Sure, let's solve this step-by-step.
First, we understand that [tex]$5,000 was invested five years ago. We're asked to find out how much this equivalent amount is today if the money earned 7% interest compounded monthly, but only over the last four years. There's an implicit assumption that no interest was earned in the first year, and the compounding starts from the second year through the fifth year.
Here's the step-by-step breakdown:
1. Initial amount:
- Principal (P) is $[/tex]5,000.
2. Interest rate:
- The annual interest rate (r) is 7%, which can be written as 0.07 in decimal form.
3. Number of compounding periods per year:
- The interest is compounded monthly (n), which means 12 times a year.
4. Number of years the interest was compounded:
- The compounding happens over 4 years (t), not 5 years as the question specifies the last four years.
The formula used for compound interest is:
[tex]\[ A = P \left(1 + \frac{r}{n}\right)^{n \cdot t} \][/tex]
By plugging the known values into the formula, we have:
[tex]\[ A = 5000 \left(1 + \frac{0.07}{12}\right)^{12 \cdot 4} \][/tex]
After calculating:
- The equivalent amount today (after four years of compounding monthly at 7%) is approximately [tex]$6,610.27.
- Additionally, if we extend the compounding to the full five years at the same rate and compounding frequency, it would be approximately $[/tex]7,088.13.
Therefore, the amount today that is equivalent to [tex]$5,000 five years ago, given the 7% interest rate compounded monthly over the last four years, is approximately $[/tex]6,610.27. If the compounding had continued for the full five years, the amount would be approximately $7,088.13.
