Suppose you want to have $700,000 for retirement in 20 years. Your account earns 8% interest.

a) How much would you need to deposit in the account each month?

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b) How much interest will you earn?


We can solve this problem using the formula for the future value of an annuity, which accounts for regular monthly deposits into an account earning compound interest.

The formula for the future value of an annuity is:

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

Where:
- $FV$ is the future value of the annuity (the amount you want to have, \$700,000),
- $P$ is the monthly deposit (what we're solving for),
- $r$ is the annual interest rate (8% or 0.08),
- $n$ is the number of times interest is compounded per year (monthly, so $n = 12$),
- $t$ is the number of years (20 years).

### a) Finding the monthly deposit $P$:

Rearrange the formula to solve for $P$:

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

Now, let's plug in the values:
- $FV = 700,000$,
- $r = 0.08$,
- $n = 12$,
- $t = 20$.

Let me calculate $P$.The required monthly deposit is approximately \$1,188.41.

### b) How much interest will you earn?

To find the total interest earned, we first calculate the total amount deposited and then subtract this from the future value.

The total amount deposited over 20 years is:

\[
\text{Total Deposits} = P \times n \times t
\]

Let me calculate the total deposits and the interest earned.The total amount deposited over 20 years is approximately \$285,219.32. The interest earned is approximately \$414,780.68.

### Summary:
- a) Monthly deposit: \$1,188.41
- b) Total interest earned: \$414,780.68

Would you like more details or have any questions about these calculations?

Here are some related questions for further exploration:
1. What if the interest rate was 6% instead of 8%?
2. How would the monthly deposit change if the goal was \$1,000,000 in 20 years?
3. What if the account compounded interest quarterly instead of monthly?
4. How would the results change if you only deposited for 15 years instead of 20?
5. What happens if you start with an initial deposit?

**Tip:** Increasing the frequency of compounding (e.g., from annual to monthly) typically increases the final value due to more frequent interest application.

Suppose you want to have $700,000 for retirement in 20 years. Your account earns 8% interest.

a) How much would you need to deposit in the account each month?

b) How much interest will you earn?

Learn how to calculate the monthly deposit needed to reach $700,000 in 20 years with an 8% interest rate. This example involves using the future value of an annuity formula and covers topics such as compound interest and total interest earned. Suitable for high school students and adults planning for retirement.

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