What is the rate of interest (in %) if simple interest earned on a certain sum for the 3 years is Rs 6,000 and compound interest earned for 2 years is Rs 4,160?

Options

A. 9
B. 8
C. 12
D. 6

Answer

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

### Step 1: Understand the given information
- Simple Interest (SI) earned in 3 years = Rs 6,000
- Compound Interest (CI) earned in 2 years = Rs 4,160

### Step 2: Calculate the Simple Interest for 2 years
Since the simple interest for 3 years is Rs 6,000, we can find the simple interest for 2 years by dividing the total simple interest by 3 and then multiplying by 2.

\[ \text{SI for 2 years} = \frac{6000}{3} \times 2 = 4000 \]

### Step 3: Find the difference between Compound Interest and Simple Interest
Now, we can find the difference between the compound interest for 2 years and the simple interest for 2 years.

\[ \text{Difference} = \text{CI} - \text{SI} = 4160 - 4000 = 160 \]

### Step 4: Use the formula to find the rate of interest
The difference between compound interest and simple interest over 2 years can be calculated using the formula:

\[ D = \frac{SI \times R \times 2}{100} \]

Where:
- \(D\) = Difference (Rs 160)
- \(SI\) = Simple Interest for 2 years (Rs 4000)
- \(R\) = Rate of interest (in %)

Rearranging the formula to find \(R\):

\[ R = \frac{D \times 100}{SI \times 2} \]

Substituting the values we have:

\[ R = \frac{160 \times 100}{4000 \times 2} \]

### Step 5: Calculate the rate of interest
Now, we can calculate \(R\):

\[ R = \frac{16000}{8000} = 2 \]

### Conclusion
The rate of interest is \(2\%\).

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