Ratios & Proportions - ACT Math

1. Convert the ratio 7:2 to a fraction.

   $\frac{7}{2}$. Write as fraction with numerator and denominator.

2. What is the value of $x$ if $\frac{5}{x} = \frac{10}{15}$?

   7.5. Cross multiply: $10x = 5 \cdot 15$.

3. What is the cross-products condition for $\frac{a}{b}=\frac{c}{d}$?

   $ad=bc$. Cross multiply: multiply diagonally opposite terms to check equality.

4. What is the missing value $x$ if $7:5=21:x$?

   $x=15$. Set up proportion: $\frac{7}{5}=\frac{21}{x}$, cross multiply.

5. What is the missing value $x$ if $4:9=x:36$?

   $x=16$. Set up proportion: $\frac{4}{9}=\frac{x}{36}$, cross multiply.

6. Determine the simplest form of the ratio 8:20.

   2:5. Divide both terms by their GCD of 4.

7. What is $x$ if $\frac{x}{12}=\frac{5}{8}$?

   $x=7.5$. Cross multiply: $8x=60$, so $x=7.5$.

8. Convert the ratio 9:3 to a fraction.

   $\frac{3}{1}$. Write as improper fraction $\frac{9}{3} = \frac{3}{1}$.

9. What is the ratio of 20 dollars to 5 dollars?

   4:1. Divide both amounts by their GCD of 5.

10. What is the ratio of 10 to 50 in simplest form?

    1:5. Divide both numbers by their GCD of 10.

11. How do you express the ratio of 15 to 5?

    3:1. Divide both terms by their GCD of 5.

12. What is $x$ if $\frac{x}{9}=\frac{4}{6}$?

    $x=6$. Cross multiply: $6x=36$, so $x=6$.

13. What is the unit rate for $7.5$ gallons used in $3$ hours?

    $2.5$ gallons per hour. Divide gallons by hours: $\frac{7.5}{3}=2.5$.

14. What is the ratio of 15 minutes to 1 hour?

    1:4. Convert 1 hour to 60 minutes, then simplify $15:60$.

15. Simplify the ratio 36 to 48.

    3:4. Divide both numbers by their GCD of 12.

16. What is the missing value $x$ if $5:8=15:x$?

    $x=24$. Set up proportion: $\frac{5}{8}=\frac{15}{x}$, cross multiply.

17. State the formula for a proportion.

    $\frac{a}{b} = \frac{c}{d}$. Two ratios set equal form a proportion.

18. Identify the cross-multiplication method for $\frac{a}{b} = \frac{c}{d}$.

    $a \times d = b \times c$. Cross-products of a proportion are equal.

19. Convert the ratio 18:6 to simplest form.

    3:1. Divide both numbers by their GCD of 6.

20. What is $y$ if $y$ varies directly with $x$ and $y=10$ when $x=4$, then $x=6$?

    $y=15$. Find constant: $k=\frac{10}{4}=2.5$, then $y=2.5(6)=15$.

21. Find the missing value in $\frac{15}{x} = \frac{45}{75}$.

    x = 25. Cross-multiply: $15 \times 75 = x \times 45$, so $x = 25$.

22. If $\frac{a}{b} = \frac{c}{d}$, which are the means?

    b and c. The means are the inner terms of the proportion.

23. What is the definition of an 'equivalent ratio'?

    Ratios that express the same relationship. Different ratios that represent the same comparison.

24. What is the ratio of 20 to 30 in simplest form?

    2:3. Divide both terms by their GCD of 10.

25. Express the ratio of 14 to 42 in simplest form.

    1:3. Divide both terms by their GCD of 14.

26. What is the inverse variation equation if $y$ varies inversely with $x$ and $y=6$ when $x=5$?

    $y=\frac{30}{x}$. Find constant $k$: $k=6 \cdot 5=30$, so $y=\frac{30}{x}$.

27. What does the ratio $a:b$ mean when written as a fraction?

    $\frac{a}{b}$. A ratio expressed as a fraction shows the first term divided by the second.

28. What is the missing value $x$ if $2.5:10=3:x$?

    $x=12$. Set up proportion: $\frac{2.5}{10}=\frac{3}{x}$, cross multiply.

29. What is the decimal form of the ratio 2:5?

    0.4. Divide $2 \div 5 = 0.4$.

30. What is $x$ if $\frac{x+4}{10}=\frac{3}{5}$?

    $x=2$. Cross multiply: $5(x+4)=30$, so $x=2$.

31. What is the ratio of $30$ minutes to $2$ hours in simplest form?

    $1:4$. Convert to same units: $30$ min to $120$ min, then simplify.

32. What is the ratio of $1.5$ meters to $60$ centimeters in simplest form?

    $5:2$. Convert to same units: $150$ cm to $60$ cm, then simplify.

33. What is the simplest form of the ratio $45:60$?

    $3:4$. Divide both terms by their GCD of 15.

34. What is $x$ if $x$ is $30\%$ of $80$?

    $x=24$. Calculate $30\%$ of $80$: $0.30 \times 80=24$.

35. What is the decimal form of the ratio 3:20?

    0.15. Divide $3 \div 20 = 0.15$.

36. In the proportion $\frac{a}{b} = \frac{c}{d}$, identify the means.

    b and c. The means are the middle terms in a proportion.

37. Find the missing term: $\frac{5}{x} = \frac{15}{9}$.

    3. Cross multiply: $15x = 5 \cdot 9$.

38. What is the direct variation equation if $y$ varies directly with $x$ and $y=12$ when $x=3$?

    $y=4x$. Find constant $k$: $k=\frac{12}{3}=4$, so $y=4x$.

39. Find the value of $x$ if $\frac{4}{x} = \frac{12}{36}$.

    x = 12. Cross-multiply: $4 \times 36 = x \times 12$, so $x = 12$.

40. What is the simplest form of the ratio $\frac{3}{4}:\frac{9}{10}$?

    $5:6$. Convert to common denominators, then simplify the ratio.

41. Express the ratio 14:21 as a fraction in simplest form.

    $\frac{2}{3}$. Simplify by dividing numerator and denominator by 7.

42. What is the simplest form of the ratio $14:49$?

    $2:7$. Divide both terms by their GCD of 7.

43. If $\frac{7}{x} = \frac{21}{42}$, what is the value of $x$?

    14. Cross multiply: $21x = 7 \cdot 42$.

44. Express the ratio 9:12 as a fraction in simplest form.

    $\frac{3}{4}$. Simplify by dividing numerator and denominator by 3.

45. Solve for $x$ if $\frac{x}{3} = \frac{5}{15}$.

    $x = 1$. Cross-multiply: $x \times 15 = 3 \times 5$, so $x = 1$

46. What is the value of $x$ if $\frac{7}{x} = \frac{14}{28}$?

    14. Cross multiply: $14x = 7 \cdot 28$.

47. What is the equivalent ratio of 6:9?

    2:3. Divide both terms by their GCD of 3.

48. If $a:b = 5:7$, what is the reciprocal ratio $b:a$?

    7:5. Reciprocal of $5:7$ is $7:5$.

49. Express the ratio 45:60 in simplest form.

    3:4. Divide both terms by their GCD of 15.

50. How is the concept of 'rate' related to ratios?

    A rate is a special type of ratio comparing different units. Rates compare quantities with different units of measurement.

51. What is the value of $x$ if $\frac{x}{6} = \frac{8}{12}$?

    4. Cross multiply and solve: $12x = 48$.

52. What is the simplified ratio of 75 to 100?

    3:4. Divide both terms by their GCD of 25.

53. What is the simplest form of the ratio $0.6:1.5$?

    $2:5$. Convert to fractions: $\frac{3}{5}:\frac{3}{2}$, then simplify.

54. Identify the extremes in $\frac{5}{8} = \frac{10}{16}$.

    5 and 16. Extremes are the first and last terms.

55. What is $y$ if $y$ varies inversely with $x$ and $y=8$ when $x=3$, then $x=12$?

    $y=2$. Find constant: $k=8 \cdot 3=24$, then $y=\frac{24}{12}=2$.

56. What is the simplest form of the ratio 50:100?

    1:2. Divide both terms by their GCD of 50.

57. What is the missing value $x$ if $x:14=9:21$?

    $x=6$. Set up proportion: $\frac{x}{14}=\frac{9}{21}$, cross multiply.

58. Determine the value of $y$ if $\frac{7}{y} = \frac{14}{28}$.

    y = 14. Cross-multiply: $7 \times 28 = y \times 14$, so $y = 14$.

59. What is $x$ if $y$ varies inversely with $x$ and $y=9$ when $x=2$, then $y=6$?

    $x=3$. Find constant: $k=9 \cdot 2=18$, then $x=\frac{18}{6}=3$.

60. What is the missing value $x$ if $3:x=12:20$?

    $x=5$. Set up proportion: $\frac{3}{x}=\frac{12}{20}$, cross multiply.

61. What is the ratio of 10 to 25 in simplest form?

    2:5. Divide both terms by their GCD of 5.

62. What is $x$ if $\frac{7}{x}=\frac{21}{30}$?

    $x=10$. Cross multiply: $7 \cdot 30=21x$, so $x=10$.

63. Find the missing number in the proportion $\frac{2}{5} = \frac{8}{x}$.

    x = 20. Cross-multiply: $2 \times x = 5 \times 8$, so $x = 20$.

64. Convert the ratio 16:4 to simplest form.

    4:1. Divide both numbers by their GCD of 4.

65. Express 0.75 as a ratio in simplest form.

    3:4. Convert $0.75$ to fraction $\frac{3}{4}$, then to ratio.

66. Express the ratio 25:100 as a fraction in simplest form.

    $\frac{1}{4}$. Simplify by dividing numerator and denominator by 25.

67. Find the value of $k$ in the proportion $\frac{9}{12} = \frac{k}{16}$.

    k = 12. Cross-multiply: $9 \times 16 = 12 \times k$, so $k = 12$.

68. What is $x$ if $\frac{x-1}{6}=\frac{3}{2}$?

    $x=10$. Cross multiply: $2(x-1)=18$, so $x=10$.

69. What is the ratio of $250$ mL to $1.5$ L in simplest form?

    $1:6$. Convert to same units: $250$ mL to $1500$ mL, then simplify.

70. What is the proportion if $x:6 = 8:12$?

    x = 4. Cross-multiply: $x \times 12 = 6 \times 8$, so $x = 4$.

71. Find the missing term: $\frac{8}{x} = \frac{16}{24}$.

    12. Cross multiply: $16x = 8 \cdot 24$.

72. Express the ratio 4:6 as a fraction in simplest form.

    $\frac{2}{3}$. Simplify by dividing numerator and denominator by 2.

73. Identify the simplest form of the ratio 9:12.

    3:4. Divide both terms by their GCD of 3.

74. What is the cross product of $\frac{2}{3} = \frac{4}{x}$?

    2x = 12. Cross multiply: $2 \cdot x = 3 \cdot 4$.

75. What is $x$ if $\frac{2x}{5}=\frac{6}{15}$?

    $x=1$. Cross multiply: $15 \cdot 2x=5 \cdot 6$, so $x=1$.

76. Express the ratio 30:120 as a fraction in simplest form.

    $\frac{1}{4}$. Simplify by dividing numerator and denominator by 30.

77. What is the simplest form of the ratio $18:24$?

    $3:4$. Divide both terms by their GCD of 6.

78. What is the result of simplifying the ratio 18:24?

    3:4. Divide both terms by their GCD of 6.

79. What is the scale factor from $10$ cm to $4$ cm (new divided by original)?

    $\frac{2}{5}$. Divide new by original: $\frac{4}{10}=\frac{2}{5}$.

80. What is the simplest form of the ratio 18:24?

    3:4. Divide both numbers by their GCD of 6.

81. Identify the extremes in the proportion $\frac{3}{5} = \frac{6}{10}$.

    3 and 10. Extremes are the outer terms in a proportion.

82. State the definition of a proportion.

    An equation that shows two ratios are equal. Two ratios set equal form a proportion.

83. What is $x$ if $\frac{x}{15}=\frac{2}{3}$?

    $x=10$. Cross multiply: $3x=30$, so $x=10$.

84. What is the percent of $45$ out of $60$?

    $75\%$. Convert to percentage: $\frac{45}{60}=0.75=75\%$.

85. What is the unit price for $12 for $8$ items?

    $1.50 per item. Divide total cost by number of items: $\frac{12}{8}=1.50$.

86. What is the definition of a 'unit rate'?

    A rate with a denominator of one. The second quantity equals 1 in a unit rate.

87. What is the ratio of 0.5 to 2.5 in simplest form?

    1:5. Convert to whole numbers: $5:25$, then divide by 5.

88. If $a:b = 6:9$, what is the reciprocal ratio $b:a$?

    9:6. Reciprocal of $6:9$ is $9:6$.

89. Simplify the ratio 48 to 64.

    3:4. Divide both numbers by their GCD of 16.

90. What is the definition of a proportion using ratios $\frac{a}{b}$ and $\frac{c}{d}$?

    $\frac{a}{b}=\frac{c}{d}$. Two ratios are equal, forming a proportion statement.

91. Find the value of $x$ if $\frac{x}{4} = \frac{9}{12}$.

    $x = 3$. Cross-multiply: $x \times 12 = 4 \times 9$, so $x = 3$.

92. What is the ratio of 8 to 32 in simplest form?

    1:4. Divide both numbers by their GCD of 8.

93. What is the unit rate (per $1$) for $180$ miles in $3$ hours?

    $60$ miles per hour. Divide distance by time: $\frac{180}{3}=60$.

94. What is the cross-multiplication result of $\frac{3}{4} = \frac{9}{x}$?

    3x = 36. Cross multiply: $3 \cdot x = 4 \cdot 9$.

95. What is the proportion if 3:4 = x:8?

    x = 6. Cross-multiply: $3 \times 8 = 4 \times x$, so $x = 6$.

96. Determine the value of $y$ in the proportion $\frac{5}{y} = \frac{10}{30}$.

    y = 15. Cross-multiply: $5 \times 30 = y \times 10$, so $y = 15$.

97. If $a:b = 3:4$, what is $b:a$?

    4:3. Reciprocal of $3:4$ is $4:3$.

98. What is the scale factor from $\triangle A$ to $\triangle B$ if sides are $6$ and $15$?

    $\frac{5}{2}$. Divide new length by original: $\frac{15}{6}=\frac{5}{2}$.

99. What is the cross product of $\frac{1}{4} = \frac{3}{x}$?

    x = 12. Cross multiply: $1 \cdot x = 4 \cdot 3$.

100. What is the ratio of 4 to 8 in simplest form?

     1:2. Divide both terms by their GCD of 4.

101. Convert the ratio 5:8 to a decimal.

     0.625. Divide $5 \div 8 = 0.625$.

102. What is $x$ if $\frac{5}{x}=\frac{25}{40}$?

     $x=8$. Cross multiply: $5 \cdot 40=25x$, so $x=8$.

103. Find the missing term in the proportion $\frac{5}{x} = \frac{10}{20}$.

     x = 10. Cross-multiply: $5 \times 20 = x \times 10$, so $x = 10$.

104. What is $x$ if $\frac{3}{x}=\frac{9}{12}$?

     $x=4$. Cross multiply: $3 \cdot 12=9x$, so $x=4$.

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