Sure, let's solve this question step-by-step.
1. Understand the Problem:
- You start with a principal amount of [tex]$100,000.
- The interest rate is 8% per year.
- We want to find out how long it will take for this amount to double.
2. Identify the Formula:
- For simple interest, the formula to find the amount after a certain time is:
\[
A = P(1 + rt)
\]
where:
- \(A\) is the final amount,
- \(P\) is the principal amount (initial investment),
- \(r\) is the interest rate (expressed as a decimal),
- \(t\) is the time in years.
3. Rearrange the Formula to Solve for Time (t):
- We know that we want to double our money, so \(A = 2P\).
- Substitute \(A\) into the formula:
\[
2P = P(1 + rt)
\]
- Simplify the equation by dividing both sides by \(P\):
\[
2 = 1 + rt
\]
- Rearrange to solve for \(t\):
\[
t = \frac{2 - 1}{r}
\]
\[
t = \frac{1}{r}
\]
4. Substitute the Interest Rate:
- The interest rate is 8%, which is 0.08 as a decimal.
- Substitute \(r = 0.08\) into the equation:
\[
t = \frac{1}{0.08}
\]
5. Calculate \(t\):
- Perform the division:
\[
t = \frac{1}{0.08} = 12.5 \text{ years}
\]
6. Final Answer:
- It will take 12.5 years for $[/tex]100,000 to double at a simple interest rate of 8%.
So, the detailed solution shows that it will take 12.5 years for your principal of \$100,000 to double at an 8% simple interest rate.
