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 $-x$. Step 3: Tripled opposite is $3(-x) = -3x$. Step 4: Decreased by 7 gives $-3x - 7$. Step 5: This equals 3 greater than twice the number: $2x + 3$. Step 6: Set up equation: $$-3x - 7 = 2x + 3$$ Step 7: Add $3x$ to both sides: $$\cancel{-3x} - 7 + 3x = 2x + 3 + 3x \Rightarrow -7 = 5x + 3$$ Step 8: Subtract 3 from both sides: $$-7 - 3 = 5x + \cancel{3} - 3 \Rightarrow -10 = 5x$$ Step 9: 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 largest find be $L$. Step 2: 80% of $L$ is 60: $$0.8L = 60$$ Step 3: Divide both sides by 0.8: $$\frac{60}{\cancel{0.8}} = \frac{0.8L}{\cancel{0.8}} \Rightarrow 75 = L$$ **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:** $4 \frac{2}{7}$ of what number is $20 \frac{1}{2}$? Step 1: Convert mixed numbers to improper fractions: $4 \frac{2}{7} = \frac{30}{7}$ $20 \frac{1}{2} = \frac{41}{2}$ Step 2: Let the number be $x$. Step 3: Equation: $$\frac{30}{7} x = \frac{41}{2}$$ Step 4: Solve for $x$: $$x = \frac{41}{2} \times \frac{7}{30} = \frac{41 \times 7}{2 \times 30} = \frac{287}{60}$$ Step 5: Simplify fraction if possible (287 and 60 share no common factors). **Answer:** $x = \frac{287}{60} \approx 4.7833$ --- 5. **Problem:** What decimal part of 20.2 is 1.01? Step 1: Find fraction: $$\frac{1.01}{20.2}$$ Step 2: Simplify by multiplying numerator and denominator by 100: $$\frac{1.01 \times 100}{20.2 \times 100} = \frac{101}{2020}$$ Step 3: Simplify fraction by dividing numerator and denominator by 101: $$\frac{\cancel{101}}{20 \times \cancel{101}} = \frac{1}{20} = 0.05$$ **Answer:** The decimal part is 0.05. --- 6. **Problem:** Solve $$2 \frac{1}{8} x - \frac{1}{5} = 5^2 \times 2^{-3}$$ Step 1: Convert mixed number: $$2 \frac{1}{8} = \frac{17}{8}$$ Step 2: Calculate right side: $$5^2 = 25, \quad 2^{-3} = \frac{1}{8}$$ So, $$25 \times \frac{1}{8} = \frac{25}{8}$$ Step 3: Equation: $$\frac{17}{8} x - \frac{1}{5} = \frac{25}{8}$$ Step 4: Add $\frac{1}{5}$ to both sides: $$\frac{17}{8} x = \frac{25}{8} + \frac{1}{5}$$ Step 5: Find common denominator 40: $$\frac{25}{8} = \frac{125}{40}, \quad \frac{1}{5} = \frac{8}{40}$$ Sum: $$\frac{125}{40} + \frac{8}{40} = \frac{133}{40}$$ Step 6: Solve for $x$: $$x = \frac{133}{40} \times \frac{8}{17} = \frac{133 \times 8}{40 \times 17} = \frac{1064}{680}$$ Step 7: Simplify numerator and denominator by 4: $$\frac{\cancel{1064}266}{\cancel{680}170} = \frac{266}{170}$$ Step 8: Simplify by 2: $$\frac{133}{85}$$ **Answer:** $x = \frac{133}{85} \approx 1.5647$ --- 7. **Problem:** Solve $$0.003k + 0.188 - 0.001k = 0.2k - 0.01$$ Step 1: Combine like terms on left: $$0.003k - 0.001k = 0.002k$$ Equation: $$0.002k + 0.188 = 0.2k - 0.01$$ Step 2: Subtract $0.002k$ from both sides: $$0.188 = 0.198k - 0.01$$ Step 3: Add 0.01 to both sides: $$0.198 = 0.198k$$ Step 4: Divide both sides by 0.198: $$k = \frac{0.198}{0.198} = 1$$ **Answer:** $k = 1$ --- 8. **Problem:** Graph on number line: $$x - 3 \not< -5$$ with domain $D = \{\text{Positive integers}\}$. Step 1: Rewrite inequality: $$x - 3 \geq -5$$ Step 2: Add 3 to both sides: $$x \geq -2$$ Step 3: Since $x$ is positive integer, $x \geq 1$. **Answer:** $x \in \{1, 2, 3, ...\}$ --- 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: $$4x - 3(5x - 18) = 10$$ Step 3: Expand: $$4x - 15x + 54 = 10$$ Step 4: Simplify: $$-11x + 54 = 10$$ Step 5: Subtract 54: $$-11x = -44$$ Step 6: Divide: $$x = 4$$ Step 7: Find $y$: $$y = 5(4) - 18 = 20 - 18 = 2$$ **Answer:** $(x,y) = (4,2)$ --- 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: $$3(-2y) + y = -10$$ Step 3: Simplify: $$-6y + y = -10 \Rightarrow -5y = -10$$ Step 4: Divide: $$y = 2$$ Step 5: Find $x$: $$x = -2(2) = -4$$ **Answer:** $(x,y) = (-4,2)$ --- 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: $$5(y - 2) + 4y = -28$$ Step 3: Expand: $$5y - 10 + 4y = -28$$ Step 4: Combine: $$9y - 10 = -28$$ Step 5: Add 10: $$9y = -18$$ Step 6: Divide: $$y = -2$$ Step 7: Find $x$: $$x = -2 - 2 = -4$$ **Answer:** $(x,y) = (-4,-2)$ --- 13. **Problem:** If $$x + \sqrt{36} = 12$$, evaluate $$x^2 - 6$$. Step 1: $$\sqrt{36} = 6$$ Step 2: $$x + 6 = 12 \Rightarrow x = 6$$ Step 3: Evaluate: $$x^2 - 6 = 6^2 - 6 = 36 - 6 = 30$$ **Answer:** 30 --- 14. **Problem:** 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. **Problem:** 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. **Problem:** Expand $$(4x + 5)^2$$ Step 1: Use formula $$(a + b)^2 = a^2 + 2ab + b^2$$ Step 2: Calculate: $$16x^2 + 2 \times 4x \times 5 + 25 = 16x^2 + 40x + 25$$ **Answer:** $16x^2 + 40x + 25$ --- 17. **Problem:** Graph $$y = 2x + 2$$ --- 18. **Problem:** Graph $$y = 2x - 2$$ --- 19. **Problem:** Determine the sets to which $$3\sqrt{2}$$ belongs. Step 1: $$\sqrt{2}$$ is irrational. Step 2: Multiplying by 3 keeps it irrational. Step 3: So, $$3\sqrt{2}$$ is irrational. Step 4: It belongs to the set of real numbers and irrational numbers. **Answer:** $$3\sqrt{2} \in \{\text{irrational numbers}, \text{real numbers}\}$$ --- 20. **Problem:** Simplify $$\frac{q}{x} - \frac{1}{x}$$ Step 1: Common denominator $x$: $$\frac{q - 1}{x}$$ **Answer:** $$\frac{q - 1}{x}$$ --- 21. **Problem:** Simplify $$\frac{\frac{a}{b}}{c}$$ Step 1: Rewrite as multiplication: $$\frac{a}{b} \times \frac{1}{c} = \frac{a}{bc}$$ **Answer:** $$\frac{a}{bc}$$ --- 22. **Problem:** Simplify $$\frac{\frac{a}{b}}{c}$$ (same as 21) **Answer:** $$\frac{a}{bc}$$ --- 23. **Problem:** Add $$\frac{a}{x - y} + \frac{b}{x + y}$$ Step 1: Common denominator: $$(x - y)(x + y) = x^2 - y^2$$ Step 2: Rewrite: $$\frac{a(x + y)}{x^2 - y^2} + \frac{b(x - y)}{x^2 - y^2} = \frac{a(x + y) + b(x - y)}{x^2 - y^2}$$ **Answer:** $$\frac{a(x + y) + b(x - y)}{x^2 - y^2}$$ --- 24. **Problem:** Simplify $$k + \frac{1}{y^2}$$ **Answer:** Expression is already simplified. --- 25. **Problem:** Simplify $$m + \frac{1}{m^2}$$ **Answer:** Expression is already simplified. --- 26. **Problem:** Factor $$12x^2 y^3 p^4 - 4x^3 y^2 p^6 + 16x^2 y p^4$$ Step 1: Find common factors: - Coefficients: 4 - $x^2$ (lowest power of $x$) - $y$ (lowest power) - $p^4$ (lowest power) Step 2: Factor out: $$4x^2 y p^4 (3 y^2 - x y p^2 + 4)$$ **Answer:** $$4x^2 y p^4 (3 y^2 - x y p^2 + 4)$$ --- 27. **Problem:** Simplify $$((3x^2)^5 m^2)^2 (x^2 y)^{-2}$$ Step 1: Simplify inside powers: $$(3x^2)^5 = 3^5 x^{10} = 243 x^{10}$$ Step 2: Raise to power 2: $$(243 x^{10} m^2)^2 = 243^2 x^{20} m^4 = 59049 x^{20} m^4$$ Step 3: Simplify $(x^2 y)^{-2} = x^{-4} y^{-2}$ Step 4: Multiply all: $$59049 x^{20} m^4 \times x^{-4} y^{-2} = 59049 x^{16} m^4 y^{-2}$$ **Answer:** $$59049 x^{16} m^4 y^{-2}$$ --- 28. **Problem:** Simplify $$\frac{(2xy)^3}{(ix)^{-2}} x^2 y p^{-4}$$ Step 1: Simplify numerator: $$(2xy)^3 = 8 x^3 y^3$$ Step 2: Simplify denominator: $$(ix)^{-2} = i^{-2} x^{-2} = \frac{1}{i^2 x^2}$$ Step 3: Division by denominator is multiplication by reciprocal: $$8 x^3 y^3 \times i^2 x^2$$ Step 4: Multiply by $x^2 y p^{-4}$: $$8 x^3 y^3 i^2 x^2 \times x^2 y p^{-4} = 8 i^2 x^{3+2+2} y^{3+1} p^{-4} = 8 i^2 x^7 y^4 p^{-4}$$ **Answer:** $$8 i^2 x^7 y^4 p^{-4}$$ --- 29. **Problem:** Simplify $$\frac{(4x^{-2})(x^2 y^0)^{-3}}{(x^{-2} y^{-1})^{-4}}$$ Step 1: Simplify powers: $$(x^2 y^0)^{-3} = x^{-6} y^0 = x^{-6}$$ $$(x^{-2} y^{-1})^{-4} = x^{8} y^{4}$$ Step 2: Numerator: $$4 x^{-2} \times x^{-6} = 4 x^{-8}$$ Step 3: Expression: $$\frac{4 x^{-8}}{x^{8} y^{4}} = 4 x^{-8 - 8} y^{-4} = 4 x^{-16} y^{-4}$$ **Answer:** $$4 x^{-16} y^{-4}$$ --- 30. **Problem:** Simplify $$(1 - 3 - 49) - (-3 - 2)^1 - \sqrt{25}$$ Step 1: Simplify inside parentheses: $$1 - 3 - 49 = -51$$ $$-3 - 2 = -5$$ Step 2: Evaluate powers and roots: $$(-5)^1 = -5$$ $$\sqrt{25} = 5$$ Step 3: Substitute: $$-51 - (-5) - 5 = -51 + 5 - 5 = -51$$ **Answer:** -51