1. **Problem Statement:** Sofea borrows a total of $15000$ from two lenders, A and B, at simple interest rates of $5\%$ and $8\%$ per annum respectively. After 1 year, the total interest on both loans is $1020$. We need to find how much she borrowed from each lender, the interest after 3 years, total repayment, and equal annual installments. 2. **Formulas and Rules:** - Simple Interest (SI) formula: $$SI = P \times r \times t$$ where $P$ is principal, $r$ is annual interest rate (in decimal), and $t$ is time in years. - Total borrowed: $$P_A + P_B = 15000$$ - Interest after 1 year: $$SI_A + SI_B = 1020$$ 3. **Step 1: Define variables:** Let $P_A$ be amount borrowed from Lender A, and $P_B$ from Lender B. 4. **Step 2: Write equations:** - Total principal: $$P_A + P_B = 15000$$ - Interest after 1 year: $$P_A \times 0.05 \times 1 + P_B \times 0.08 \times 1 = 1020$$ 5. **Step 3: Solve system of equations:** From first equation: $$P_B = 15000 - P_A$$ Substitute into second: $$0.05 P_A + 0.08 (15000 - P_A) = 1020$$ $$0.05 P_A + 1200 - 0.08 P_A = 1020$$ $$-0.03 P_A + 1200 = 1020$$ $$-0.03 P_A = 1020 - 1200 = -180$$ $$P_A = \frac{-180}{-0.03} = 6000$$ 6. **Step 4: Find $P_B$:** $$P_B = 15000 - 6000 = 9000$$ 7. **Step 5: Calculate interest after 3 years:** - Lender A: $$SI_A = 6000 \times 0.05 \times 3 = 900$$ - Lender B: $$SI_B = 9000 \times 0.08 \times 3 = 2160$$ 8. **Step 6: Total interest after 3 years:** $$SI_{total} = 900 + 2160 = 3060$$ 9. **Step 7: Total amount to repay after 3 years:** $$Amount = Principal + Interest = 15000 + 3060 = 18060$$ 10. **Step 8: Equal annual installments over 3 years:** $$Installment = \frac{18060}{3} = 6020$$ **Final answers:** - Borrowed from Lender A: $6000$ - Borrowed from Lender B: $9000$ - Interest to Lender A after 3 years: $900$ - Interest to Lender B after 3 years: $2160$ - Total repayment after 3 years: $18060$ - Annual installment: $6020$