A sum of money becomes 3 times in 10 years at the rate of compound interest (compounded annually). In how many years will it become 81 times?

Options

A. 40 years
B. 30 years
C. 35 years
D. 50 years

Answer

To solve the problem step by step, we will use the properties of compound interest.

### Step 1: Understand the given information
We are told that a sum of money becomes 3 times in 10 years at a certain rate of compound interest. This means:
- Initial amount (Principal, P) becomes 3P in 10 years.

### Step 2: Set up the compound interest formula
The formula for compound interest is given by:
\[ A = P \left(1 + \frac{r}{100}\right)^t \]
Where:
- \( A \) is the amount after time \( t \),
- \( P \) is the principal amount,
- \( r \) is the rate of interest,
- \( t \) is the time in years.

### Step 3: Apply the information to the formula
From the information given:
\[ 3P = P \left(1 + \frac{r}{100}\right)^{10} \]

### Step 4: Simplify the equation
Dividing both sides by \( P \) (assuming \( P \neq 0 \)):
\[ 3 = \left(1 + \frac{r}{100}\right)^{10} \]

### Step 5: Take the 10th root
To find \( 1 + \frac{r}{100} \), we take the 10th root of both sides:
\[ 1 + \frac{r}{100} = 3^{\frac{1}{10}} \]

### Step 6: Calculate \( r \)
Subtract 1 from both sides:
\[ \frac{r}{100} = 3^{\frac{1}{10}} - 1 \]
Multiply by 100:
\[ r = 100 \left(3^{\frac{1}{10}} - 1\right) \]

### Step 7: Set up the equation for becoming 81 times
Now we need to find out how many years it will take for the principal to become 81 times:
\[ 81P = P \left(1 + \frac{r}{100}\right)^t \]

### Step 8: Simplify the equation
Dividing both sides by \( P \):
\[ 81 = \left(1 + \frac{r}{100}\right)^t \]

### Step 9: Substitute \( 1 + \frac{r}{100} \)
From Step 5, we know:
\[ 1 + \frac{r}{100} = 3^{\frac{1}{10}} \]
Thus:
\[ 81 = \left(3^{\frac{1}{10}}\right)^t \]

### Step 10: Express 81 as a power of 3
Since \( 81 = 3^4 \), we can rewrite the equation:
\[ 3^4 = \left(3^{\frac{1}{10}}\right)^t \]

### Step 11: Equate the exponents
Since the bases are the same, we can equate the exponents:
\[ 4 = \frac{t}{10} \]

### Step 12: Solve for \( t \)
Multiply both sides by 10:
\[ t = 40 \]

### Conclusion
The sum of money will become 81 times in **40 years**.

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