Question

Morales Bank offers you 3.9% interest compounded monthly. How long will it take for your money to triple if you use Morales Bank?

a) 9 years
b) 12 years
c) 20 years
d) 30 years

Answer 1

Let's solve this question step by step.

Given:
- Interest rate (\(r\)) = 3.9%, which is 0.039 in decimal form.
- Compounded monthly, which means the number of times interest is compounded per year (\(n\)) = 12.
- We want to triple the investment, so \(A = 3P\).
We can use the compound interest formula to find the time (\(t\)) it takes for an investment to grow to a certain amount:
\[A = P \left(1 + \frac{r}{n}\right)^{nt}\]
Since \(A = 3P\), we can set \(P\) to 1 (as its value will cancel out) and substitute the values we have:
\[3 = \left(1 + \frac{0.039}{12}\right)^{12t}\]
This means:
\[3 = \left(1 + \frac{0.039}{12}\right)^{12t}\]
We'll isolate \(t\) by using logarithms. Taking a natural logarithm (ln) on both sides, we get:
\[ln(3) = ln\left(\left(1 + \frac{0.039}{12}\right)^{12t}\right)\]
Recall that logarithm properties allow us to bring down the exponent in front of the log:
\[ln(3) = 12t \cdot ln\left(1 + \frac{0.039}{12}\right)\]
Now, we solve for \(t\):
\[t = \frac{ln(3)}{12 \cdot ln\left(1 + \frac{0.039}{12}\right)}\]
We will calculate each part of the right-hand side step by step:
First:
\[1 + \frac{0.039}{12} = 1 + \frac{0.00325}{1} \approx 1.00325\]
Then calculate the natural log of this value:
\[ln(1.00325) \approx 0.00324393\]
And the natural log of 3 is:
\[ln(3) \approx 1.09861229\]
Finally, divide to get \(t\):
\[t \approx \frac{1.09861229}{12 \cdot 0.00324393}\]
\[t \approx \frac{1.09861229}{0.03892716}\]
\[t \approx 28.21975306\]
So, it will take roughly 28.22 years for the investment to triple at a 3.9% annual interest rate compounded monthly.
The closest option is d) 30 years.

Answer 2

To calculate how long it takes to triple your money with a 3.9% interest rate compounded monthly, use the compound interest formula. Solve for time t by setting the final amount to triple the principal and applying logarithms to isolate t. The answer will correspond to one of the given multiple-choice options.

Step-by-step explanation:

To determine how long it will take for your money to triple with interest compounded monthly at 3.9%, we can use the formula for compound interest. The formula is A = P(1 + r/n)^(nt), where:

• A is the amount of money accumulated after n years, including interest.
• P is the principal amount (the initial amount of money).
• 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 to triple the initial investment, we can set A to 3*P (where P is the principal), solve for t, and plug in the values: r = 0.039 (interest rate), n = 12 (compounded monthly).

The formula will look like this:

3P = P(1 + 0.039/12)^(12t)

We can divide both sides by P to get:

3 = (1 + 0.039/12)^(12t)

Next, we will use the logarithm to solve for t:

ln(3) = 12t * ln(1 + 0.039/12)

t = ln(3) / (12 * ln(1 + 0.039/12))

Calculating the value for t will give us the number of years required to triple the investment. The answer will fall into one of the provided multiple-choice answers.