1. **State the problem:** A vintage car was valued at 650000 five years ago. For the first 3 years, its value depreciates by 2% per year compounded annually. For the next 2 years, its value increases by 5% per year compounded annually. We need to find the current value of the car. 2. **Formula for compound interest/depreciation:** The value after $n$ years with an annual rate $r$ (expressed as a decimal) is given by: $$ V = P(1 + r)^n $$ where $P$ is the initial value. 3. **Calculate depreciation for the first 3 years:** The depreciation rate is 2%, so $r = -0.02$. Initial value $P = 650000$. $$ V_3 = 650000(1 - 0.02)^3 = 650000(0.98)^3 $$ Calculate: $$ (0.98)^3 = 0.98 \times 0.98 \times 0.98 = 0.941192 $$ So, $$ V_3 = 650000 \times 0.941192 = 611774.8 $$ 4. **Calculate appreciation for the next 2 years:** The appreciation rate is 5%, so $r = 0.05$. Starting value after 3 years is $V_3 = 611774.8$. $$ V_5 = 611774.8(1 + 0.05)^2 = 611774.8(1.05)^2 $$ Calculate: $$ (1.05)^2 = 1.1025 $$ So, $$ V_5 = 611774.8 \times 1.1025 = 674784.3 $$ 5. **Final answer:** The current value of the car after 5 years is approximately **674784.3**.