A small business owner contributes $4,000 at the end of each quarter to a retirement account that earns 10% compounded quarterly.

(a) How long will it be until the account is worth at least $150,000? (Round your answer UP to the nearest quarter.)

 ____ quarters

(b) Suppose when the account reaches $150,000, the business owner increases the contributions to $6,000 at the end of each quarter. What will the total value of the account be after 15 more years? (Round your answer to the nearest dollar.)
$ _____

Let's solve this updated problem in two parts, as you described:

### Part (a): Time to Reach $150,000 with Quarterly Contributions of $4,000

We already calculated this earlier. With the quarterly contribution of $4,000 at 10% interest compounded quarterly, the account will reach $150,000 after **27 quarters**.

### Part (b): Total Value of the Account After 15 More Years with Increased Contributions of $6,000

Given:
- New quarterly contribution ($R$) = $6,000
- Quarterly interest rate ($i$) = 2.5% = 0.025
- Number of periods ($n$) = 15 years $\times 4$ quarters per year = 60 quarters
- Initial amount ($P$) = $150,000

We use the same future value formula combining the initial amount compounded and the future value of the new annuity contributions:

\[
FV_{\text{total}} = P \times (1 + i)^n + R \times \frac{(1 + i)^n - 1}{i}
\]

Plugging in the updated values:

\[
FV_{\text{total}} = 150,000 \times (1.025)^{60} + 6,000 \times \frac{(1.025)^{60} - 1}{0.025}
\]

Let's calculate the total future value of the account.The total value of the account after 15 more years, with increased contributions of $6,000 at the end of each quarter, will be **$1,475,918**.

Let me know if you need further clarifications or if you have any questions!

Here are some related questions that might interest you:

1. What if the contributions were made at the beginning of each quarter instead of at the end?
2. How would the total amount change if the interest rate increased or decreased?
3. What would the account value be if contributions stopped after reaching $150,000?
4. How much should the business owner contribute each quarter to reach $2,000,000 in 15 years?
5. What effect would changing the compounding frequency to monthly or annually have on the total value?

**Tip:** Consistently increasing contributions, even by a small amount, can significantly boost the future value of an investment due to compounding effects over time.

Learn how to calculate the future value of a retirement account with compounded interest and increased contributions. Solve for the time to reach $150,000 and determine the total account value after 15 years with increased quarterly contributions. Detailed step-by-step solution provided.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation