 60 = 5 \]
So, we have:
\[ 5 = \frac{P \cdot R^2}{10000} \]
(Equation 1)

### Step 4: Use the alternative formula for C.I and S.I
The difference can also be expressed as:
\[ C.I - S.I = \frac{S.I \cdot R}{200} \]
Substituting the known S.I:
\[ 5 = \frac{60 \cdot R}{200} \]

### Step 5: Solve for R
Rearranging the equation gives:
\[ 5 = \frac{60R}{200} \]
\[ 5 = \frac{3R}{10} \]
Multiplying both sides by 10:
\[ 50 = 3R \]
Dividing by 3:
\[ R = \frac{50}{3} \approx 16.67\% \]

### Step 6: Substitute R back to find P
Now we substitute R back into Equation 1:
\[ 5 = \frac{P \cdot \left(\frac{50}{3}\right)^2}{10000} \]
Calculating \( \left(\frac{50}{3}\right)^2 \):
\[ \left(\frac{50}{3}\right)^2 = \frac{2500}{9} \]
Substituting this back:
\[ 5 = \frac{P \cdot \frac{2500}{9}}{10000} \]
\[ 5 = \frac{P \cdot 2500}{90000} \]
Multiplying both sides by 90000:
\[ 450000 = P \cdot 2500 \]
Dividing by 2500:
\[ P = \frac{450000}{2500} = 180 \]

### Final Answers
- The sum of money (P) = Rs. 180
- The rate of interest (R) = 16.67%