Ratios & Proportions - PSAT Math

1. What is the definition of the ratio $a:b$ in fraction form (with $b \ne 0$)?

   $\frac{a}{b}$. A ratio $a:b$ is expressed as a fraction with $a$ in the numerator and $b$ in the denominator.

2. Which equation represents inverse variation between $y$ and $x$ with constant $k$?

   $y=\frac{k}{x}$. In inverse variation, $y$ equals a constant divided by $x$.

3. A map scale is $1$ inch $=5$ miles. What distance is $3.6$ inches?

   $18$ miles. Multiply: $3.6 \times 5 = 18$ miles.

4. What is the unit rate for $\frac{a}{b}$ when $b$ represents the number of units?

   $\frac{a}{b}$ per $1$ unit. Divide the total amount $a$ by the number of units $b$.

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

   $ad=bc$. Cross-multiply to check if ratios are equal.

6. A map scale is $1$ inch:$50$ miles. What distance is shown by $3.5$ inches?

   $175$ miles. Multiply map distance by scale: $3.5 \times 50 = 175$ miles.

7. Identify the unit rate for $\frac{x}{y}$ when $y$ is the number of units.

   $\frac{x}{y}$ per $1$ unit. Divide the first quantity by the second to get rate per unit.

8. What is the definition of a ratio $a:b$ in fraction form?

   $\frac{a}{b}$. A ratio compares two quantities as a fraction.

9. What is the definition of a constant of proportionality $k$ in $y=kx$?

   $k=\frac{y}{x}$. The constant $k$ is the ratio of $y$ to $x$ in a proportional relationship.

10. Identify whether the relationship is proportional: points $(2,6)$ and $(5,15)$ in $y=kx$.

    Proportional; $k=3$. Check if $\frac{y}{x}$ is constant: $\frac{6}{2}=\frac{15}{5}=3$.

11. What is the property of equivalent ratios when multiplying both terms by $c$?

    $a:b=(ac):(bc)$. Multiplying both terms of a ratio by the same number preserves the ratio.

12. If $y$ is proportional to $x$ with $k=\frac{5}{2}$, what is $y$ when $x=8$?

    $y=20$. Use $y = kx$ with $k = \frac{5}{2}$: $y = \frac{5}{2} \times 8 = 20$.

13. A map scale is $1$ inch:$20$ miles. How many miles does $3.5$ inches represent?

    $70$ miles. Multiply map distance by scale factor: $3.5 \times 20 = 70$ miles.

14. Which option is proportional to $x$ if the table includes $(x,y)=(2,6)$ and $(5,15)$?

    $y=3x$. Check if $\frac{y}{x}$ is constant: $\frac{6}{2} = \frac{15}{5} = 3$, so $y = 3x$.

15. What is the constant of proportionality $k$ if $y$ is proportional to $x$ and $y = kx$?

    $k = \frac{y}{x}$. In $y = kx$, solve for $k$ by dividing both sides by $x$.

16. Identify the simplified form of the ratio $12:18$.

    $2:3$. Divide both terms by their GCD of $6$.

17. Two quantities are in the ratio $4:7$ with total $55$. What is the larger quantity?

    $35$. Total parts = $4+7=11$; larger quantity = $\frac{7}{11} \times 55 = 35$.

18. What is the direct variation form if $y$ varies directly with $x$ and $y=18$ when $x=6$?

    $y=3x$. Find $k$: $18 = k(6)$, so $k = 3$ and $y = 3x$.

19. What is the part-to-whole fraction for a part-to-part ratio $a:b$ (total $a+b$)?

    $\frac{a}{a+b}$. Part $a$ over total $(a+b)$ gives the part-to-whole fraction.

20. If a scale factor is $\frac{3}{2}$ and the original length is $8$, what is the new length?

    $12$. New length = original × scale factor = $8 \times \frac{3}{2} = 12$.

21. A mixture has ratio water:juice $=4:1$. What fraction of the mixture is juice?

    $\frac{1}{5}$. Juice is $1$ part out of $4+1=5$ total parts.

22. Identify $x$ if $\frac{x}{12}=\frac{5}{3}$.

    $x=20$. Cross-multiply: $3x = 60$, so $x = 20$.

