Shawn invested one half of his savings in a bond that paid simple interest for 2 years and received Rs.550 as interest. He invested the remaining in a bond that paid compound interest, interest being compounded annually, for the same 2 years at the same rate of interest and received Rs.605 as interest. What was the value of his total savings before investing in thesetwo bonds?

Options

A. A. 2750
B. B. 3320
C. C. 3590
D. TD. 4000

Explanation

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

### Step 1: Understand the Investment
Let’s denote Shawn's total savings as \( S \). According to the problem, he invested half of his savings in a bond that paid simple interest. Therefore, the amount invested in the simple interest bond is:
\[ \text{Amount in Simple Interest Bond} = \frac{S}{2} \]

### Step 2: Calculate Simple Interest
The simple interest earned from this bond over 2 years is given as Rs. 550. The formula for simple interest is:
\[ \text{SI} = \frac{P \times R \times T}{100} \]
Where:
- \( P \) = Principal amount (which is \( \frac{S}{2} \))
- \( R \) = Rate of interest (unknown)
- \( T \) = Time (2 years)

Substituting the values we know:
\[ 550 = \frac{\frac{S}{2} \times R \times 2}{100} \]
This simplifies to:
\[ 550 = \frac{S \times R}{100} \]
From this, we can express \( S \times R \):
\[ S \times R = 55000 \quad \text{(Equation 1)} \]

### Step 3: Calculate Compound Interest
Shawn invested the remaining half of his savings in a bond that paid compound interest, which earned him Rs. 605 over the same 2 years. The amount invested in the compound interest bond is also:
\[ \text{Amount in Compound Interest Bond} = \frac{S}{2} \]
The formula for compound interest is:
\[ \text{CI} = P \left(1 + \frac{R}{100}\right)^T - P \]
Substituting the values we know:
\[ 605 = \frac{S}{2} \left(1 + \frac{R}{100}\right)^2 - \frac{S}{2} \]
This simplifies to:
\[ 605 = \frac{S}{2} \left[\left(1 + \frac{R}{100}\right)^2 - 1\right] \]
Expanding the square:
\[ \left(1 + \frac{R}{100}\right)^2 = 1 + 2\frac{R}{100} + \left(\frac{R}{100}\right)^2 \]
Thus:
\[ 605 = \frac{S}{2} \left[2\frac{R}{100} + \left(\frac{R}{100}\right)^2\right] \]
This can be simplified to:
\[ 605 = \frac{S}{100} \left[R + \frac{R^2}{100}\right] \]
From this, we can express \( S \) in terms of \( R \):
\[ S \left[R + \frac{R^2}{100}\right] = 60500 \quad \text{(Equation 2)} \]

### Step 4: Solve the Equations
Now we have two equations:
1. \( S \times R = 55000 \)
2. \( S \left[R + \frac{R^2}{100}\right] = 60500 \)

From Equation 1, we can express \( R \) in terms of \( S \):
\[ R = \frac{55000}{S} \]

Substituting \( R \) into Equation 2:
\[ S \left[\frac{55000}{S} + \frac{\left(\frac{55000}{S}\right)^2}{100}\right] = 60500 \]
This simplifies to:
\[ 55000 + \frac{55000^2}{100S} = 60500 \]
Rearranging gives:
\[ \frac{55000^2}{100S} = 60500 - 55000 \]
\[ \frac{55000^2}{100S} = 5500 \]
Cross-multiplying:
\[ 55000^2 = 5500 \times 100S \]
\[ 55000^2 = 550000S \]
Solving for \( S \):
\[ S = \frac{55000^2}{550000} \]
\[ S = \frac{3025000000}{550000} = 5500 \]

### Final Answer
Thus, the value of Shawn's total savings before investing in the two bonds is:
\[ \boxed{11000} \]