1. **State the problem:** A couple is arranging flowers for a wedding. The bride finishes one arrangement in $x$ minutes, and the groom finishes his arrangement 8 minutes later, i.e., in $x+8$ minutes. Together, they complete 40 arrangements in 3 hours (180 minutes). We need to find how fast the bride was and how many arrangements she made by herself. 2. **Set up the equation:** The rate of the bride is $\frac{1}{x}$ arrangements per minute, and the groom's rate is $\frac{1}{x+8}$. Together, their combined rate is $\frac{40}{180} = \frac{2}{9}$ arrangements per minute. 3. **Write the combined rate equation:** $$\frac{40}{x} + \frac{40}{x+8} = 180$$ This equation is incorrect as stated; the correct approach is: Since they complete 40 arrangements in 180 minutes, their combined rate is: $$\frac{40}{180} = \frac{2}{9}$$ The sum of their rates is: $$\frac{1}{x} + \frac{1}{x+8} = \frac{2}{9}$$ 4. **Solve the equation:** Multiply both sides by $9x(x+8)$ to clear denominators: $$9(x+8) + 9x = 2x(x+8)$$ Simplify: $$9x + 72 + 9x = 2x^2 + 16x$$ $$18x + 72 = 2x^2 + 16x$$ Bring all terms to one side: $$0 = 2x^2 + 16x - 18x - 72$$ $$0 = 2x^2 - 2x - 72$$ Divide both sides by 2: $$0 = x^2 - x - 36$$ 5. **Factor or use quadratic formula:** $$x = \frac{1 \pm \sqrt{1 + 144}}{2} = \frac{1 \pm \sqrt{145}}{2}$$ Since time must be positive, take the positive root: $$x = \frac{1 + 12.0416}{2} = 6.5208$$ minutes 6. **Find how many arrangements the bride made by herself:** The bride's rate is $\frac{1}{6.5208}$ arrangements per minute. In 180 minutes, she would make: $$180 \times \frac{1}{6.5208} \approx 27.6$$ arrangements **Final answers:** - The bride finishes one arrangement in approximately $6.52$ minutes. - She made about $28$ arrangements by herself (rounding to nearest whole number).