Simple Payment 160E49

1. **State the problem:** A debtor owes $20,000$ at the end of $2$ years and $50,000$ at the end of $7$ years. The interest rate is $7\%$ effective per year. We want to find a single payment at the end of $4$ years that will settle the debt. 2. **Formula and concept:** We use the concept of the time value of money with simple interest. The idea is to find the equivalent value of all debts at the same time point (here, at year $4$) and then find the single payment that equals this total. Simple interest formula for amount $A$ after $t$ years at rate $i$ is: $$ A = P(1 + it) $$ where $P$ is the principal. 3. **Calculate the equivalent value of each debt at year $4$:** - For the $20,000$ owed at year $2$, we find its value at year $4$ by applying simple interest for $2$ more years: $$ 20,000 \times (1 + 0.07 \times (4 - 2)) = 20,000 \times (1 + 0.07 \times 2) = 20,000 \times 1.14 = 22,800 $$ - For the $50,000$ owed at year $7$, we find its value at year $4$ by discounting back $3$ years (since $7 - 4 = 3$) using simple interest: $$ P = \frac{50,000}{1 + 0.07 \times (7 - 4)} = \frac{50,000}{1 + 0.21} = \frac{50,000}{1.21} \approx 41,322.31 $$ 4. **Sum the equivalent amounts at year $4$:** $$ 22,800 + 41,322.31 = 64,122.31 $$ 5. **Conclusion:** The simple payment at the end of $4$ years that will settle the debt is approximately $64,122.31$.