23. What is the scale factor from a length $8$ to a length $12$?

    $\frac{3}{2}$. Scale factor is $\frac{\text{new}}{\text{old}}=\frac{12}{8}=\frac{3}{2}$.

24. If $y$ is proportional to $x$ and $y=18$ when $x=6$, what is $y$ when $x=10$?

    $y=30$. Find $k = 3$, then $y = 3(10) = 30$.

25. Find the unit rate for $12 for $8$ pounds.

    $1.50 per pound. Divide total cost by quantity: $\frac{\$12}{8}=\$1.50$ per pound.

26. Find the percent $p$ if the part is $18$ and the whole is $60$.

    $p=30\%$. Use $p = \frac{\text{part}}{\text{whole}} \times 100\% = \frac{18}{60} \times 100\% = 30\%$.

27. What is the constant of proportionality $k$ if $y$ is proportional to $x$?

    $k=\frac{y}{x}$. The constant $k$ is the ratio of $y$ to $x$ in direct variation.

28. What is the percent equivalent of the ratio $\frac{p}{100}$?

    $p\%$. Percent means "per hundred," so $\frac{p}{100} = p\%$.

29. What is the equation of proportionality if $y$ is proportional to $x$ with $k=7$?

    $y=7x$. Direct proportionality means $y = kx$ where $k$ is the constant.

30. Find $k$ if $y$ is proportional to $x$ and $(x,y)=(6,15)$.

    $k=\frac{5}{2}$. Calculate $k = \frac{y}{x} = \frac{15}{6} = \frac{5}{2}$.

31. Identify whether the relationship is proportional: points $(1,2)$ and $(3,6)$.

    Yes, $\frac{y}{x}=2$ for both points. Both points have the same ratio $\frac{y}{x}=2$.

32. What is the cross-multiplication rule for $\frac{a}{b}=\frac{c}{d}$?

    $ad=bc$. Cross-multiply by multiplying the numerator of each fraction by the denominator of the other.

33. What is the formula for a part-to-whole fraction for the other part $b$ out of $a+b$?

    $\frac{b}{a+b}$. Part $b$ over the total $(a+b)$ gives its part-to-whole fraction.

34. Identify whether the table shows proportionality: $(x,y)=(2,6),(3,9),(5,15)$.

    Yes, $\frac{y}{x}=3$ for all pairs. Check: $\frac{6}{2}=\frac{9}{3}=\frac{15}{5}=3$, constant ratio confirms proportionality.

35. Identify the simplified form of the ratio $18:24$.

    $3:4$. Divide both terms by their GCD of $6$: $18÷6:24÷6 = 3:4$.

36. If lengths scale by $\frac{3}{2}$, what factor scales the area?

    $\left(\frac{3}{2}\right)^2=\frac{9}{4}$. Area scales by the square of the linear scale factor.

37. Identify $k$ if $y$ varies directly with $x$ and $y=14$ when $x=7$.

    $k=2$. Direct variation: $k=\frac{y}{x}=\frac{14}{7}=2$.

38. Find $x$ if $\frac{x}{9}=\frac{14}{21}$.

    $x=6$. Simplify $\frac{14}{21} = \frac{2}{3}$, then $\frac{x}{9} = \frac{2}{3}$ gives $x = 6$.

39. What is the ratio of boys to girls if there are $12$ boys and $18$ girls, simplified?

    $2:3$. Simplify $12:18$ by dividing by GCD $6$.

40. Find $y$ if $y$ varies inversely with $x$, $k=24$, and $x=6$.

    $y=4$. Use $y = \frac{k}{x}$: $y = \frac{24}{6} = 4$.

41. State the cross-multiplication rule for $\frac{a}{b}=\frac{c}{d}$.

    $ad=bc$ (with $b,d\ne 0$). Cross-multiply to get the product of extremes equals the product of means.

42. A map scale is $1:50{,}000$. What distance on the map represents $100{,}000$ cm in reality?

    $2$ cm. Map distance = real distance ÷ scale = $\frac{100{,}000}{50{,}000} = 2$ cm.

