Out of Rs 8000, Gopal invested: a certain sum in scheme A, and the remaining sum in scheme B for two years. Both the schemes offer compound interest (compound annually). The rate of interest of scheme A and Bare 10 pcpa and 20 pcpa respectively. If the total amount accrued by him after two years from both the schemes together was Rs 10,600, what was the amount invested in scheme B?

Options

A. Rs 4800
B. Rs 5200
C. Rs 3600
D. Rs 4000

Answer

To solve the problem step by step, we will break down the information given and use the formula for compound interest.

### Step 1: Define the Variables
Let:
- \( X \) = amount invested in Scheme A
- \( Y \) = amount invested in Scheme B

From the problem, we know:
- Total investment: \( X + Y = 8000 \) (Equation 1)

### Step 2: Write the Compound Interest Formula
The formula for the amount \( A \) after \( t \) years with principal \( P \) and rate \( R \) is:
\[ A = P \left(1 + \frac{R}{100}\right)^t \]

### Step 3: Calculate Amounts for Each Scheme
For Scheme A (10% per annum for 2 years):
\[ A_A = X \left(1 + \frac{10}{100}\right)^2 = X \left(1.1\right)^2 = X \times 1.21 \]

For Scheme B (20% per annum for 2 years):
\[ A_B = Y \left(1 + \frac{20}{100}\right)^2 = Y \left(1.2\right)^2 = Y \times 1.44 \]

### Step 4: Total Amount After 2 Years
According to the problem, the total amount after 2 years from both schemes is Rs 10,600:
\[ A_A + A_B = 10600 \]
Substituting the expressions for \( A_A \) and \( A_B \):
\[ 1.21X + 1.44Y = 10600 \quad \text{(Equation 2)} \]

### Step 5: Solve the Equations
Now we have two equations:
1. \( X + Y = 8000 \) (Equation 1)
2. \( 1.21X + 1.44Y = 10600 \) (Equation 2)

From Equation 1, we can express \( Y \) in terms of \( X \):
\[ Y = 8000 - X \]

### Step 6: Substitute \( Y \) in Equation 2
Substituting \( Y \) in Equation 2:
\[ 1.21X + 1.44(8000 - X) = 10600 \]
Expanding this:
\[ 1.21X + 11552 - 1.44X = 10600 \]
Combining like terms:
\[ -0.23X + 11552 = 10600 \]
Rearranging gives:
\[ -0.23X = 10600 - 11552 \]
\[ -0.23X = -952 \]

### Step 7: Solve for \( X \)
Dividing both sides by -0.23:
\[ X = \frac{952}{0.23} \approx 4147.83 \]

### Step 8: Find \( Y \)
Now substituting \( X \) back to find \( Y \):
\[ Y = 8000 - X \approx 8000 - 4147.83 \approx 3852.17 \]

### Step 9: Rounding to Nearest Option
Since the options provided are whole numbers, we check the closest option to our calculated value for \( Y \):
- The closest option is Rs 4,000.

Thus, the amount invested in Scheme B is **Rs 4,000**.