1. **Problem:** The opposite of a number is tripled and then decreased by 7. The result is 3 greater than twice the number. Find the number. Step 1: Let the number be $x$. Step 2: The opposite of $x$ is $-x$. Tripled is $3(-x) = -3x$. Step 3: Decreased by 7 gives $-3x - 7$. Step 4: This equals 3 greater than twice the number: $2x + 3$. Step 5: Set up the equation: $$-3x - 7 = 2x + 3$$ Step 6: Add $3x$ to both sides: $$\cancel{-3x} - 7 + 3x = 2x + 3 + 3x \Rightarrow -7 = 5x + 3$$ Step 7: Subtract 3 from both sides: $$-7 - 3 = 5x + \cancel{3} - 3 \Rightarrow -10 = 5x$$ Step 8: Divide both sides by 5: $$\frac{-10}{\cancel{5}} = \frac{5x}{\cancel{5}} \Rightarrow -2 = x$$ **Answer:** $x = -2$ 2. **Problem:** Rubella found 60 escargots, which is 80% of her largest find. Find the largest find. Step 1: Let the largest find be $L$. Step 2: 80% of $L$ is 60: $$0.8L = 60$$ Step 3: Divide both sides by 0.8: $$\frac{0.8L}{0.8} = \frac{60}{0.8} \Rightarrow L = 75$$ **Answer:** Largest find is 75 escargots. 3. **Problem:** 37% of 300 people had oil for lamps. Find how many had oil. Step 1: Calculate 37% of 300: $$0.37 \times 300 = 111$$ **Answer:** 111 people had oil. 4. **Problem:** $4 \frac{2}{9}$ of what number is $20 \frac{1}{2}$? Step 1: Convert mixed numbers to improper fractions: $$4 \frac{2}{9} = \frac{40}{9}, \quad 20 \frac{1}{2} = \frac{41}{2}$$ Step 2: Let the number be $x$: $$\frac{40}{9} x = \frac{41}{2}$$ Step 3: Solve for $x$: $$x = \frac{41}{2} \times \frac{9}{40} = \frac{41 \times 9}{2 \times 40} = \frac{369}{80} = 4 \frac{49}{80}$$ **Answer:** $x = 4 \frac{49}{80}$ 5. **Problem:** What decimal part of 20.2 is 1.01? Step 1: Let the decimal part be $d$: $$d \times 20.2 = 1.01$$ Step 2: Solve for $d$: $$d = \frac{1.01}{20.2} = 0.05$$ **Answer:** 0.05 6. **Problem:** Solve $2 \frac{7}{8} x - 5 = 5^2 (-2)$ Step 1: Convert $2 \frac{7}{8}$ to improper fraction: $$2 \frac{7}{8} = \frac{23}{8}$$ Step 2: Calculate right side: $$5^2 (-2) = 25 \times (-2) = -50$$ Step 3: Equation: $$\frac{23}{8} x - 5 = -50$$ Step 4: Add 5 to both sides: $$\frac{23}{8} x = -50 + 5 = -45$$ Step 5: Multiply both sides by reciprocal $\frac{8}{23}$: $$x = -45 \times \frac{8}{23} = \frac{-360}{23} = -15 \frac{15}{23}$$ **Answer:** $x = -15 \frac{15}{23}$ 7. **Problem:** Solve $0.003k + 0.188 - 0.001k = 0.2k - 0.01$ Step 1: Combine like terms on left: $$0.003k - 0.001k + 0.188 = 0.002k + 0.188$$ Step 2: Equation: $$0.002k + 0.188 = 0.2k - 0.01$$ Step 3: Subtract $0.002k$ from both sides: $$0.188 = 0.198k - 0.01$$ Step 4: Add 0.01 to both sides: $$0.198 = 0.198k$$ Step 5: Divide both sides by 0.198: $$k = \frac{0.198}{0.198} = 1$$ **Answer:** $k = 1$ 8. **Problem:** Solve inequality $x - 3 < -5$ with domain $D = \{\text{Positive integers}\}$ Step 1: Add 3 to both sides: $$x < -5 + 3$$ $$x < -2$$ Step 2: Since $x$ must be positive integers, no positive integer is less than -2. **Answer:** No solution in positive integers. 9. **Problem:** Solve system: $$\begin{cases} 5x - y = 18 \\ 4x - 3y = 10 \end{cases}$$ Step 1: From first equation: $$y = 5x - 18$$ Step 2: Substitute into second: $$4x - 3(5x - 18) = 10$$ $$4x - 15x + 54 = 10$$ $$-11x + 54 = 10$$ Step 3: Subtract 54: $$-11x = 10 - 54 = -44$$ Step 4: Divide by -11: $$x = \frac{-44}{-11} = 4$$ Step 5: Find $y$: $$y = 5(4) - 18 = 20 - 18 = 2$$ **Answer:** $x=4$, $y=2$ 10. **Problem:** Solve system: $$\begin{cases} x + 2y = 0 \\ 3x + y = -10 \end{cases}$$ Step 1: From first: $$x = -2y$$ Step 2: Substitute into second: $$3(-2y) + y = -10$$ $$-6y + y = -10$$ $$-5y = -10$$ Step 3: Divide: $$y = 2$$ Step 4: Find $x$: $$x = -2(2) = -4$$ **Answer:** $x=-4$, $y=2$ 11. **Problem:** Solve system: $$\begin{cases} 5x + 4y = -28 \\ x - y = -2 \end{cases}$$ Step 1: From second: $$x = y - 2$$ Step 2: Substitute into first: $$5(y - 2) + 4y = -28$$ $$5y - 10 + 4y = -28$$ $$9y - 10 = -28$$ Step 3: Add 10: $$9y = -18$$ Step 4: Divide: $$y = -2$$ Step 5: Find $x$: $$x = -2 - 2 = -4$$ **Answer:** $x=-4$, $y=-2$ 13. **Problem:** If $x + \sqrt{36} = 12$, evaluate $x^2 - 6$. Step 1: $\sqrt{36} = 6$ Step 2: Solve for $x$: $$x + 6 = 12 \Rightarrow x = 6$$ Step 3: Calculate: $$x^2 - 6 = 6^2 - 6 = 36 - 6 = 30$$ **Answer:** 30 14. **Simplify:** $5\sqrt{20} - 6\sqrt{32}$ Step 1: Simplify radicals: $$\sqrt{20} = \sqrt{4 \times 5} = 2\sqrt{5}$$ $$\sqrt{32} = \sqrt{16 \times 2} = 4\sqrt{2}$$ Step 2: Substitute: $$5 \times 2\sqrt{5} - 6 \times 4\sqrt{2} = 10\sqrt{5} - 24\sqrt{2}$$ **Answer:** $10\sqrt{5} - 24\sqrt{2}$ 15. **Simplify:** $2\sqrt{45} - 3\sqrt{20}$ Step 1: Simplify radicals: $$\sqrt{45} = \sqrt{9 \times 5} = 3\sqrt{5}$$ $$\sqrt{20} = 2\sqrt{5}$$ Step 2: Substitute: $$2 \times 3\sqrt{5} - 3 \times 2\sqrt{5} = 6\sqrt{5} - 6\sqrt{5} = 0$$ **Answer:** 0 16. **Expand:** $(4x + 5)^2$ Step 1: Use formula $(a+b)^2 = a^2 + 2ab + b^2$ Step 2: $$ (4x)^2 + 2 \times 4x \times 5 + 5^2 = 16x^2 + 40x + 25$$ **Answer:** $16x^2 + 40x + 25$ 17. **Graph:** $y = 2x + 2$ This is a straight line with slope 2 and y-intercept 2. 19. **Question:** $3\sqrt{2} \in$ which sets? Answer: $3\sqrt{2}$ is an irrational number, so it belongs to the set of real numbers $\mathbb{R}$ but not rational numbers $\mathbb{Q}$. 20. **Simplify:** $\frac{a}{x} - \frac{1}{x} = \frac{a - 1}{x}$ 21. **Simplify:** $\frac{a}{c} - \frac{b}{c} = \frac{a - b}{c}$ 22. **Simplify:** Same as 21, $\frac{a}{c} - \frac{b}{c} = \frac{a - b}{c}$ 23. **Add:** $\frac{a}{xy} + \frac{b}{x + y}$ (cannot combine further without common denominator) 24. **Add:** $k + \frac{1}{x^2}$ (cannot combine further) 26. **Factor:** $12x^3 y^3 p^4 - 4x^3 y^2 p^6 + 16 x y^3 p^6$ Step 1: Find common factors: $$4 x y^2 p^4$$ Step 2: Factor out: $$4 x y^2 p^4 (3 x^2 y - x^2 p^2 + 4 y p^2)$$ 27. **Simplify:** $(3x^2 y^3 m^3)(x^2 y)^{-2}$ Step 1: Apply negative exponent: $$(x^2 y)^{-2} = x^{-4} y^{-2}$$ Step 2: Multiply: $$3 x^2 y^3 m^3 \times x^{-4} y^{-2} = 3 x^{2-4} y^{3-2} m^3 = 3 x^{-2} y^{1} m^3 = \frac{3 y m^3}{x^2}$$ 28. **Simplify:** $\frac{2 x y l (x y)^{-2}}{x^2 y^3 p^{-4}}$ Step 1: Simplify numerator: $$(x y)^{-2} = x^{-2} y^{-2}$$ Numerator: $$2 x y l x^{-2} y^{-2} = 2 l x^{1-2} y^{1-2} = 2 l x^{-1} y^{-1}$$ Step 2: Denominator: $$x^2 y^3 p^{-4} = x^2 y^3 p^{-4}$$ Step 3: Divide numerator by denominator: $$\frac{2 l x^{-1} y^{-1}}{x^2 y^3 p^{-4}} = 2 l x^{-1-2} y^{-1-3} p^{4} = 2 l x^{-3} y^{-4} p^{4} = \frac{2 l p^{4}}{x^{3} y^{4}}$$ 29. **Simplify:** $\frac{(4 x^{-2})(x^2)(y^3)}{x^{-3} y^{-1}}$ Step 1: Multiply numerator: $$4 x^{-2} x^{2} y^{3} = 4 x^{0} y^{3} = 4 y^{3}$$ Step 2: Denominator: $$x^{-3} y^{-1}$$ Step 3: Divide: $$\frac{4 y^{3}}{x^{-3} y^{-1}} = 4 y^{3 - (-1)} x^{0 - (-3)} = 4 y^{4} x^{3} = 4 x^{3} y^{4}$$ 30. **Simplify:** $[(1 - 3 - 4^{0}) - (-3 - 2)] - \sqrt{25}$ Step 1: Calculate inside brackets: $$4^{0} = 1$$ Step 2: $$1 - 3 - 1 = -3$$ Step 3: $$-3 - 2 = -5$$ Step 4: $$(-3) - (-5) = -3 + 5 = 2$$ Step 5: $$2 - \sqrt{25} = 2 - 5 = -3$$ **Answer:** -3