43. What is the definition of the ratio $a:b$ in fraction form?

    $\frac{a}{b}$. A ratio $a:b$ is expressed as a fraction with $a$ in the numerator and $b$ in the denominator.

44. What is the ratio $0.6:1$ written as a simplified whole-number ratio?

    $3:5$. Multiply both parts by $5$ to eliminate decimals: $0.6×5:1×5=3:5$.

45. What is the percent equivalent of the ratio $\frac{p}{100}$?

    $p\%$. Percent means per hundred, so $\frac{p}{100} = p\%$.

46. What is the scale factor from an original length $L$ to a new length $L'$?

    $\frac{L'}{L}$. Scale factor is the ratio of new length to original length.

47. Find the unit rate: What is $84$ miles in $3$ hours in miles per hour?

    $28$ miles per hour. Divide total miles by hours: $\frac{84}{3} = 28$ miles per hour.

48. What is the missing value $x$ in the proportion $\frac{x}{14}=\frac{3}{7}$?

    $x=6$. Cross-multiply: $7x=42$, so $x=6$.

49. Find the unit rate: $18$ miles in $3$ hours.

    $6$ miles per hour. Divide total miles by total hours: $\frac{18}{3} = 6$ miles per hour.

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

    $\frac{a}{b}=\frac{c}{d}$. A proportion states that two ratios are equal.

51. What is the percent form of the ratio $3:20$?

    $15\%$. Convert ratio to fraction: $\frac{3}{20}=0.15=15\%$.

52. Find $x$ if $\frac{3}{5}=\frac{x}{20}$.

    $x=12$. Cross-multiply: $3 \cdot 20 = 5x$, so $60 = 5x$.

53. Identify the constant of proportionality in the table: $x=2,4,6$ and $y=10,20,30$.

    $k=5$. Each $y$-value equals $5$ times its $x$-value.

54. Find $x$ if $\frac{x}{9}=\frac{4}{3}$.

    $x=12$. Cross-multiply: $3x = 36$, so $x = 12$.

55. What is the unit rate for $\frac{a}{b}$ units per 1 unit of $b$?

    $\frac{a}{b}$ per $1$ (with $b\ne 0$). Unit rate expresses how many units of $a$ per one unit of $b$.

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

    $x=10$. Cross-multiply: $21x=210$, so $x=10$.

57. A shirt is discounted $20\%$ from $35. What is the sale price?

    $28. Sale price = $35 - 0.20(35) = 35 - 7 = \$28$.

58. What is the scale factor from a drawing length $6$ to an actual length $15$?

    $\frac{5}{2}$. Scale factor = $\frac{\text{actual}}{\text{drawing}}=\frac{15}{6}=\frac{5}{2}$.

59. A class has boys:girls $=2:3$. What fraction of the class is girls?

    $\frac{3}{5}$. Girls are $3$ parts out of $2+3=5$ total parts.

60. A recipe uses flour:sugar $=2:3$. If flour is $8$ cups, what is sugar?

    $12$ cups. Set up proportion: $\frac{2}{3} = \frac{8}{s}$, so $s = 12$.

61. A recipe uses flour:sugar $=2:3$. If flour is $8$ cups, how many cups of sugar are needed?

    $12$ cups. Set up proportion: $\frac{2}{3} = \frac{8}{x}$, cross-multiply to get $x = 12$.

62. Identify the constant of proportionality $k$ if $y$ is proportional to $x$.

    $k=\frac{y}{x}$. The constant $k$ is the ratio of $y$ to $x$.

63. A class has boys:girls $=3:5$ and $24$ students total. How many boys are in the class?

    $9$. Total parts = $3+5=8$, boys = $\frac{3}{8} × 24 = 9$.

64. State the formula for converting a part-to-whole ratio $p:w$ to a fraction.

    $\frac{p}{w}$ (with $w\ne 0$). Part-to-whole ratio becomes a fraction with part as numerator, whole as denominator.

65. If a recipe uses flour:sugar $=4:1$, how much sugar is needed for $12$ cups of flour?

    $3$ cups. Set up proportion: $\frac{4}{1} = \frac{12}{x}$, so $x = 3$.

66. What is the cross-multiplication result of $\frac{a}{b}=\frac{c}{d}$ (with $b,d\ne 0$)?

    $ad=bc$. Cross-multiply by multiplying the numerator of each fraction by the other's denominator.

