Dilip invested 100 for three years at a compound interest rate of 10%. Haru also invested 100 for three years at simple interest rate of x%. At the end of three years, both received the same amount of money. What is the value of x?

Options

A. 11.03
B. 11.3
C. 11.05
D. 10.93

Answer

To find the value of \( x \) such that both Dilip and Haru received the same amount after three years, we can follow these steps:

### Step 1: Calculate the amount Dilip receives from his compound interest investment.

Dilip invested \( P = 100 \) at a compound interest rate of \( R = 10\% \) for \( T = 3 \) years. The formula for compound interest is:

\[ A = P \left(1 + \frac{R}{100}\right)^T \]

Substituting the values:

\[ A = 100 \left(1 + \frac{10}{100}\right)^3 \]

### Step 2: Simplify the expression.

Calculating \( 1 + \frac{10}{100} = 1.1 \):

\[ A = 100 \left(1.1\right)^3 \]

Now, calculate \( (1.1)^3 \):

\[ (1.1)^3 = 1.1 \times 1.1 \times 1.1 = 1.331 \]

Thus,

\[ A = 100 \times 1.331 = 133.1 \]

### Step 3: Set up the equation for Haru's simple interest investment.

Haru also invested \( P = 100 \) at a simple interest rate of \( x\% \) for \( T = 3 \) years. The formula for simple interest is:

\[ A = P + \left(\frac{P \cdot R \cdot T}{100}\right) \]

Substituting the values:

\[ A = 100 + \left(\frac{100 \cdot x \cdot 3}{100}\right) \]

This simplifies to:

\[ A = 100 + 3x \]

### Step 4: Set the amounts equal to each other.

Since both received the same amount:

\[ 133.1 = 100 + 3x \]

### Step 5: Solve for \( x \).

Subtract 100 from both sides:

\[ 133.1 - 100 = 3x \]

\[ 33.1 = 3x \]

Now, divide by 3:

\[ x = \frac{33.1}{3} \approx 11.03 \]

### Conclusion:

The value of \( x \) is approximately \( 11.03\% \).

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