Kayla starts an IRA (Individual Retirement Account) at the age of 22

to save for retirement. She deposits $450 each month. The IRA has an average annual interest rate of 5% compounded monthly. How much money will she have saved when she retires at the age of 65? Round your answer to the nearest cent, if necessary

To solve this problem, we will use the **future value formula for compound interest with regular monthly deposits**, which is:

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

Where:
- $A$ is the future value of the investment
- $P$ is the monthly deposit
- $r$ is the annual interest rate (in decimal form)
- $n$ is the number of times the interest is compounded per year (monthly compounding, so $n = 12$)
- $t$ is the number of years the money is invested

### Step 1: Identify the Variables
- $P = 450$ (monthly deposit)
- $r = 5\%$ = 0.05 (as a decimal)
- $n = 12$ (compounded monthly)
- Kayla invests from age 22 to age 65, so $t = 65 - 22 = 43$ years

### Step 2: Plug the Values into the Formula
\[
A = 450 \times \frac{(1 + 0.05/12)^{12 \times 43} - 1}{0.05/12}
\]

Now, let's calculate this step-by-step.

### Step 3: Perform the Calculation
First, calculate the value inside the parentheses:

\[
1 + \frac{0.05}{12} = 1.0041667
\]

Next, calculate the exponent:

\[
12 \times 43 = 516
\]

Now, raise $1.0041667$ to the power of $516$:

\[
(1.0041667)^{516} = 7.6974
\]

Subtract 1:

\[
7.6974 - 1 = 6.6974
\]

Now, divide by $0.05/12 = 0.0041667$:

\[
\frac{6.6974}{0.0041667} = 1606.84
\]

Finally, multiply by the monthly deposit of $450:

\[
A = 450 \times 1606.84 = 723078.00
\]

### Step 4: Conclusion
At the age of 65, Kayla will have saved approximately **$723,078.00** in her IRA.

Would you like more information on how compound interest works over time?

Here are some related questions:
1. How much would she save if she deposits $500 per month instead of $450?
2. How does the total amount change if the interest rate is 6% instead of 5%?
3. What would be the result if she started saving at age 30 instead of 22?
4. How much interest will she earn in total over the 43 years?
5. How does the compounding frequency (e.g., daily, quarterly) affect the future value?

**Tip:** Starting early with regular contributions to a retirement account maximizes the effect of compound interest over time.

Kayla starts an IRA (Individual Retirement Account) at the age of 22 to save for retirement. She deposits $450 each month. The IRA has an average annual interest rate of 5% compounded monthly. How much money will she have saved when she retires at the age of 65? Round your answer to the nearest cent, if necessary.

Learn how to calculate the future value of Kayla's IRA savings with monthly deposits of $450 at a 5% annual interest rate compounded monthly. This detailed solution shows step-by-step calculations of compound interest over 43 years, resulting in a total savings of approximately $723,078. Perfect for high school students studying compound interest and exponential growth.

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