If the compound interest on a certain sum of money for 2 years is Rs. 164 whereas the simple interest for 3 years at the same rate is Rs. 240 on the same sum, find the sum.

Options

A. A)Rs. 1200
B. B)Rs. 1400
C. C)Rs. 1600
D. D)Rs. 1800

Answer

To solve the problem step by step, we will find the principal amount (P) using the information provided about simple interest (SI) and compound interest (CI).

### Step 1: Understand the given data
- Compound interest for 2 years (CI) = Rs. 164
- Simple interest for 3 years (SI) = Rs. 240

### Step 2: Calculate the simple interest for 1 year
The simple interest for 3 years is Rs. 240, so the simple interest for 1 year can be calculated as follows:
\[ \text{SI for 1 year} = \frac{\text{Total SI for 3 years}}{3} = \frac{240}{3} = Rs. 80 \]

### Step 3: Calculate the total simple interest for 2 years
Since the simple interest for 1 year is Rs. 80, the total simple interest for 2 years will be:
\[ \text{SI for 2 years} = 80 \times 2 = Rs. 160 \]

### Step 4: Relate compound interest to simple interest
The compound interest for 2 years is Rs. 164. The total interest from simple interest for 2 years is Rs. 160. The difference between the compound interest and simple interest gives us the interest earned on the interest from the first year:
\[ \text{Interest on interest} = \text{CI} - \text{SI for 2 years} = 164 - 160 = Rs. 4 \]

### Step 5: Calculate the rate of interest
The interest on the interest (Rs. 4) is earned on the simple interest of Rs. 80 (the interest for the first year). We can set up the equation to find the rate of interest (R):
\[ \text{Interest on SI} = \frac{P \times R \times T}{100} \]
Where:
- P = Rs. 80 (the interest from the first year)
- R = rate of interest
- T = time (1 year)

Substituting the values:
\[ 4 = \frac{80 \times R \times 1}{100} \]
\[ 4 = \frac{80R}{100} \]
\[ 4 = 0.8R \]
\[ R = \frac{4}{0.8} = 5\% \]

### Step 6: Calculate the principal amount
Now that we have the rate of interest (R = 5%), we can use the simple interest formula to find the principal (P):
\[ \text{SI for 3 years} = \frac{P \times R \times T}{100} \]
Substituting the values:
\[ 240 = \frac{P \times 5 \times 3}{100} \]
\[ 240 = \frac{15P}{100} \]
\[ 240 = 0.15P \]
\[ P = \frac{240}{0.15} = 1600 \]

### Final Answer
The principal amount is Rs. 1600.

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