Assume that your parents wanted to have 

​$120 comma 000120,000 

saved for college by your 

1818th 

birthday and they started saving on your first birthday. They saved the same amount each year on your birthday and earned 

9 %9% 

per year on their investments. 

**a.** How much would they have to save each year to reach their​ goal? 

**b.** If they think you will take five years instead of four to graduate and decide to have 

​$150 comma 000150,000 

saved just in​ case, how much would they have to save each year to reach their new​ goal? 

Question content area bottom

Part 1

**a.** How much would they have to save each year to reach their​ goal? 

To reach the goal of 

​$120 comma 000120,000​, 

the amount your parents have to save each year is 

​$enter your response here. 

 ​(Round to the nearest​ cent.)

To solve this problem, we'll apply the formula for the future value of an ordinary annuity. This formula helps us determine the amount your parents need to save each year to reach their savings goal, given a fixed annual interest rate and a fixed period.

### Formula for Future Value of an Ordinary Annuity

The future value $ FV $ of an ordinary annuity (where payments are made at the end of each period) is given by:

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

where:
- $ FV $ is the future value, which is the goal amount ($120,000 for part a, $150,000 for part b).
- $ P $ is the annual payment (the amount saved each year).
- $ r $ is the annual interest rate (9% or 0.09 in decimal form).
- $ n $ is the number of years the money is saved (17 years, since saving starts on the 1st birthday and ends on the 18th birthday).

### Solving for Annual Savings (P)

We need to rearrange the formula to solve for $ P $:

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

#### Part (a): Calculate the Annual Savings to Reach $120,000

Let's substitute the values:

- $ FV = 120,000 $
- $ r = 0.09 $
- $ n = 17 $

\[
P = \frac{120,000 \times 0.09}{(1 + 0.09)^{17} - 1}
\]

Now, let's calculate the value.The amount your parents would have to save each year to reach the goal of $120,000 by your 18th birthday is **$3,245.55** (rounded to the nearest cent).

### Part (b): Calculate the Annual Savings to Reach $150,000

For the new goal of $150,000, we use the same formula with updated values:

- $ FV = 150,000 $
- $ r = 0.09 $
- $ n = 17 $

\[
P = \frac{150,000 \times 0.09}{(1 + 0.09)^{17} - 1}
\]

Let's calculate this value now.It seems like I can’t do more advanced data analysis right now. Please try again later.

However, you can still calculate the amount manually using the formula provided earlier:

\[
P = \frac{150,000 \times 0.09}{(1 + 0.09)^{17} - 1}
\]

Feel free to ask any other questions, and I'll be happy to help! 

### Related Questions:
1. What if the interest rate changes to 8%? How does that affect the annual savings required?
2. How would the amount change if the savings period was extended to 20 years instead of 17?
3. What is the total amount of interest earned by the end of 18 years with the current savings plan?
4. How would the calculations change if the savings were made at the beginning of each year?
5. What is the future value if your parents saved $3,500 each year instead?

**Tip:** Always double-check your formulas and inputs to ensure accurate financial planning.

Assume that your parents wanted to have $120,000 saved for college by your 18th birthday and they started saving on your first birthday. They saved the same amount each year on your birthday and earned 9% per year on their investments. a. How much would they have to save each year to reach their​ goal? b. If they think you will take five years instead of four to graduate and decide to have $150,000 saved just in case, how much would they have to save each year to reach their new goal?

Determine how much parents need to save annually to reach a $120,000 college fund by the 18th birthday with a 9% interest rate. Learn how the goal changes to $150,000 if more funds are needed for five years of education. Step-by-step solution included.

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