1. **State the problem:** A bride and groom are arranging flowers. The bride finishes one arrangement in $x$ minutes, and the groom finishes one 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 is (value of $x$) and how many arrangements she made by herself. 2. **Write the formula:** The rate of work for the bride is $\frac{1}{x}$ arrangements per minute, and for the groom is $\frac{1}{x+8}$. Together, their combined rate is $\frac{40}{180} = \frac{2}{9}$ arrangements per minute. 3. **Set up the equation:** $$\frac{1}{x} + \frac{1}{x+8} = \frac{2}{9}$$ 4. **Solve the equation:** Multiply both sides by $x(x+8)$ to clear denominators: $$ (x+8) + x = \frac{2}{9} x (x+8) $$ Simplify left side: $$ 2x + 8 = \frac{2}{9} (x^2 + 8x) $$ Multiply both sides by 9: $$ 9(2x + 8) = 2(x^2 + 8x) $$ $$ 18x + 72 = 2x^2 + 16x $$ Bring all terms to one side: $$ 0 = 2x^2 + 16x - 18x - 72 $$ $$ 0 = 2x^2 - 2x - 72 $$ Divide entire equation by 2: $$ x^2 - x - 36 = 0 $$ 5. **Factor the quadratic:** $$ (x - 6)(x + 6) = 0 $$ So, $x = 6$ or $x = -6$. Since time cannot be negative, $x = 6$ minutes. 6. **Interpret the result:** The bride takes 6 minutes per arrangement. 7. **Calculate how many arrangements the bride made alone:** The bride's rate is $\frac{1}{6}$ arrangements per minute. In 180 minutes, she would make: $$ 180 \times \frac{1}{6} = 30 $$ arrangements. **Final answers:** - Bride's time per arrangement: $6$ minutes - Number of arrangements bride made alone: $30$