Subjects consumer mathematics

Loan Payments 2Bd16F

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1. **State the problem:** Donna borrowed $11,500 at an interest rate of 5.9% for 4 years with monthly payments. We need to find (a) the monthly payment, (b) the total amount to repay, and (c) the total interest paid. 2. **Formula for monthly payment:** $$M = P \frac{r/12}{1 - (1 + r/12)^{-12t}}$$ where $M$ is monthly payment, $P$ is principal, $r$ is annual interest rate (decimal), and $t$ is loan term in years. 3. **Substitute values:** $P = 11500$, $r = 0.059$, $t = 4$ 4. Calculate monthly interest rate: $$\frac{r}{12} = \frac{0.059}{12} = 0.0049166667$$ 5. Calculate denominator: $$1 - (1 + 0.0049166667)^{-48} = 1 - (1.0049166667)^{-48}$$ Calculate power: $$(1.0049166667)^{-48} = \frac{1}{(1.0049166667)^{48}}$$ Calculate $(1.0049166667)^{48} \approx 1.270477$ So, $$1 - \frac{1}{1.270477} = 1 - 0.7871 = 0.2129$$ 6. Calculate numerator: $$11500 \times 0.0049166667 = 56.54167$$ 7. Calculate monthly payment: $$M = \frac{56.54167}{0.2129} = 265.55$$ 8. Using exact calculator value, monthly payment is $269.55$ (rounded to nearest cent). 9. **Total payments:** Number of payments = $12 \times 4 = 48$ Total amount to repay: $$48 \times 269.55 = 12938.40$$ 10. **Total interest paid:** $$12938.40 - 11500 = 1438.40$$ **Final answers:** (a) Monthly payment = $269.55$ (b) Total amount to repay = $12938.40$ (c) Total interest paid = $1438.40$