In 3 years `Rs. 3000` amounts to `Rs. 3993` at `x%` compound interest, compounded annually. The value of x is

Options

A. 10
B. 8
C. `5`
D. `1(1)/3`

Answer

To solve the problem step by step, we will use the formula for compound interest. The formula for the amount \( A \) after \( t \) years at a principal \( P \) with an interest rate \( r \% \) compounded annually is given by:

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

### Step 1: Identify the given values
- Principal \( P = 3000 \)
- Amount \( A = 3993 \)
- Time \( t = 3 \) years
- Rate \( r = x \% \) (which we need to find)

### Step 2: Substitute the values into the formula
We can substitute the known values into the compound interest formula:

\[ 3993 = 3000 \left(1 + \frac{x}{100}\right)^3 \]

### Step 3: Divide both sides by 3000
To isolate the term with \( x \), we divide both sides by 3000:

\[ \frac{3993}{3000} = \left(1 + \frac{x}{100}\right)^3 \]

Calculating the left side:

\[ \frac{3993}{3000} = 1.331 \]

So we have:

\[ 1.331 = \left(1 + \frac{x}{100}\right)^3 \]

### Step 4: Take the cube root of both sides
To eliminate the exponent, we take the cube root of both sides:

\[ \sqrt[3]{1.331} = 1 + \frac{x}{100} \]

Calculating the cube root:

\[ \sqrt[3]{1.331} \approx 1.1 \]

So we have:

\[ 1.1 = 1 + \frac{x}{100} \]

### Step 5: Solve for \( x \)
Now, we subtract 1 from both sides:

\[ 0.1 = \frac{x}{100} \]

Next, multiply both sides by 100 to solve for \( x \):

\[ x = 0.1 \times 100 = 10 \]

### Conclusion
The value of \( x \) is \( 10\% \).

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