1. **Problem statement:** We have two amounts with interest over 5 years: 28000 and 40000. We need to find: (a) The annual interest rate. (b) The interest earned. (c) The principal amount that yields 8000 interest in 5 years. 2. **Formula used:** For compound interest, the amount after $n$ years is given by: $$ A = P(1 + r)^n $$ where $A$ is the amount, $P$ is the principal, $r$ is the annual interest rate (in decimal), and $n$ is the number of years. 3. **Step (a): Find the annual interest rate $r$** Given: $$ A_1 = 28000, \quad A_2 = 40000, \quad n = 5 $$ Assuming the principal $P$ is the same for both cases, we have: $$ 28000 = P(1 + r)^5 $$ $$ 40000 = P(1 + r)^5 $$ But these cannot both be true for the same $P$ and $r$ unless the problem means these are two different principal amounts or two different cases. Since the problem is ambiguous, we interpret that 28000 and 40000 are amounts for two different principal amounts after 5 years. Alternatively, if 28000 is the principal and 40000 is the amount after 5 years, then: $$ P = 28000, \quad A = 40000, \quad n = 5 $$ Then: $$ 40000 = 28000(1 + r)^5 $$ Divide both sides by 28000: $$ \frac{40000}{28000} = (1 + r)^5 $$ $$ \frac{40000}{28000} = \frac{40}{28} = \frac{10}{7} \approx 1.42857 $$ So: $$ (1 + r)^5 = 1.42857 $$ Take the fifth root: $$ 1 + r = (1.42857)^{\frac{1}{5}} $$ Calculate: $$ 1 + r \approx 1.0732 $$ So: $$ r \approx 0.0732 = 7.32\% $$ 4. **Step (b): Find the interest earned** Interest $I$ is: $$ I = A - P = 40000 - 28000 = 12000 $$ 5. **Step (c): Find the principal $P$ that yields 8000 interest in 5 years** Given interest $I = 8000$, amount after 5 years is: $$ A = P + I = P + 8000 $$ Using the formula: $$ A = P(1 + r)^5 $$ Substitute $A$: $$ P + 8000 = P(1 + r)^5 $$ Rearranged: $$ P(1 + r)^5 - P = 8000 $$ $$ P[(1 + r)^5 - 1] = 8000 $$ Using $r = 0.0732$ and $(1 + r)^5 = 1.42857$: $$ P(1.42857 - 1) = 8000 $$ $$ P(0.42857) = 8000 $$ $$ P = \frac{8000}{0.42857} \approx 18666.67 $$ **Final answers:** - (a) Annual interest rate $r \approx 7.32\%$ - (b) Interest earned $I = 12000$ - (c) Principal for 8000 interest in 5 years $P \approx 18666.67$