ou would like to have $650,000 when you retire in 25 years. How much should you invest each quarter if you can earn a rate of 7.5% compounded quarterly?

a) How much should you deposit each quarter?

$Incorrect

b) How much total money will you put into the account?

$Incorrect

c) How much total interest will you earn?


To solve this problem, we need to calculate the quarterly deposit amount using the **future value of an ordinary annuity formula**, where regular deposits are made at the end of each quarter. The formula for future value is:

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

Where:
- $FV$ is the future value of the annuity (in this case, \$650,000).
- $P$ is the quarterly deposit.
- $r$ is the interest rate per period (quarterly interest rate).
- $n$ is the total number of periods (total number of quarters over 25 years).

### a) How much should you deposit each quarter?

First, we need to calculate $r$ and $n$:
- The annual interest rate is 7.5%, but since it's compounded quarterly, we divide by 4:
  \[
  r = \frac{7.5\%}{4} = \frac{0.075}{4} = 0.01875
  \]
- The number of quarters in 25 years is:
  \[
  n = 25 \times 4 = 100
  \]

Now, plug in the values into the future value formula to solve for $P$:

\[
650,000 = P \times \frac{{(1 + 0.01875)^{100} - 1}}{0.01875}
\]

Now I will calculate $P$.The quarterly deposit required is approximately **\$2,253.36**.

### b) How much total money will you put into the account?

To find this, we multiply the quarterly deposit by the total number of quarters:
\[
\text{Total Deposits} = P \times n = 2253.36 \times 100
\]

Let me calculate that.The total amount of money you will put into the account is approximately **\$225,336.19**.

### c) How much total interest will you earn?

To find the total interest earned, we subtract the total deposits from the future value:

\[
\text{Total Interest} = FV - \text{Total Deposits} = 650,000 - 225,336.19
\]

Let me calculate that.The total interest earned will be approximately **\$424,663.81**.

### Summary:
- a) Quarterly deposit: **\$2,253.36**
- b) Total amount deposited: **\$225,336.19**
- c) Total interest earned: **\$424,663.81**

Would you like any further details or clarifications?

Here are some related questions:
1. What happens if the interest rate changes to 8%?
2. How would the result differ if you contributed monthly instead of quarterly?
3. What if the investment term was 30 years instead of 25?
4. How does changing the compounding frequency affect the future value?
5. What would be the effect of making the deposits at the beginning of each quarter?

**Tip:** Small increases in the interest rate can result in significantly higher returns over long periods due to compound interest.

You would like to have $650,000 when you retire in 25 years. How much should you invest each quarter if you can earn a rate of 7.5% compounded quarterly?

 a) How much should you deposit each quarter?
 b) How much total money will you put into the account?
 c) How much total interest will you earn?

Learn how to calculate the quarterly investment needed to reach $650,000 in 25 years at a 7.5% annual interest rate compounded quarterly. Detailed explanation of future value of annuities, including how much total money you'll invest and the total interest earned. Suitable for advanced high school students.

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