\begin{tabular}{|l}\hline You decide to borrow $$ \$ 27,000 $$ to put in a new swimming pool. You obtain a loan for 30 \ years, making equal monthly payments. After how many years will the greatest amount of \ your payment be going toward principal? \ \hline 2 years \ $$ \square 25 $$ years \ 29 years \ 1 year \ \hline\end{tabular}

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To solve the problem, we need to understand how loan amortization works. When you take out a loan, your monthly payments consist of two parts: interest and principal. In the early years of the loan, a larger portion of your payment goes toward interest, and as time goes on, more of your payment goes toward paying down the principal.

### Step 1: Understand the Loan Structure
- **Loan Amount (P)**: $$ 27,000 $$
- **Loan Term (n)**: 30 years
- **Monthly Payments**: The loan will be paid off in equal monthly installments.

### Step 2: Determine the Payment Structure
The monthly payment can be calculated using the formula for an amortizing loan:

$$
M = P \frac{r(1 + r)^N}{(1 + r)^N - 1}
$$

Where:
- $$ M $$ = monthly payment
- $$ P $$ = principal (loan amount)
- $$ r $$ = monthly interest rate (annual rate divided by 12)
- $$ N $$ = total number of payments (loan term in months)

### Step 3: Analyze the Payment Distribution
The amount of each payment that goes toward interest decreases over time, while the amount that goes toward principal increases. The point at which the greatest amount of the payment goes toward principal is typically around the midpoint of the loan term.

### Step 4: Calculate the Midpoint
For a 30-year loan:
- Total payments = $$ 30 \times 12 = 360 $$ months
- Midpoint = $$ \frac{360}{2} = 180 $$ months

This means that after 15 years (or 180 months), the payments will start to shift significantly toward principal.

### Conclusion
Since the question asks after how many years the greatest amount of your payment will be going toward principal, we can conclude that this will occur after approximately 15 years.

However, since the options provided are:
- 2 years
- 25 years
- 29 years
- 1 year

None of these options directly match our calculated midpoint of 15 years. The closest option that reflects a significant shift toward principal payments would be **25 years**, as by this time, the majority of the payment will be going toward the principal.

Thus, the answer is:

**25 years**.
After 25 years, the majority of your monthly payments will go toward the principal.

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