1. **Problem:** If 18% of the goats crossed the bridge and that number is 45, how many goats were there in total? **Step 1:** Let the total number of goats be $x$. **Step 2:** Write the equation for 18% of $x$ equals 45: $$0.18x = 45$$ **Step 3:** Solve for $x$ by dividing both sides by 0.18: $$x = \frac{45}{0.18}$$ $$x = \cancel{\frac{45}{0.18}} = 250$$ **Answer:** There were 250 goats in total. 2. **Problem:** Find Jaime's weighted average for scores 75, 80, 88, 93 with weights 1, 2, 3, 4. **Step 1:** Multiply each score by its weight: $$75 \times 1 = 75$$ $$80 \times 2 = 160$$ $$88 \times 3 = 264$$ $$93 \times 4 = 372$$ **Step 2:** Sum the weighted scores: $$75 + 160 + 264 + 372 = 871$$ **Step 3:** Sum the weights: $$1 + 2 + 3 + 4 = 10$$ **Step 4:** Calculate weighted average: $$\frac{871}{10} = 87.1$$ **Answer:** Jaime's weighted average is 87.1. 3. **Problem:** A 130% increase results in 1610 dolls. Find the original number. **Step 1:** Let original number be $x$. **Step 2:** A 130% increase means the new amount is $x + 1.3x = 2.3x$. **Step 3:** Set up equation: $$2.3x = 1610$$ **Step 4:** Solve for $x$: $$x = \frac{1610}{2.3}$$ $$x = \cancel{\frac{1610}{2.3}} = 700$$ **Answer:** There were originally 700 dolls. 4. **Simplify:** $5\sqrt{80}$ $$\sqrt{80} = \sqrt{16 \times 5} = 4\sqrt{5}$$ $$5 \times 4\sqrt{5} = 20\sqrt{5}$$ 5. **Simplify:** $3\sqrt{120}$ $$\sqrt{120} = \sqrt{4 \times 30} = 2\sqrt{30}$$ $$3 \times 2\sqrt{30} = 6\sqrt{30}$$ 6. **Add:** $7\sqrt{5} - \sqrt{5} + 5\sqrt{3} - 3\sqrt{3}$ $$ (7 - 1)\sqrt{5} + (5 - 3)\sqrt{3} = 6\sqrt{5} + 2\sqrt{3}$$ 8. **Multiply:** $(3p - 4)(2p + 5)$ $$3p \times 2p = 6p^2$$ $$3p \times 5 = 15p$$ $$-4 \times 2p = -8p$$ $$-4 \times 5 = -20$$ $$6p^2 + 15p - 8p - 20 = 6p^2 + 7p - 20$$ 9. **Graph:** $x = -\frac{1}{2}$ is a vertical line crossing x-axis at $-0.5$. 10. **Graph:** $y = -\frac{1}{2}x$ is a line with slope $-0.5$ through origin. 11. **Graph:** $2y = x - 8$ or $y = \frac{1}{2}x - 4$ is a line with slope $0.5$ and y-intercept $-4$. 12. **Graph on number line:** Solve $x + 3 > -7$ with $D = \{\text{positive integers}\}$ $$x > -10$$ Since $x$ is positive integer, solution is all positive integers. 13. **Solve system:** $$\begin{cases} x + y = 10 \\ -x + y = 0 \end{cases}$$ Add equations: $$x + y - x + y = 10 + 0$$ $$2y = 10$$ $$y = 5$$ Substitute $y=5$ into $x + y = 10$: $$x + 5 = 10$$ $$x = 5$$ 14. **Solve system:** $$\begin{cases} 3x - 3y = 3 \\ x - 5y = -3 \end{cases}$$ Multiply second equation by 3: $$3x - 15y = -9$$ Subtract first equation: $$(3x - 15y) - (3x - 3y) = -9 - 3$$ $$-12y = -12$$ $$y = 1$$ Substitute $y=1$ into $x - 5y = -3$: $$x - 5 = -3$$ $$x = 2$$ 15. **Solve system:** $$\begin{cases} 3x - y = 8 \\ x - 3y = -8 \end{cases}$$ Multiply second equation by 3: $$3x - 9y = -24$$ Subtract first equation: $$(3x - 9y) - (3x - y) = -24 - 8$$ $$-8y = -32$$ $$y = 4$$ Substitute $y=4$ into $3x - y = 8$: $$3x - 4 = 8$$ $$3x = 12$$ $$x = 4$$ 16. **Simplify:** $\frac{a/x}{1/a^2}$ $$= \frac{a}{x} \times \frac{a^2}{1} = \frac{a^3}{x}$$ 17. **Simplify:** $\frac{a}{a+b} / a$ $$= \frac{a}{a+b} \times \frac{1}{a} = \frac{\cancel{a}}{(a+b) \cancel{a}} = \frac{1}{a+b}$$ 18. **Simplify:** $\frac{x/y}{1/y}$ $$= \frac{x}{y} \times \frac{y}{1} = x$$ 19. **Add:** $\frac{a}{x+y} + \frac{5}{x^2}$ Cannot combine further; answer is $\frac{a}{x+y} + \frac{5}{x^2}$. 