The compound interest (compounded annually) on 2300 for 2 years at 'R' percent per annum is 1012. If 3100 is invested for '2+y' years in scheme B, which offers simple interest at 'R-8 per cent per annum the amount received from scheme B is 4960, what is the value of y?

Options

A. 3
B. 5
C. 2
D. 1

Answer

To solve the given problem step by step, we will break down the calculations for both schemes A and B.

### Step 1: Calculate the Rate of Interest (R) from Scheme A
We know that the compound interest (CI) for 2 years on a principal of 2300 at R% is 1012.

Using the formula for compound interest:
\[ A = P \left(1 + \frac{R}{100}\right)^t \]
Where:
- \(A\) is the total amount after time \(t\),
- \(P\) is the principal amount,
- \(R\) is the rate of interest,
- \(t\) is the time in years.

The total amount \(A\) can be calculated as:
\[ A = P + CI = 2300 + 1012 = 3312 \]

Now we can set up the equation:
\[ 3312 = 2300 \left(1 + \frac{R}{100}\right)^2 \]

### Step 2: Simplify the Equation
Dividing both sides by 2300:
\[ \frac{3312}{2300} = \left(1 + \frac{R}{100}\right)^2 \]
Calculating the left side:
\[ \frac{3312}{2300} = 1.44 \]
So we have:
\[ 1.44 = \left(1 + \frac{R}{100}\right)^2 \]

### Step 3: Take the Square Root
Taking the square root of both sides:
\[ \sqrt{1.44} = 1 + \frac{R}{100} \]
Calculating the square root:
\[ 1.2 = 1 + \frac{R}{100} \]

### Step 4: Solve for R
Subtracting 1 from both sides:
\[ 0.2 = \frac{R}{100} \]
Multiplying both sides by 100:
\[ R = 20 \]

### Step 5: Calculate Simple Interest for Scheme B
Now we know \(R = 20\). In scheme B, the principal is 3100, the rate is \(R - 8 = 20 - 8 = 12\%\), and the total amount received is 4960.

First, calculate the interest earned:
\[ \text{Interest} = \text{Total Amount} - \text{Principal} = 4960 - 3100 = 1860 \]

### Step 6: Set Up the Simple Interest Formula
The formula for simple interest (SI) is:
\[ SI = \frac{P \times R \times T}{100} \]
Where:
- \(SI\) is the simple interest,
- \(P\) is the principal,
- \(R\) is the rate of interest,
- \(T\) is the time in years.

Substituting the known values:
\[ 1860 = \frac{3100 \times 12 \times (2 + y)}{100} \]

### Step 7: Simplify the Equation
Multiplying both sides by 100:
\[ 186000 = 3100 \times 12 \times (2 + y) \]
Calculating \(3100 \times 12\):
\[ 3100 \times 12 = 37200 \]
So we have:
\[ 186000 = 37200 \times (2 + y) \]

### Step 8: Solve for \(2 + y\)
Dividing both sides by 37200:
\[ \frac{186000}{37200} = 2 + y \]
Calculating the left side:
\[ 5 = 2 + y \]

### Step 9: Solve for y
Subtracting 2 from both sides:
\[ y = 5 - 2 = 3 \]

Thus, the value of \(y\) is **3**.