At age 25​, someone sets up an IRA​ (individual retirement​ account) with an APR of 5​%. At the end of each month he deposits ​$20 in the account. How much will the IRA contain when he retires at age​ 65?  Compare that amount to the total deposits made over the time period. 
After retirement the IRA will contain ​$____.  ​(Do not round until the final answer. Then round to the nearest cent as​ needed.)

This is a problem of future value of an ordinary annuity, where monthly deposits are made into an account that earns compound interest. The formula to calculate the future value $ A $ of an ordinary annuity is:

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

Where:
- $ P $ is the monthly deposit (in this case, $20),
- $ r $ is the annual interest rate (5%, or 0.05),
- $ n $ is the number of times interest is compounded per year (monthly, so 12),
- $ t $ is the number of years the deposits are made (from age 25 to age 65, which is 40 years).

### Step 1: Plug in the values into the formula.
We know:
- $ P = 20 $,
- $ r = 0.05 $,
- $ n = 12 $,
- $ t = 40 $.

Now we calculate the future value of the IRA when the person retires.

\[
A = 20 \times \frac{\left(1 + \frac{0.05}{12}\right)^{12 \times 40} - 1}{\frac{0.05}{12}}
\]

I'll compute this step-by-step.The IRA will contain approximately **$30,520.40** when the person retires at age 65.

### Step 2: Compare with total deposits made.
The total amount deposited is simply the monthly deposit multiplied by the number of months over 40 years:

\[
\text{Total deposits} = 20 \times 12 \times 40 = 9,600
\]

The total deposits made are **$9,600**.

### Comparison:
- The account will grow to **$30,520.40**.
- Total deposits were **$9,600**.
  
The amount in the account at retirement is more than 3 times the total deposits made, thanks to the effect of compound interest.

Would you like more details, or have any questions? Here are 5 related questions:

1. What would happen if the interest rate was increased to 6%?
2. How would the future value change if monthly deposits were $50 instead of $20?
3. What if the deposits were made annually instead of monthly?
4. How would starting 5 years earlier (at age 20) affect the future value?
5. What happens if you deposit an additional lump sum at the beginning?

**Tip:** The earlier you start saving and the more frequently you contribute, the more compound interest can work in your favor!

At age 25​, someone sets up an IRA​ (individual retirement​ account) with an APR of 5​%. At the end of each month he deposits ​$20 in the account. How much will the IRA contain when he retires at age​ 65? Compare that amount to the total deposits made over the time period.

Learn how to calculate the future value of an IRA with 5% APR, monthly deposits of $20, and a 40-year growth period. This problem uses compound interest and the future value of an annuity formula, providing key insights into retirement planning. Suitable for high school students in Grades 10-12.

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