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%.
### 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%.
