At what percent per annum will Rs 3000/- amounts Rs 3993/- in 3 years if the Interest is compounded annually?

Options

A. `9%`
B. `10%`
C. `11%`
D. `13%`

Answer

To find the rate of interest per annum at which Rs 3000 amounts to Rs 3993 in 3 years with compound interest, we can use the compound interest formula:

**Step 1: Write down the compound interest formula.**
The formula for compound interest is given by:
\[ A = P \left(1 + \frac{R}{100}\right)^T \]
where:
- \( A \) = Amount after time \( T \)
- \( P \) = Principal amount (initial investment)
- \( R \) = Rate of interest per annum
- \( T \) = Time in years

**Hint:** Remember that the amount \( A \) is what you will have after the interest is applied over the time period.

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**Step 2: Substitute the known values into the formula.**
Given:
- \( A = 3993 \)
- \( P = 3000 \)
- \( T = 3 \)

Substituting these values into the formula:
\[ 3993 = 3000 \left(1 + \frac{R}{100}\right)^3 \]

**Hint:** Make sure to keep track of your units and values as you substitute them into the formula.

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**Step 3: Divide both sides by the principal amount.**
To isolate the term with \( R \):
\[ \frac{3993}{3000} = \left(1 + \frac{R}{100}\right)^3 \]

Calculating the left side:
\[ 1.331 = \left(1 + \frac{R}{100}\right)^3 \]

**Hint:** This step simplifies the equation, making it easier to solve for \( R \).

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**Step 4: Take the cube root of both sides.**
To eliminate the exponent, take the cube root:
\[ 1 + \frac{R}{100} = \sqrt[3]{1.331} \]

Calculating the cube root:
\[ 1 + \frac{R}{100} = 1.1 \]

**Hint:** Remember that taking the cube root will give you a value that you can then manipulate to find \( R \).

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**Step 5: Solve for \( R \).**
Now, subtract 1 from both sides:
\[ \frac{R}{100} = 1.1 - 1 \]
\[ \frac{R}{100} = 0.1 \]

Now, multiply both sides by 100 to find \( R \):
\[ R = 0.1 \times 100 \]
\[ R = 10 \]

**Hint:** Always check your calculations to ensure accuracy in your final result.

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**Final Answer:**
The rate of interest per annum is **10%**.

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