67. What is the proportion equation that states $\frac{a}{b}$ equals $\frac{c}{d}$?

    $\frac{a}{b}=\frac{c}{d}$. A proportion states that two ratios are equal.

68. Identify the equivalent ratio to $3:5$ with second term $20$.

    $12:20$. Multiply both terms by $4$ since $5 \times 4 = 20$.

69. If a scale drawing uses $1$ inch for $5$ feet, what actual length corresponds to $7$ inches?

    $35$ feet. Multiply: $7$ inches × $5$ feet/inch = $35$ feet.

70. Identify the means in the proportion $\frac{a}{b}=\frac{c}{d}$.

    $b$ and $c$. The means are the inner terms when written as $a:b::c:d$.

71. A recipe uses flour:sugar $=3:2$. If flour is $12$ cups, what is sugar in cups?

    $8$ cups. Set up proportion: $\frac{3}{2} = \frac{12}{s}$, so $s = 8$ cups.

72. What is the percent equivalent of the ratio $3:20$?

    $15\%$. Convert ratio to percent: $\frac{3}{20} = 0.15 = 15\%$.

73. If $y$ is proportional to $x$ and $y=18$ when $x=6$, what is $k$ in $y=kx$?

    $3$. Substitute values: $18 = k(6)$, so $k = \frac{18}{6} = 3$.

74. Identify whether the table shows proportionality: $(x,y)=(2,5),(4,10),(6,16)$.

    No, ratios $\frac{y}{x}$ are not constant. Check: $\frac{5}{2}=2.5$, $\frac{10}{4}=2.5$, but $\frac{16}{6}=2.67$, not constant.

75. Find $x$ if $7:21=x:12$.

    $x=4$. Set up proportion $\frac{7}{21} = \frac{x}{12}$; simplify to get $x = 4$.

76. What is the unit rate for $150$ miles in $3$ hours?

    $50$ miles per hour. Divide total distance by time: $\frac{150}{3} = 50$.

77. What is the scale factor from an original length $L$ to a new length $L'$?

    $\frac{L'}{L}$. Scale factor is the ratio of new to original length.

78. If $y$ is proportional to $x$ with $k=\frac{5}{2}$, what is $y$ when $x=8$?

    $20$. Substitute: $y = \frac{5}{2}(8) = \frac{40}{2} = 20$.

79. Find $x$ if $\frac{x}{9}=\frac{4}{3}$.

    $x=12$. Cross-multiply: $3x = 36$, so $x = 12$.

80. What is the inverse variation form if $y$ varies inversely with $x$ and $y=4$ when $x=3$?

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

81. What is the ratio of areas of similar figures with linear scale factor $k$?

    $k^2:1$. Area scales with the square of the linear scale factor.

82. A class has boys:girls $=4:5$ and $27$ students total. How many boys are there?

    $12$. Boys are $\frac{4}{9}$ of total: $\frac{4}{9} \times 27 = 12$.

83. What is the constant of proportionality $k$ if $y= kx$ and $(x,y)=(4,10)$?

    $k=\frac{5}{2}$. Substitute and solve: $10 = k(4)$, so $k = \frac{10}{4} = \frac{5}{2}$.

84. What is the new length if an $18$ cm segment is scaled by factor $\frac{5}{3}$?

    $30$ cm. Multiply: $18 \times \frac{5}{3}=30$.

85. What is the unit rate for $\frac{a}{b}$ (with $b>0$) interpreted as $a$ per $b$ units?

    $\frac{a}{b}$ per $1$ unit. Divide the ratio by the denominator to find the rate per one unit.

86. What must be true about the graph of a proportional relationship?

    It is a line through $(0,0)$. Proportional relationships always pass through the origin.

87. What is a proportion in terms of two ratios?

    An equation stating two ratios are equal. A proportion sets two ratios equal, like $\frac{a}{b}=\frac{c}{d}$.

88. Find $y$ if $y$ is proportional to $x$ with $y=18$ when $x=6$, and $x=10$.

    $y=30$. Find $k=3$ from first values, then $y=3(10)=30$.

