1. **State the problem:** We need to demonstrate that the project with cash flows at years 1 to 4 has internal rates of return (IRRs) of 0%, 100%, and 200%. 2. **Recall the IRR definition:** The IRR is the discount rate $r$ that makes the net present value (NPV) of all cash flows equal to zero: $$\text{NPV} = \sum_{t=1}^n \frac{CF_t}{(1+r)^t} = 0$$ where $CF_t$ is the cash flow at year $t$. 3. **Write the NPV equation for this project:** $$-1200 \times \frac{1}{(1+r)^1} + 7200 \times \frac{1}{(1+r)^2} - 13200 \times \frac{1}{(1+r)^3} + 7200 \times \frac{1}{(1+r)^4} = 0$$ 4. **Test $r=0\%$ (i.e., $r=0$):** $$-1200 + 7200 - 13200 + 7200 = (-1200 + 7200) + (-13200 + 7200) = 6000 - 6000 = 0$$ So, NPV = 0 at $r=0\%$. 5. **Test $r=100\%$ (i.e., $r=1$):** Calculate each term: $$-1200 \times \frac{1}{2} = -600$$ $$7200 \times \frac{1}{4} = 1800$$ $$-13200 \times \frac{1}{8} = -1650$$ $$7200 \times \frac{1}{16} = 450$$ Sum: $$-600 + 1800 - 1650 + 450 = (1200) + (-1200) = 0$$ So, NPV = 0 at $r=100\%$. 6. **Test $r=200\%$ (i.e., $r=2$):** Calculate each term: $$-1200 \times \frac{1}{3} = -400$$ $$7200 \times \frac{1}{9} = 800$$ $$-13200 \times \frac{1}{27} = -488.89$$ $$7200 \times \frac{1}{81} = 88.89$$ Sum: $$-400 + 800 - 488.89 + 88.89 = 400 - 400 = 0$$ So, NPV = 0 at $r=200\%$. 7. **Conclusion:** The project has IRRs at 0%, 100%, and 200% because the NPV equals zero at these discount rates.