1. **State the problem:** Katrina takes 7 hours to fill the pie alone. Together, Katrina and Mei take 3 hours. We need to find how long Mei takes alone. 2. **Use the work rate formula:** Work rate is the reciprocal of time. If $t$ is the time to fill the pie, the rate is $\frac{1}{t}$ pie per hour. 3. **Set up the equation:** Katrina's rate is $\frac{1}{7}$. Mei's rate is $\frac{1}{x}$ where $x$ is the unknown time for Mei alone. Together, their combined rate is $\frac{1}{3}$. So, $$\frac{1}{7} + \frac{1}{x} = \frac{1}{3}$$ 4. **Solve for $x$:** Subtract $\frac{1}{7}$ from both sides: $$\frac{1}{x} = \frac{1}{3} - \frac{1}{7}$$ Find common denominator 21: $$\frac{1}{x} = \frac{7}{21} - \frac{3}{21} = \frac{4}{21}$$ 5. **Invert both sides to solve for $x$:** $$x = \frac{21}{4}$$ 6. **Simplify:** $$x = 5.25$$ hours **Answer:** Mei takes 5.25 hours to fill the pie alone.