You take out an installment loan to purchase a time-share costing \( \$ 18,000 \). You make a down payment of \( \$ 2,700 \) and finance the balance by making monthly payments of \( \$ 762 \) for 24 months. Use Table \( 13-1 \) to find the APR. Select one: a. \( 17.50 \% \) O. \( 17.75 \% \) c. \( 18.00 \% \) d. \( 18.25 \% \) Clear my choice

We begin by noting that the time‐share costs \(\$18{,}000\) and a down payment of \(\$2{,}700\) is made. Thus the amount financed (present value) is \[ P = 18000 - 2700 = 15300. \] The monthly payment is given as \(\$762\) over 24 months. The standard formula for an installment (annuity) payment is \[ A = \frac{P \cdot r(1+r)^n}{(1+r)^n-1}, \] where \(A\) is the monthly payment, \(P=15300\) is the financed amount, \(n=24\) is the number of months, and \(r\) is the monthly interest rate. Our equation becomes \[ 762 = \frac{15300\cdot r(1+r)^{24}}{(1+r)^{24}-1}. \] The goal is to solve for \(r\) and then convert it to an Annual Percentage Rate (APR). Since the APR (when compounded monthly) is given by \[ \text{APR} = 12r, \] we need to find the value of \(r\) that satisfies the above equation. This equation is nonlinear in \(r\), so we use trial and error (or a financial calculator/table) to pinpoint \(r\). Let’s test an APR of \(18\%\). Then the monthly rate is \[ r=\frac{0.18}{12}=0.015. \] Calculate \((1+r)^{24}\): \[ (1.015)^{24} \approx 1.429. \] Then, the payment as computed by the formula would be \[ A = \frac{15300\cdot 0.015\cdot 1.429}{1.429-1} = \frac{15300\cdot 0.021435}{0.429} \approx \frac{327.95}{0.429}\approx 764.6. \] This payment is slightly higher than \(\$762\), indicating that with a \(18\%\) APR the payment overshoots a bit. Now try an APR of \(17.75\%\). The monthly rate is \[ r=\frac{0.1775}{12}\approx 0.01479. \] Now, compute \((1+r)^{24}\): \[ (1.01479)^{24} \approx 1.422. \] The denominator then is \[ 1.422-1=0.422. \] The numerator is \[ 15300 \cdot 0.01479 \cdot 1.422 \approx 15300 \cdot 0.02102 \approx 321.9. \] Thus, the payment becomes \[ A \approx \frac{321.9}{0.422} \approx 762.3. \] This value is extremely close to \(\$762\), confirming that the APR is about \(17.75\%\). Therefore, the answer is \( \boxed{17.75\%} \).