The number of​ periods, N​, is the number of years multiplied by 12.​ Therefore,
Number of periods equals 20 times 12 equals 240.
Part 12
​Therefore, with monthly deposits and compounding for 20 ​years:
​Enter:
240
0.66667
0
minus383.33
N
​I/Y
PV
PMT
FV
Solve​ for:
225 comma 790.32
With monthly investments and monthly compounding​ interest, after 20 ​years, Shaylea would have ​$225 comma 790.32.  

This problem involves calculating the future value (FV) of a series of monthly investments using compound interest. Based on the details provided, the variables are defined as follows:

- **N (Number of periods)**: 240 (20 years × 12 months per year)
- **I/Y (Interest rate per period)**: 0.66667% (This is the monthly interest rate, which is likely derived from an annual interest rate divided by 12)
- **PV (Present Value)**: 0 (Assumes no initial lump-sum investment)
- **PMT (Payment per period)**: -383.33 (Monthly payment or deposit, treated as negative since it represents an outflow of money)
- **FV (Future Value)**: $225,790.32 (This is the final value after 20 years of monthly compounding)

The equation being used is the standard formula for the future value of an annuity:

\[
FV = PMT \times \frac{(1 + r)^N - 1}{r}
\]

Where:
- $ PMT $ is the monthly payment (-383.33)
- $ r $ is the monthly interest rate (0.0066667)
- $ N $ is the total number of periods (240)

Let's walk through the calculation step by step:

1. **Substitute values into the formula**:

\[
FV = -383.33 \times \frac{(1 + 0.0066667)^{240} - 1}{0.0066667}
\]

2. **Calculate the result**.

Let me calculate this for you:The calculated future value (FV) is approximately $-225,790.32$, which matches the given value of $225,790.32$ (the negative sign simply indicates that the payment is an outflow). This means that after 20 years of monthly deposits and compounding at the given rate, Shaylea would accumulate $225,790.32.

Would you like more details or have any other questions?

Here are 5 related questions you might want to explore:
1. How is the monthly interest rate derived from an annual interest rate?
2. What happens if the interest rate changes midway through the 20 years?
3. How would the future value change if the monthly payment was increased?
4. What is the difference between monthly compounding and yearly compounding?
5. How can inflation affect the real value of $225,790.32 after 20 years?

**Tip**: Always ensure the interest rate matches the compounding period (monthly deposits should use monthly interest rates).

The number of periods, N, is the number of years multiplied by 12. Therefore, Number of periods equals 20 times 12 equals 240. Therefore, with monthly deposits and compounding for 20 years: Enter: 240 (N) 0.66667 (I/Y) 0 (PV) -383.33 (PMT) FV Solve for: 225,790.32. With monthly investments and monthly compounding interest, after 20 years, Shaylea would have $225,790.32.

Explore how to calculate the future value of monthly investments compounded over 20 years using the annuity formula. This problem covers essential financial mathematics concepts, including compound interest and the time value of money. Suitable for students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation