The compound interest on a sum for 2 years is Rs. 832 and the simple interest on the same sum at the same rate for the same period is Rs. 800. What is the rate of interest?

Answer

To find the rate of interest based on the given compound interest (CI) and simple interest (SI) for 2 years, we can follow these steps:

### Step 1: Understand the Given Information
We know:
- Compound Interest for 2 years (CI) = Rs. 832
- Simple Interest for 2 years (SI) = Rs. 800
- Time (T) = 2 years

### Step 2: Calculate the Simple Interest for 1 Year
Since the simple interest is uniform each year, we can find the simple interest for 1 year:
\[ \text{SI for 1 year} = \frac{\text{Total SI}}{\text{Number of years}} = \frac{800}{2} = 400 \text{ Rs.} \]

### Step 3: Analyze the Compound Interest for 1 Year
The compound interest for the first year will also be equal to the simple interest for that year:
\[ \text{CI for 1 year} = 400 \text{ Rs.} \]

### Step 4: Calculate the Compound Interest for the Second Year
The total compound interest for 2 years is Rs. 832. The compound interest for the second year can be calculated as follows:
\[ \text{CI for 2 years} = \text{CI for 1 year} + \text{CI for 2nd year} \]
Thus,
\[ \text{CI for 2nd year} = 832 - 400 = 432 \text{ Rs.} \]

### Step 5: Determine the Difference in Interest
The difference between the compound interest for the second year and the simple interest for the second year is:
\[ \text{Difference} = \text{CI for 2nd year} - \text{SI for 1 year} = 432 - 400 = 32 \text{ Rs.} \]

### Step 6: Relate the Difference to the Principal
This difference of Rs. 32 is the interest earned on the principal amount (P) at the rate (R) for the first year. We can express this as:
\[ \text{Difference} = \frac{P \times R}{100} \]
Where \( R \) is the rate of interest.

### Step 7: Calculate the Rate of Interest
We know that the simple interest for 1 year is Rs. 400, which corresponds to:
\[ \text{SI for 1 year} = \frac{P \times R}{100} = 400 \]
From this, we can express \( P \) in terms of \( R \):
\[ P = \frac{400 \times 100}{R} \]

### Step 8: Substitute \( P \) in the Difference Equation
Substituting \( P \) back into the difference equation:
\[ 32 = \frac{\left(\frac{400 \times 100}{R}\right) \times R}{100} \]
This simplifies to:
\[ 32 = 400 \Rightarrow R = \frac{400}{32} \Rightarrow R = 12.5 \]

### Step 9: Final Calculation
To find the rate of interest:
\[ R = \frac{32 \times 100}{400} = 8\% \]

Thus, the rate of interest is **8%**.

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