A retired couple plans to supplement their Social Security with interest earned by a 

​$180 comma 000180,000 

retirement fund. 

a. If the fund compounds interest monthly at an APR of 

7.57.5​%, 

which the couple takes out at the end of the year for spending in the next​ year, how much average interest is generated each​ month? 

b. Suppose the APR drops to 

66​%. 

What is the resulting average interest payment each​ month? 

c. Estimate the APR needed to generate 

​$14001400 

each month in average interest.

Question content area bottom

Part 1

a. The average interest generated each month is about 

​$enter your response here.

​(Do not round until the final answer. Then round to the nearest dollar as​ needed.)

To solve these parts, we will use the formula for compound interest, which takes into account monthly compounding. The formula is:

\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]

Where:
- $A$ is the amount of money accumulated after $t$ years, including interest.
- $P$ is the principal amount (initial investment).
- $r$ is the annual interest rate (in decimal form).
- $n$ is the number of times interest is compounded per year (monthly means $n = 12$).
- $t$ is the time the money is invested or borrowed for, in years.

### Part (a)
For this part, we need to calculate the interest generated each month for an APR of 7.5%, compounded monthly, from a principal of $180,000.

Given:
- $P = 180,000$
- $r = 7.5\%$ or 0.075
- $n = 12$ (since compounding is monthly)
- $t = 1$ year

The interest earned each month is the difference between the accumulated amount after one year and the principal, divided by 12 months.

Let's calculate this step-by-step.

### Part (b)
For part (b), the only change is that the APR is now 6%. So we will repeat the calculation using $r = 6\%$ or 0.06.

### Part (c)
To estimate the APR needed to generate $1400 each month in average interest, we need to find the interest rate that gives a monthly interest payment of $1,400. We will use the compound interest formula and solve for $r$.

Let me calculate all these values.### Part (a):
The average interest generated each month with a 7.5% APR is approximately **$1,164**.

### Part (b):
The average interest generated each month with a 6% APR is approximately **$925**.

### Part (c):
To estimate the APR needed to generate $1,400 each month, we will solve for the interest rate using the compound interest formula. Let me calculate this next.It seems like I can’t do more advanced data analysis right now. Please try again later. 

Would you like to proceed with other parts of the question or ask about something else? Here are some related questions you could explore:

1. How does compounding frequency affect interest generation?
2. What are the differences between simple and compound interest?
3. How do changing interest rates impact long-term investments?
4. How can you estimate future value for different APR scenarios?
5. What are some strategies to maximize retirement fund growth?

**Tip:** Understanding how interest compounds can help you make better financial decisions, especially with long-term savings.

A retired couple plans to supplement their Social Security with interest earned by a $180,000 retirement fund. a. If the fund compounds interest monthly at an APR of 7.5%, which the couple takes out at the end of the year for spending in the next year, how much average interest is generated each month? b. Suppose the APR drops to 6%. What is the resulting average interest payment each month? c. Estimate the APR needed to generate $1,400 each month in average interest.

Calculate monthly interest for a $180,000 retirement fund compounded monthly at an APR of 7.5%, as well as at a reduced APR of 6%. Learn how to estimate the APR needed to generate $1,400 in monthly interest. This problem involves compound interest, algebra, and interest rate calculations, suitable for advanced high school or college-level students.

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