Alright, let's work through this problem step-by-step:
1. Determine the total period in months for which Jason is planning to save:
- Jason wants to save for [tex]\(5\)[/tex] years.
- There are 12 months in a year.
- So, the total period in months is calculated as:
[tex]\[
\text{Total months} = 5 \times 12 = 60 \text{ months}
\][/tex]
2. Account for the months Jason missed:
- Jason starts saving after a delay of 3 months.
- Additionally, he misses the last 3 months of payments.
- Therefore, the total months missed is:
[tex]\[
\text{Missed months} = 3 + 3 = 6 \text{ months}
\][/tex]
- The effective saving period in months will then be:
[tex]\[
\text{Effective months} = 60 - 6 = 54 \text{ months}
\][/tex]
3. Monthly savings:
- Let's assume Jason saves [tex]$1 per month.
4. Interest rate:
- The annual interest rate is 9%.
- To find the monthly interest rate, we need to divide the annual interest rate by 12, as it's compounded monthly. Hence:
\[
\text{Monthly interest rate} = \frac{9\%}{12} = 0.75\% = 0.0075
\]
5. Future value of a series:
- The future value formula for a series of monthly savings considering a monthly compounded interest rate is:
\[
FV = P \times \left( \frac{(1 + r)^n - 1}{r} \right)
\]
Where:
- \( FV \) is the future value.
- \( P \) is the monthly saving amount ($[/tex]1 in this case).
- [tex]\( r \)[/tex] is the monthly interest rate (0.0075).
- [tex]\( n \)[/tex] is the number of effective saving months (54).
6. Calculate the total accumulation:
- Plugging in the values into the future value formula:
[tex]\[
FV = 1 \times \left( \frac{(1 + 0.0075)^{54} - 1}{0.0075} \right)
\][/tex]
7. Compute the final result:
- The result from the calculation is:
[tex]\[
FV \approx 66.27179562469507
\][/tex]
So, after saving [tex]$1 per month effectively over 54 months with a monthly interest rate of 0.75%, Jason will have approximately $[/tex]66.27 at the end of 5 years, taking into account his delays and missed payments.
