P invested Rs. X in a scheme for 2 year which offered simple at the rate of 15% per annum and Q invested Rs. (X + 2500} in another scheme for same period of time, which offered compound interest at the rate of 20% per annum. If from both scheme P and Q got total interest of Rs. 32550, then find the value of X?

Options

A. A)41500
B. B)42500
C. C)40500
D. D)40000

Answer

To solve the problem step by step, we will break down the information provided and calculate the required value of \( X \).

### Step 1: Understand the Investments
- P invested \( Rs. X \) at a simple interest rate of 15% per annum for 2 years.
- Q invested \( Rs. (X + 2500) \) at a compound interest rate of 20% per annum for the same period of 2 years.

### Step 2: Calculate the Simple Interest for P
The formula for simple interest (SI) is:
\[ SI = \frac{P \times R \times T}{100} \]
Where:
- \( P \) = Principal amount (investment)
- \( R \) = Rate of interest per annum
- \( T \) = Time in years

For P:
- \( P = X \)
- \( R = 15\% \)
- \( T = 2 \)

Calculating the simple interest for P:
\[ SI_P = \frac{X \times 15 \times 2}{100} = \frac{30X}{100} = 0.3X \]

### Step 3: Calculate the Compound Interest for Q
The formula for compound interest (CI) is:
\[ CI = P \left(1 + \frac{R}{100}\right)^T - P \]

For Q:
- \( P = X + 2500 \)
- \( R = 20\% \)
- \( T = 2 \)

Calculating the compound interest for Q:
\[ CI_Q = (X + 2500) \left(1 + \frac{20}{100}\right)^2 - (X + 2500) \]
\[ = (X + 2500) \left(1.2\right)^2 - (X + 2500) \]
\[ = (X + 2500) \times 1.44 - (X + 2500) \]
\[ = 1.44(X + 2500) - (X + 2500) \]
\[ = 1.44X + 3600 - X - 2500 \]
\[ = 0.44X + 1100 \]

### Step 4: Set Up the Equation for Total Interest
According to the problem, the total interest from both schemes is Rs. 32,550:
\[ SI_P + CI_Q = 32550 \]
Substituting the values we calculated:
\[ 0.3X + (0.44X + 1100) = 32550 \]

### Step 5: Simplify the Equation
Combine like terms:
\[ 0.3X + 0.44X + 1100 = 32550 \]
\[ 0.74X + 1100 = 32550 \]

### Step 6: Solve for X
Subtract 1100 from both sides:
\[ 0.74X = 32550 - 1100 \]
\[ 0.74X = 31450 \]

Now, divide both sides by 0.74:
\[ X = \frac{31450}{0.74} \approx 42500 \]

### Conclusion
The value of \( X \) is \( Rs. 42500 \).