If simple interest on some amount for 2 years is 280 and compound interest on same amount for 2 years is 294 then find the rate of interest?

Options

A. `9% `
B. `10.5%`
C. `9.5%`
D. `10%`

Answer

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

### Step 1: Understand the given information
- Simple Interest (SI) for 2 years = Rs. 280
- Compound Interest (CI) for 2 years = Rs. 294
- Time (T) = 2 years

### Step 2: Calculate the Simple Interest for 1 year
Since the total simple interest for 2 years is Rs. 280, the simple interest for 1 year is:
\[ \text{SI for 1 year} = \frac{\text{Total SI}}{2} = \frac{280}{2} = Rs. 140 \]

### Step 3: Calculate the Compound Interest for the first year
The compound interest for the first year is equal to the simple interest for the first year:
\[ \text{CI for 1st year} = \text{SI for 1st year} = Rs. 140 \]

### Step 4: Calculate the Compound Interest for the second year
The total compound interest for 2 years is Rs. 294. Therefore, the compound interest for the second year can be calculated as:
\[ \text{CI for 2nd year} = \text{Total CI} - \text{CI for 1st year} = 294 - 140 = Rs. 154 \]

### Step 5: Set up equations for Principal (P) and Rate (R)
Using the formula for simple interest:
\[ \text{SI} = \frac{P \times R \times T}{100} \]
For the first year:
\[ 140 = \frac{P \times R \times 1}{100} \quad \text{(Equation 1)} \]
This simplifies to:
\[ P \times R = 14000 \quad \text{(Multiply both sides by 100)} \]

For the second year, the compound interest is calculated on the principal plus the interest from the first year:
\[ 154 = \frac{(P + 140) \times R}{100} \quad \text{(Equation 2)} \]
This simplifies to:
\[ (P + 140) \times R = 15400 \quad \text{(Multiply both sides by 100)} \]

### Step 6: Substitute Equation 1 into Equation 2
From Equation 1, we know \( P \times R = 14000 \). We can express \( P \) in terms of \( R \):
\[ P = \frac{14000}{R} \]
Now substitute \( P \) into Equation 2:
\[ \left(\frac{14000}{R} + 140\right) \times R = 15400 \]
Expanding this gives:
\[ 14000 + 140R = 15400 \]

### Step 7: Solve for R
Rearranging the equation:
\[ 140R = 15400 - 14000 \]
\[ 140R = 1400 \]
Dividing both sides by 140:
\[ R = \frac{1400}{140} = 10 \]

### Conclusion
The rate of interest (R) is **10%**.

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