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To solve this problem, we can use the formula for the future value of an ordinary annuity, as you are making monthly payments into an account that earns compound interest:

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

Where:
- $FV$ is the future value of the annuity (the total amount in the IRA when you retire).
- $P$ is the monthly payment (in this case, $50).
- $r$ is the monthly interest rate (annual rate of 5% divided by 12 months, so $r = \frac{0.05}{12} = 0.004167$).
- $n$ is the total number of payments (you are saving from age 25 to 65, so for 40 years, with monthly contributions, $n = 40 \times 12 = 480$ payments).

### Step 1: Calculate the future value (a)
Substitute the values into the future value formula:

\[
FV = 50 \times \frac{{(1 + 0.004167)^{480} - 1}}{0.004167}
\]

Now we calculate step by step:

\[
FV = 50 \times \frac{{(1.004167)^{480} - 1}}{0.004167}
\]

\[
FV = 50 \times \frac{{7.106698 - 1}}{0.004167}
\]

\[
FV = 50 \times \frac{6.106698}{0.004167}
\]

\[
FV = 50 \times 1465.484
\]

\[
FV \approx 73,274.22
\]

So, at retirement, you will have approximately **$73,274.22** in the IRA.

### Step 2: Find the interest (b)
The total interest earned is the difference between the future value and the total amount of money deposited.

The total amount deposited is:
\[
50 \times 480 = 24,000
\]

The interest earned is:
\[
73,274.22 - 24,000 = 49,274.22
\]

So, the interest earned is approximately **$49,274.22**.

---

Would you like more details or have any questions?  
Here are 5 related questions you might want to explore:

1. What happens to the future value if the interest rate increases to 6%?
2. How would the outcome change if you deposit $100 instead of $50 per month?
3. What if the interest is compounded quarterly instead of monthly?
4. How does delaying the start of contributions by 5 years affect the final amount?
5. What if you increase the contribution by 3% each year to adjust for inflation?

**Tip**: The power of compound interest comes from time. The earlier you start saving, the more your money grows!

Suppose that at age 25, you decide to save for retirement by depositing $50 at the end of each month in an IRA that pays 5% interest compounded monthly. How much will you have from the IRA when you retire at age 65? Find the interest.

Learn how to calculate the future value of an IRA when making $50 monthly deposits for 40 years with a 5% interest rate compounded monthly. Understand how to find the total interest earned and the future value at retirement.

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