Student Loan Payment 8532Df

1. **State the problem:** Calculate the required monthly payment for a student loan of $50,000 with an APR of 9% over 20 years. 2. **Formula used:** The monthly payment $M$ for a loan is given by the amortization formula: $$M = P \times \frac{r(1+r)^n}{(1+r)^n - 1}$$ where: - $P$ is the loan principal ($50,000$), - $r$ is the monthly interest rate (APR divided by 12), - $n$ is the total number of payments (years times 12). 3. **Calculate monthly interest rate:** $$r = \frac{9\%}{12} = \frac{0.09}{12} = 0.0075$$ 4. **Calculate total number of payments:** $$n = 20 \times 12 = 240$$ 5. **Substitute values into the formula:** $$M = 50000 \times \frac{0.0075(1+0.0075)^{240}}{(1+0.0075)^{240} - 1}$$ 6. **Calculate $(1 + r)^n$:** $$ (1 + 0.0075)^{240} = 1.0075^{240} \approx 6.022575 $$ 7. **Calculate numerator:** $$0.0075 \times 6.022575 = 0.0451693$$ 8. **Calculate denominator:** $$6.022575 - 1 = 5.022575$$ 9. **Calculate fraction:** $$\frac{0.0451693}{5.022575} \approx 0.008991$$ 10. **Calculate monthly payment:** $$M = 50000 \times 0.008991 = 449.55$$ **Answer:** The required monthly payment is approximately $449.55$.