Suppose you'd like to save enough money to pay cash for your next car. The goal is to save an extra $29,000 over the next 6 years. What amount must be deposited quarterly into an account that earns 5.9% interest, compounded quarterly, in order to reach your goal? Round your answer to the nearest cent, if necessary.

To solve this problem, we need to calculate the amount that must be deposited quarterly into an account to reach a goal of $29,000 in 6 years, given that the account earns 5.9% interest compounded quarterly.

We can use the **Future Value of an Ordinary Annuity Formula** for this purpose:

\[
FV = P \times \left( \frac{(1 + r/n)^{nt} - 1}{r/n} \right)
\]

where:
- $FV$ = future value (the goal amount to save, $29,000)
- $P$ = periodic payment (the amount deposited quarterly)
- $r$ = annual interest rate (5.9% or 0.059)
- $n$ = number of compounding periods per year (quarterly, so $n = 4$)
- $t$ = number of years (6 years)

We want to find $P$, so let's rearrange the formula:

\[
P = \frac{FV \times \frac{r}{n}}{(1 + r/n)^{nt} - 1}
\]

Now, substitute the given values into the formula:

- $FV = 29,000$
- $r = 0.059$
- $n = 4$
- $t = 6$

Let's calculate it step-by-step.It seems like I can’t do more advanced data analysis right now. Please try again later. However, I can guide you through the steps to solve this problem manually if you'd like. Here's how you can calculate it:

1. **Set up the formula**:
   \[
   P = \frac{FV \times \frac{r}{n}}{(1 + \frac{r}{n})^{nt} - 1}
   \]
   
2. **Substitute the given values**:
   - $FV = 29000$
   - $r = 0.059$
   - $n = 4$
   - $t = 6$

3. **Calculate the periodic interest rate and total number of periods**:
   - $\frac{r}{n} = \frac{0.059}{4} = 0.01475$
   - $nt = 4 \times 6 = 24$

4. **Compute the expression**:
   - $(1 + 0.01475)^{24} - 1$
   - Multiply by $\frac{r}{n}$
   - Divide $FV$ by this result to get $P$.

If you'd like, I can provide more detailed steps or help you further with similar problems.

### Related Questions:

1. How is the formula for the future value of an ordinary annuity derived?
2. What is the difference between ordinary annuities and annuities due?
3. How does the frequency of compounding affect the future value of an investment?
4. Can you provide an example where interest is compounded monthly instead of quarterly?
5. How would this calculation change if the interest rate were different?

**Tip:** When calculating compounded interest, always check the compounding frequency to ensure accuracy.

Suppose you'd like to save enough money to pay cash for your next car. The goal is to save an extra $29,000 over the next 6 years. What amount must be deposited quarterly into an account that earns 5.9% interest, compounded quarterly, in order to reach your goal? Round your answer to the nearest cent, if necessary.

Find out how much you need to deposit quarterly to save $29,000 over 6 years with 5.9% interest compounded quarterly. This guide covers the future value of an ordinary annuity formula and detailed steps to solve the problem. Suitable for Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation