1. **Problem:** Simplify the ratio 36:90 to its lowest terms. **Formula:** To simplify a ratio, divide both terms by their greatest common divisor (GCD). **Work:** GCD of 36 and 90 is 18. $$\frac{36}{\cancel{18}} : \frac{90}{\cancel{18}} = 2 : 5$$ **Answer:** The ratio in lowest terms is 2:5. 2. **Problem:** Find the sale price after a 30% discount on a $50 hoodie. **Formula:** Sale price = Regular price - (Discount rate × Regular price) **Work:** $$30\% = 0.30$$ $$\text{Discount} = 0.30 \times 50 = 15$$ $$\text{Sale price} = 50 - 15 = 35$$ **Answer:** The sale price is 35. 3. **Problem:** Simplify $\frac{5}{8} \div \frac{2}{3}$ to lowest terms. **Formula:** Division of fractions: $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$ **Work:** $$\frac{5}{8} \times \frac{3}{2} = \frac{5 \times 3}{8 \times 2} = \frac{15}{16}$$ **Answer:** $\frac{15}{16}$ 4. **Problem:** Simplify $6 \frac{1}{4} \times \frac{3}{8}$ to lowest terms. **Formula:** Convert mixed number to improper fraction, then multiply. **Work:** $$6 \frac{1}{4} = \frac{25}{4}$$ $$\frac{25}{4} \times \frac{3}{8} = \frac{75}{32}$$ **Answer:** $\frac{75}{32}$ or $2 \frac{11}{32}$ 5. **Problem:** Lydia spends $24 on entertainment, which is $\frac{2}{5}$ of her clothes expenses. Find clothes expenses. **Formula:** Entertainment = $\frac{2}{5}$ × Clothes **Work:** $$24 = \frac{2}{5} \times \text{Clothes}$$ $$\text{Clothes} = \frac{24}{\frac{2}{5}} = 24 \times \frac{5}{2} = 60$$ **Answer:** Lydia spends 60 on clothes. 6. **Problem:** Find volume of right triangular prism with base area 50 cm² and height 8 cm. **Formula:** Volume = Base area × Height **Work:** $$V = 50 \times 8 = 400$$ **Answer:** Volume is 400 cm³. 7. **Problem:** Solve $3(x - 4) = 15$ for $x$. **Formula:** Distribute and isolate $x$. **Work:** $$3x - 12 = 15$$ $$3x = 15 + 12 = 27$$ $$x = \frac{27}{3} = 9$$ **Answer:** $x = 9$ 8. **Problem:** Delivery cost $c = 4m + 5$, find cost for $m=9$ pieces. **Work:** $$c = 4 \times 9 + 5 = 36 + 5 = 41$$ **Answer:** Cost is 41. 9. **Problem:** Find two whole numbers between which $\sqrt{51}$ lies. **Work:** $$7^2 = 49 < 51 < 64 = 8^2$$ **Answer:** Between 7 and 8. 10. **Problem:** Find area of square with side 8 cm. **Formula:** Area = side² **Work:** $$8^2 = 64$$ **Answer:** 64 cm². 11. **Problem:** If 240 grams cost 4, how many grams for 3? **Formula:** Use ratio. **Work:** $$\frac{240}{4} = \frac{x}{3}$$ $$x = \frac{240}{4} \times 3 = 180$$ **Answer:** 180 grams. 12. **Problem:** Use Pythagorean theorem to find unknown square area. **Formula:** $a^2 + b^2 = c^2$ **Work:** Known squares: 49 and 81 $$49 + 81 = 130$$ **Answer:** Unknown square area is 130. 13. **Problem:** Calculate $6 \times 4 - (-7)$. **Work:** $$24 + 7 = 31$$ **Answer:** 31. 14. **Problem:** Volume of block 80 cm × 20 cm × 30 cm. **Formula:** Volume = length × width × height **Work:** $$80 \times 20 \times 30 = 48000$$ **Answer:** 48000 cm³. 15. **Problem:** Find $\frac{3}{5}$ of 30. **Work:** $$\frac{3}{5} \times 30 = 18$$ **Answer:** 18. 16. **Problem:** Temperature drops 4°C every 2 hours, find change after 8 hours. **Work:** $$\frac{4}{2} = 2 \text{°C per hour}$$ $$2 \times 8 = 16$$ **Answer:** Temperature drops 16°C. 17. **Problem:** 1/4 of cards are rare; 10 rare cards total. Find total cards. **Work:** $$\frac{1}{4} \times \text{Total} = 10$$ $$\text{Total} = 10 \times 4 = 40$$ **Answer:** 40 cards. 18. **Problem:** Express $\frac{6}{15}$ as decimal. **Work:** $$\frac{6}{15} = 0.4$$ **Answer:** 0.4 19. **Problem:** Distance $d=rt$, find time $t$ for 1200 km at 800 km/h. **Formula:** $t = \frac{d}{r}$ **Work:** $$t = \frac{1200}{800} = 1.5$$ **Answer:** 1.5 hours. 