Answer :

To determine how long it will take to triple an investment of [tex]$2,500 at an annual interest rate of 4.75%, we need to use the compound interest formula. Let's break down the steps to solve this problem: ### Step 1: Understand the Compound Interest Formula The compound interest formula is: \[ A = P (1 + r)^t \] where: - \(A\) is the amount of money accumulated after \(t\) years, including interest. - \(P\) is the principal amount (initial investment). - \(r\) is the annual interest rate (as a decimal). - \(t\) is the time the money is invested for in years. ### Step 2: Identify Known Values Here, we are given: - The principal \(P = 2500\) dollars. - The annual interest rate \(r = 4.75\%\). - The target amount \(A\), which is three times the principal, so \(A = 3 \times 2500 = 7500\) dollars. The goal is to find \(t\), the number of years it will take for the investment to grow to $[/tex]7,500.

### Step 3: Rearrange the Formula to Solve for [tex]\(t\)[/tex]

To find [tex]\(t\)[/tex], we start with the compound interest formula and solve for [tex]\(t\)[/tex]:

[tex]\[ 7500 = 2500 (1 + 0.0475)^t \][/tex]

First, divide both sides by 2500 to isolate the growth factor [tex]\((1 + r)^t\)[/tex]:

[tex]\[ \frac{7500}{2500} = (1 + 0.0475)^t \][/tex]

Simplify the left side:

[tex]\[ 3 = (1.0475)^t \][/tex]

### Step 4: Use Logarithms to Solve for [tex]\(t\)[/tex]

To solve for [tex]\(t\)[/tex], take the natural logarithm (ln) of both sides:

[tex]\[ \ln(3) = \ln((1.0475)^t) \][/tex]

Using the logarithm power rule [tex]\( \ln(a^b) = b \ln(a) \)[/tex], we get:

[tex]\[ \ln(3) = t \ln(1.0475) \][/tex]

Now, solve for [tex]\(t\)[/tex]:

[tex]\[ t = \frac{\ln(3)}{\ln(1.0475)} \][/tex]

### Step 5: Calculate the Answer

Evaluating the above expression, we find:

[tex]\[ t \approx 23.67 \][/tex]

Therefore, it will take approximately 23.67 years for the $2,500 investment to triple at an annual interest rate of 4.75%.