Purav invested, equal sums in schemes A and B respectively for 4 years and 2 years respectively Scheme A offers simple interest at the rate of 12 pcpa and scheme B offers compound interest (compounded annually) -at 10 pcpa. If the total interest earned by him from both the schemes together is ₹2208, what is the interest earned by him from scheme B ?

Options

A. ₹642
B. ₹664
C. ₹672
D. ₹716

Answer

To solve the problem step by step, we will follow these calculations:

### Step 1: Define the Variables
Let the amount invested in each scheme (A and B) be \( x \).

### Step 2: Calculate Simple Interest from Scheme A
The formula for Simple Interest (SI) is:
\[ \text{SI} = \frac{P \times R \times T}{100} \]
Where:
- \( P \) is the principal amount (which is \( x \)),
- \( R \) is the rate of interest (12% for Scheme A),
- \( T \) is the time (4 years for Scheme A).

So, the Simple Interest from Scheme A is:
\[ \text{SI}_A = \frac{x \times 12 \times 4}{100} = \frac{48x}{100} = \frac{12x}{25} \]

### Step 3: Calculate Compound Interest from Scheme B
The formula for Compound Interest (CI) is:
\[ \text{CI} = A - P \]
Where \( A \) is the amount after time \( T \) and is calculated as:
\[ A = P \left(1 + \frac{R}{100}\right)^T \]
For Scheme B:
- \( P = x \),
- \( R = 10\% \),
- \( T = 2 \) years.

Calculating \( A \):
\[ A_B = x \left(1 + \frac{10}{100}\right)^2 = x \left(1 + 0.1\right)^2 = x \left(1.1\right)^2 = x \times 1.21 = 1.21x \]
Now, calculating the Compound Interest:
\[ \text{CI}_B = A_B - P = 1.21x - x = 0.21x \]

### Step 4: Total Interest from Both Schemes
The total interest earned from both schemes is given as ₹2208:
\[ \text{SI}_A + \text{CI}_B = 2208 \]
Substituting the values we calculated:
\[ \frac{12x}{25} + 0.21x = 2208 \]

### Step 5: Solve for \( x \)
To combine the terms, we first convert \( 0.21x \) to a fraction:
\[ 0.21x = \frac{21x}{100} \]
Now, we need a common denominator to combine:
The least common multiple of 25 and 100 is 100:
\[ \frac{12x}{25} = \frac{48x}{100} \]
So the equation becomes:
\[ \frac{48x}{100} + \frac{21x}{100} = 2208 \]
Combining the fractions:
\[ \frac{69x}{100} = 2208 \]
Multiplying both sides by 100:
\[ 69x = 220800 \]
Now, divide by 69:
\[ x = \frac{220800}{69} = 3200 \]

### Step 6: Calculate the Interest Earned from Scheme B
Now that we have \( x \), we can find the interest from Scheme B:
\[ \text{CI}_B = 0.21x = 0.21 \times 3200 = 672 \]

### Final Answer
The interest earned by Purav from Scheme B is ₹672.

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