A sum of money becomes 1.331 times in 3 years as compound interest. The rate of interest is

Options

A. `8%`
B. `7.5% `
C. `10% `
D. `50% `

Answer

To solve the problem step by step, we will follow the concept of compound interest.

### Step 1: Understand the problem
We know that a sum of money becomes 1.331 times in 3 years as compound interest. We need to find the rate of interest.

### Step 2: Define the principal amount
Let's assume the principal amount (P) is 1000X. This is a convenient choice because it allows us to work with whole numbers.

### Step 3: Calculate the amount after 3 years
According to the problem, the amount (A) after 3 years is 1.331 times the principal. Therefore:
\[ A = 1.331 \times 1000X = 1331X \]

### Step 4: Use the compound interest formula
The formula for the amount in compound interest is:
\[ A = P \left(1 + \frac{r}{100}\right)^n \]
where:
- A = final amount
- P = principal amount
- r = rate of interest
- n = number of years

Substituting the values we have:
\[ 1331X = 1000X \left(1 + \frac{r}{100}\right)^3 \]

### Step 5: Simplify the equation
We can divide both sides by 1000X (assuming X is not zero):
\[ \frac{1331X}{1000X} = \left(1 + \frac{r}{100}\right)^3 \]
This simplifies to:
\[ \frac{1331}{1000} = \left(1 + \frac{r}{100}\right)^3 \]

### Step 6: Calculate the left-hand side
Calculating \( \frac{1331}{1000} \):
\[ \frac{1331}{1000} = 1.331 \]

### Step 7: Take the cube root
To find \( 1 + \frac{r}{100} \), we take the cube root of both sides:
\[ 1 + \frac{r}{100} = \sqrt[3]{1.331} \]

### Step 8: Calculate the cube root
The cube root of 1.331 is:
\[ \sqrt[3]{1.331} = 1.1 \]

### Step 9: Solve for r
Now we can solve for \( r \):
\[ 1 + \frac{r}{100} = 1.1 \]
Subtracting 1 from both sides gives:
\[ \frac{r}{100} = 0.1 \]

Multiplying both sides by 100:
\[ r = 10 \]

### Final Answer
The rate of interest is **10%**.

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