1. **Problem statement:** A man borrows 5400 and repays 300 per month plus 1.5% interest on the unpaid balance each month. Find the total cost of the loan over 18 months. 2. **Understanding the problem:** Each month, the unpaid balance accrues 1.5% interest, then the man pays 300 to reduce the balance. We need to find the total amount paid over 18 months. 3. **Formula and approach:** Let $B_n$ be the balance after $n$ months. Initially, $B_0=5400$. Each month: $$B_n = B_{n-1} \times (1 + 0.015) - 300$$ We iterate this for $n=1$ to $18$. 4. **Calculations:** Calculate $B_1$: $$B_1 = 5400 \times 1.015 - 300 = 5481 - 300 = 5181$$ Calculate $B_2$: $$B_2 = 5181 \times 1.015 - 300 = 5458.715 - 300 = 5158.715$$ Continue this process up to $B_{18}$. 5. **Summation of payments:** The man pays 300 each month for 18 months, so total principal payments = $300 \times 18 = 5400$. 6. **Interest payments:** The total cost of the loan is the sum of all payments minus the original loan amount. Since the loan is fully repaid after 18 months, total payments = $300 \times 18 = 5400$ plus the interest paid monthly. But since the balance reduces each month, the interest is included in the monthly payments. 7. **Using the formula for the balance after $n$ months in a loan with fixed payments and interest rate:** $$B_n = P \times (1 + r)^n - PMT \times \frac{(1 + r)^n - 1}{r}$$ Where: - $P=5400$ (initial principal) - $r=0.015$ (monthly interest rate) - $PMT=300$ (monthly payment) - $n=18$ Calculate: $$B_{18} = 5400 \times (1.015)^{18} - 300 \times \frac{(1.015)^{18} - 1}{0.015}$$ Calculate $(1.015)^{18}$: $$ (1.015)^{18} \approx 1.30477$$ Calculate numerator: $$1.30477 - 1 = 0.30477$$ Calculate denominator: $$0.015$$ Calculate fraction: $$\frac{0.30477}{0.015} = 20.318$$ Calculate $B_{18}$: $$B_{18} = 5400 \times 1.30477 - 300 \times 20.318 = 7045.76 - 6095.4 = 950.36$$ Since the balance is not zero, the loan is not fully repaid after 18 months with these payments. 8. **Total amount paid:** Total payments = $300 \times 18 = 5400$ Remaining balance = 950.36 Total cost = payments + remaining balance = $5400 + 950.36 = 6350.36$ 9. **Answer:** The total cost of the loan over 18 months is approximately 6350.36.