20. **Problem:** Solve $\frac{2}{3} = \frac{x}{60}$ for $x$. **Work:** $$x = \frac{2}{3} \times 60 = 40$$ **Answer:** $x=40$ 21. **Problem:** Find 150% of 80. **Work:** $$150\% = 1.5$$ $$1.5 \times 80 = 120$$ **Answer:** 120 22. **Problem:** Simplify $\frac{2}{3} \times \frac{3}{4} \div \frac{1}{2}$. **Work:** $$\frac{2}{3} \times \frac{3}{4} = \frac{6}{12} = \frac{1}{2}$$ $$\frac{1}{2} \div \frac{1}{2} = 1$$ **Answer:** 1 23. **Problem:** Evaluate $(-22) - (-12)$. **Work:** $$-22 + 12 = -10$$ **Answer:** -10 24. **Problem:** Evaluate $(-10) \times (-6) \times (-8)$. **Work:** $$(-10) \times (-6) = 60$$ $$60 \times (-8) = -480$$ **Answer:** -480 25. **Problem:** Evaluate $(-28) \div (-7) \times (-2)$. **Work:** $$-28 \div -7 = 4$$ $$4 \times -2 = -8$$ **Answer:** -8 26. **Problem:** Approximate $\sqrt{27}$ to nearest whole number. **Work:** $$5^2=25 < 27 < 36=6^2$$ Closer to 5. **Answer:** 5 27. **Problem:** Evaluate $14^2$. **Work:** $$14 \times 14 = 196$$ **Answer:** 196 28. **Problem:** Solve $2x + 4 = -8$. **Work:** $$2x = -8 - 4 = -12$$ $$x = \frac{-12}{2} = -6$$ **Answer:** $x = -6$ 29. **Problem:** Solve $67 = 4x + 3$. **Work:** $$4x = 67 - 3 = 64$$ $$x = \frac{64}{4} = 16$$ **Answer:** $x=16$ 30. **Problem:** Solve $\frac{x}{5} - 4 = -2$. **Work:** $$\frac{x}{5} = -2 + 4 = 2$$ $$x = 2 \times 5 = 10$$ **Answer:** $x=10$ 31. **Problem:** Find 125% of 60. **Work:** $$125\% = 1.25$$ $$1.25 \times 60 = 75$$ **Answer:** 75 32. **Problem:** Simplify $\frac{1}{4} + \frac{3}{8}$. **Work:** $$\frac{1}{4} = \frac{2}{8}$$ $$\frac{2}{8} + \frac{3}{8} = \frac{5}{8}$$ **Answer:** $\frac{5}{8}$ 33. **Problem:** Simplify $\frac{2}{5} \times \frac{3}{4}$. **Work:** $$\frac{6}{20} = \frac{3}{10}$$ **Answer:** $\frac{3}{10}$ 34. **Problem:** Simplify $\frac{5}{8} \div \frac{2}{5}$. **Work:** $$\frac{5}{8} \times \frac{5}{2} = \frac{25}{16}$$ **Answer:** $\frac{25}{16}$ 35. **Problem:** Simplify $\frac{3}{4} \div \frac{5}{9}$. **Work:** $$\frac{3}{4} \times \frac{9}{5} = \frac{27}{20}$$ **Answer:** $\frac{27}{20}$ 36. **Problem:** Simplify $\frac{11}{12} - \frac{1}{2} \times \left(\frac{1}{4} + \frac{2}{3}\right)$. **Work:** $$\frac{1}{4} + \frac{2}{3} = \frac{3}{12} + \frac{8}{12} = \frac{11}{12}$$ $$\frac{1}{2} \times \frac{11}{12} = \frac{11}{24}$$ $$\frac{11}{12} - \frac{11}{24} = \frac{22}{24} - \frac{11}{24} = \frac{11}{24}$$ **Answer:** $\frac{11}{24}$ 37. **Problem:** Evaluate $(-5) + 7$. **Work:** $$2$$ **Answer:** 2 38. **Problem:** Evaluate $7 \times (-5)$. **Work:** $$-35$$ **Answer:** -35 39. **Problem:** Evaluate $(-8) \times (-9)$. **Work:** $$72$$ **Answer:** 72 40. **Problem:** Evaluate $32 \div (-4)$. **Work:** $$-8$$ **Answer:** -8 41. **Problem:** Evaluate $(-4) \times 3 \times (-5)$. **Work:** $$(-4) \times 3 = -12$$ $$-12 \times (-5) = 60$$ **Answer:** 60 42. **Problem:** Evaluate $16 \div (-8) \times 6$. **Work:** $$16 \div (-8) = -2$$ $$-2 \times 6 = -12$$ **Answer:** -12 43. **Problem:** Order from smallest to greatest: $\frac{5}{6} \times 6$, 12.5% of 24, $\frac{2}{3} \times 9$, 20% of 20. **Work:** $$\frac{5}{6} \times 6 = 5$$ $$12.5\% \times 24 = 3$$ $$\frac{2}{3} \times 9 = 6$$ $$20\% \times 20 = 4$$ Order: 3, 4, 5, 6 **Answer:** 12.5% of 24, 20% of 20, $\frac{5}{6} \times 6$, $\frac{2}{3} \times 9$ 44. **Problem:** Approximate $\sqrt{66}$ to nearest whole number. **Work:** $$8^2=64 < 66 < 81=9^2$$ Closer to 8. **Answer:** 8 45. **Problem:** Evaluate $17^2$. **Work:** $$17 \times 17 = 289$$ **Answer:** 289 46. **Problem:** Solve $12 = -4x - 12$. **Work:** $$-4x = 12 + 12 = 24$$ $$x = \frac{24}{-4} = -6$$ **Answer:** $x = -6$ 47. **Problem:** Solve $5(x + 2) = 40$. **Work:** $$x + 2 = \frac{40}{5} = 8$$ $$x = 8 - 2 = 6$$ **Answer:** $x=6$ 48. **Problem:** Solve $-2(7x - 1) = -40$. **Work:** $$-14x + 2 = -40$$ $$-14x = -42$$ $$x = 3$$ **Answer:** $x=3$ 49. **Problem:** Solve $-80 = -4(3x + 2)$. **Work:** $$-80 = -12x - 8$$ $$-12x = -72$$ $$x = 6$$ **Answer:** $x=6$ 50. **Problem:** Solve $\frac{x}{4} + 9 = 20$. **Work:** $$\frac{x}{4} = 11$$ $$x = 44$$ **Answer:** $x=44$