20. **Add:** $1 + \frac{a}{b}$ $$= \frac{b}{b} + \frac{a}{b} = \frac{a + b}{b}$$ 21. **Add:** $x + \frac{1}{x}$ Cannot combine; answer is $x + \frac{1}{x}$. 22. **Factor:** $4x^2 y^2 z - 8x^2 y^2 z^5$ Factor out $4x^2 y^2 z$: $$4x^2 y^2 z (1 - 2z^4)$$ 23. **Expand:** $-3x^{-2} y^2 \left( \frac{y^{-2}}{x^{-2}} + 4x^2 y \right)$ Simplify inside parentheses: $$\frac{y^{-2}}{x^{-2}} = y^{-2} x^{2}$$ So inside is: $$y^{-2} x^{2} + 4x^{2} y$$ Multiply by $-3x^{-2} y^{2}$: $$-3x^{-2} y^{2} \times y^{-2} x^{2} = -3 x^{-2 + 2} y^{2 - 2} = -3$$ $$-3x^{-2} y^{2} \times 4x^{2} y = -12 x^{-2 + 2} y^{2 + 1} = -12 y^{3}$$ Final: $$-3 - 12 y^{3}$$ 24. **Simplify:** $\frac{4kx - 4kx^2}{4kx}$ Factor numerator: $$4kx(1 - x)$$ Divide: $$\frac{4kx(1 - x)}{4kx} = \cancel{\frac{4kx}{4kx}} (1 - x) = 1 - x$$ 25. **Simplify:** $\frac{m^0 (p^{-2})^2 x^2 y^4}{(y^{-2})^2 y y^0 x^{-2}}$ Simplify powers: $$m^0 = 1$$ $$(p^{-2})^2 = p^{-4}$$ $$(y^{-2})^2 = y^{-4}$$ Substitute: $$\frac{p^{-4} x^{2} y^{4}}{y^{-4} y x^{-2}} = \frac{p^{-4} x^{2} y^{4}}{y^{-3} x^{-2}}$$ Rewrite denominator: $$y^{-3} x^{-2} = \frac{1}{y^{3} x^{2}}$$ So division becomes multiplication: $$p^{-4} x^{2} y^{4} \times y^{3} x^{2} = p^{-4} x^{4} y^{7}$$ 26. **Simplify:** $\left( \frac{2x^{-2} y}{p} \right)^2 \left( \frac{p^{2} x}{2} \right)^{-2}$ First term squared: $$\frac{4 x^{-4} y^{2}}{p^{2}}$$ Second term to power -2: $$\left( \frac{p^{2} x}{2} \right)^{-2} = \left( \frac{2}{p^{2} x} \right)^{2} = \frac{4}{p^{4} x^{2}}$$ Multiply both: $$\frac{4 x^{-4} y^{2}}{p^{2}} \times \frac{4}{p^{4} x^{2}} = \frac{16 y^{2} x^{-4 - 2}}{p^{2 + 4}} = \frac{16 y^{2} x^{-6}}{p^{6}}$$ 27. **Simplify:** $\frac{x^{2} x^{-2} x^{0} y^{2}}{y^{2} (x^{-4})^{2}}$ Simplify numerator: $$x^{2 - 2 + 0} y^{2} = y^{2}$$ Simplify denominator: $$(x^{-4})^{2} = x^{-8}$$ So denominator is: $$y^{2} x^{-8}$$ Divide: $$\frac{y^{2}}{y^{2} x^{-8}} = x^{8}$$ 28. **Simplify by adding like terms:** $$\frac{3a^{2} x}{m} + \frac{5x m^{-1}}{a^{2}} - \frac{4 a a x^{-1}}{x^{-2} m}$$ Rewrite terms: $$\frac{3 a^{2} x}{m} + \frac{5 x}{a^{2} m} - \frac{4 a^{2} x^{-1}}{x^{-2} m}$$ Simplify last denominator: $$\frac{4 a^{2} x^{-1}}{x^{-2} m} = \frac{4 a^{2} x^{-1}}{x^{-2} m} = \frac{4 a^{2} x^{-1} x^{2}}{m} = \frac{4 a^{2} x^{1}}{m}$$ So expression is: $$\frac{3 a^{2} x}{m} + \frac{5 x}{a^{2} m} - \frac{4 a^{2} x}{m} = \frac{(3 a^{2} x - 4 a^{2} x)}{m} + \frac{5 x}{a^{2} m} = \frac{- a^{2} x}{m} + \frac{5 x}{a^{2} m}$$ 29. **Evaluate:** $-x^{-4} - x^{2}(x - m)$ for $x = -2$ and $m + 3 = 6$ Find $m$: $$m = 3$$ Calculate each term: $$-x^{-4} = -(-2)^{-4} = -\frac{1}{(-2)^4} = -\frac{1}{16}$$ $$x^{2}(x - m) = (-2)^2 (-2 - 3) = 4 \times (-5) = -20$$ Sum: $$-\frac{1}{16} - (-20) = -\frac{1}{16} + 20 = \frac{320}{16} - \frac{1}{16} = \frac{319}{16}$$ 30. **Simplify:** $-3^{0} - 3(-2 - 2^{0})(-8^{0} - 5) + \sqrt{16}$ Calculate powers: $$3^{0} = 1$$ $$2^{0} = 1$$ $$-8^{0} = -1$$ (Note: $-8^{0} = -(8^{0}) = -1$) $$\sqrt{16} = 4$$ Simplify inside parentheses: $$-2 - 1 = -3$$ $$-1 - 5 = -6$$ Calculate: $$-1 - 3(-3)(-6) + 4 = -1 - 3 \times 18 + 4 = -1 - 54 + 4 = -51$$