Luis has $180,000 in his retirement account at his present company. Because he is assuming a position with another company, Luis is planning to "roll over" his assets to a new account. Luis also plans to put $2000/quarter into the new account until his retirement 20 years from now. If the new account earns interest at the rate of 2.5%/year compounded quarterly, how much will Luis have in his account at the time of his retirement? Hint: Use the compound interest formula and the annuity formula. (Round your answer to the nearest cent.)

To determine how much Luis will have in his account at the time of his retirement, we need to take into account two components: 

1. The initial $180,000 that is rolled over into the new account, which will grow based on compound interest.
2. The regular quarterly contributions of $2,000, which will also accumulate interest.

### Step 1: Compound Interest for the Initial Deposit
For the initial $180,000 that Luis rolls over, we can use the **compound interest formula**:

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

Where:
- $A$ is the amount in the account after interest.
- $P = 180,000$ (the initial principal).
- $r = 0.025$ (annual interest rate of 2.5%).
- $n = 4$ (since the interest is compounded quarterly).
- $t = 20$ years (time until retirement).

Substitute the values:

\[
A = 180,000 \left(1 + \frac{0.025}{4}\right)^{4 \times 20}
\]

\[
A = 180,000 \left(1 + 0.00625\right)^{80}
\]

\[
A = 180,000 (1.00625)^{80}
\]

Now, we compute $ (1.00625)^{80} $ and multiply by 180,000.

\[
A = 180,000 \times 1.568568
\]

\[
A = 282,342.24
\]

So, the amount from the initial deposit at retirement will be **$282,342.24**.

### Step 2: Future Value of Quarterly Contributions (Annuity Formula)
Next, we calculate the future value of the quarterly contributions using the **future value of an annuity formula**:

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

Where:
- $FV$ is the future value of the annuity (quarterly contributions).
- $P = 2,000$ (quarterly contribution).
- $r = 0.025$ (annual interest rate).
- $n = 4$ (compounded quarterly).
- $t = 20$ years (time until retirement).

Substitute the values:

\[
FV = 2,000 \times \frac{\left(1 + \frac{0.025}{4}\right)^{4 \times 20} - 1}{\frac{0.025}{4}}
\]

\[
FV = 2,000 \times \frac{(1.00625)^{80} - 1}{0.00625}
\]

From earlier, we know $ (1.00625)^{80} = 1.568568 $, so:

\[
FV = 2,000 \times \frac{1.568568 - 1}{0.00625}
\]

\[
FV = 2,000 \times \frac{0.568568}{0.00625}
\]

\[
FV = 2,000 \times 90.971
\]

\[
FV = 181,942.37
\]

So, the future value of the quarterly contributions is **$181,942.37**.

### Step 3: Total Amount at Retirement
Finally, to find the total amount in Luis's retirement account, we add the amount from the initial deposit and the future value of the quarterly contributions:

\[
\text{Total} = 282,342.24 + 181,942.37 = 464,284.61
\]

Thus, the total amount Luis will have in his account at the time of his retirement is **$464,284.61**.

---

Would you like more details on the calculation or have any further questions?

### 5 Related Questions:
1. How would the total change if the interest rate were 3% instead of 2.5%?
2. What happens to the total if Luis contributes $2,500 per quarter instead of $2,000?
3. How much will Luis have in 10 years instead of 20 years?
4. What if Luis decides to stop the quarterly contributions after 10 years? How much would he accumulate?
5. How would the calculation change if the interest were compounded monthly instead of quarterly?

### Tip:
When working with compound interest problems, ensure you adjust the number of compounding periods ($n$) and the interest rate per period carefully for accurate results!

Luis has $180,000 in his retirement account at his present company. Because he is assuming a position with another company, Luis is planning to 'roll over' his assets to a new account. Luis also plans to put $2000/quarter into the new account until his retirement 20 years from now. If the new account earns interest at the rate of 2.5%/year compounded quarterly, how much will Luis have in his account at the time of his retirement?

Learn how to calculate the total retirement savings with compound interest and quarterly contributions. Using a $180,000 initial deposit, 2.5% annual interest compounded quarterly, and $2,000 quarterly contributions, this problem shows how to determine the future value over 20 years. Suitable for high school students studying finance and compound interest calculations.

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