Questions: Lucy deposits some money into a new savings account that earns 13.25% interest compounded monthly. How long will it take for the money to double? Round your answer to the nearest month. years and months

Solution Steps

To determine how long it will take for Lucy's money to double with monthly compounding interest, we can use the formula for compound interest:

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

where:

• \( A \) is the amount of money accumulated after n years, including interest.
• \( P \) is the principal amount (initial deposit).
• \( r \) is the annual interest rate (decimal).
• \( n \) is the number of times that interest is compounded per year.
• \( t \) is the time the money is invested for in years.

Since we want the money to double, \( A = 2P \). We need to solve for \( t \) when \( r = 0.1325 \) and \( n = 12 \).

1. Set up the equation \( 2P = P \left(1 + \frac{0.1325}{12}\right)^{12t} \).
2. Simplify to find \( t \) by taking the natural logarithm of both sides.
3. Solve for \( t \) and convert it to years and months.

To find out how long it will take for Lucy's money to double with monthly compounding interest, we use the compound interest formula:

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

where:

• \( A \) is the final amount (which is \( 2P \) since the money doubles),
• \( P \) is the initial principal,
• \( r = 0.1325 \) is the annual interest rate,
• \( n = 12 \) is the number of compounding periods per year,
• \( t \) is the time in years.

Since \( A = 2P \), we can set up the equation:

\[ 2P = P \left(1 + \frac{0.1325}{12}\right)^{12t} \]

Dividing both sides by \( P \) gives:

\[ 2 = \left(1 + \frac{0.1325}{12}\right)^{12t} \]

To solve for \( t \), take the natural logarithm of both sides:

\[ \ln(2) = 12t \cdot \ln\left(1 + \frac{0.1325}{12}\right) \]

Solving for \( t \) gives:

\[ t = \frac{\ln(2)}{12 \cdot \ln\left(1 + \frac{0.1325}{12}\right)} \]

Calculating this expression yields:

\[ t \approx 5.2601 \text{ years} \]

The time \( t \) in years is approximately 5.2601. This can be broken down into:

• Years: \( \lfloor 5.2601 \rfloor = 5 \)
• Months: \( (5.2601 - 5) \times 12 \approx 3.1212 \), which rounds to 3 months.

Final Answer