89. What is $y$ if $y$ varies directly with $x$ and $y=14$ when $x=7$, then $x=10$?

    $y=20$. Find $k=2$ from first pair, then $y=2(10)=20$.

90. What is the cross-products condition for $\frac{a}{b}=\frac{c}{d}$ with $b\neq 0$ and $d\neq 0$?

    $ad=bc$. Cross-multiply the proportion to get this equality.

91. What is the percent equivalent of the ratio $1:4$ expressed as $\frac{1}{4}$?

    $25\%$. Convert fraction to percent: $\frac{1}{4} = 0.25 = 25\%$.

92. Find the missing value: $\frac{7}{x}=\frac{21}{30}$.

    $x=10$. Cross-multiply: $21x = 7 \cdot 30 = 210$, so $x = 10$.

93. What is the constant of proportionality $k$ in the equation $y=kx$ (with $x\neq 0$)?

    $k=\frac{y}{x}$. Solve for $k$ by dividing both sides by $x$.

94. What is the definition of a ratio comparing quantity $a$ to quantity $b$?

    $\frac{a}{b}$, read as $a:b$ or $a$ to $b$. A ratio compares two quantities by division.

95. Find $k$ if $y$ varies directly with $x$ and $(x,y)=(6,15)$.

    $k=\frac{5}{2}$. From $y = kx$: $15 = k \cdot 6$, so $k = \frac{15}{6} = \frac{5}{2}$.

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

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

97. What does it mean for two ratios to be proportional?

    They are equal: $\frac{a}{b}=\frac{c}{d}$ with $b,d\ne 0$. Two ratios are proportional when their cross-products are equal.

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

    $3:4$. Divide both parts by their GCD: $18÷6=3$, $24÷6=4$.

99. Find $x$ if $\frac{7}{x}=\frac{14}{20}$.

    $x=10$. Cross-multiply: $14x = 140$, so $x = 10$.

100. What is the missing value $x$ in the proportion $\frac{9}{12}=\frac{x}{20}$?

     $x=15$. Cross-multiply: $9×20=12x$, so $x=\frac{180}{12}=15$.

101. What is the part-to-whole fraction for $3$ red marbles out of $12$ total, simplified?

     $\frac{1}{4}$. Part-to-whole: $\frac{3}{12}=\frac{1}{4}$ after simplifying.

102. What does it mean for two ratios to be equivalent?

     They have the same value as fractions. Equivalent ratios simplify to the same fraction when reduced.

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

     $x=\frac{15}{2}$. Cross-multiply: $8x=60$, so $x=\frac{60}{8}=\frac{15}{2}$.

104. What is the definition of the ratio $a:b$ written as a fraction?

     $\frac{a}{b}$. A ratio $a:b$ is expressed as the fraction $\frac{a}{b}$.

105. Find $x$ if $\frac{3}{x}=\frac{9}{12}$.

     $x=4$. Cross-multiply: $3 \cdot 12 = 9x$, so $36 = 9x$ and $x = 4$.

106. Find $x$ if $y$ varies inversely with $x$, $y=3$, and $k=30$.

     $x=10$. From $xy = k$: $x \cdot 3 = 30$, so $x = 10$.

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

     $ad=bc$. Cross-multiply to check if two ratios are equal.

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

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

109. Identify $y$ if $y$ varies directly with $x$, $k=7$, and $x=3$.

     $y=21$. Substitute into $y=kx$: $y=7(3)=21$.

110. What is the definition of a scale factor from an original length $L$ to a new length $L'$?

     $\text{scale factor}=\frac{L'}{L}$. Scale factor is the ratio of new to original length.

111. A class has girls:boys $=5:3$ and $24$ students total. How many girls are in the class?

     $15$. Girls are $\frac{5}{8}$ of total: $\frac{5}{8} \times 24 = 15$.

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

     $ad = bc$. Cross-multiply to get the products equal when ratios are equal.

113. What is the new length after scaling $12$ by a factor of $\frac{3}{2}$?

     $18$. Multiply original by scale factor: $12×\frac{3}{2}=18$.

114. A map uses scale $1:50{,}000$. What distance (km) is $3$ cm on the map?

     $1.5\text{ km}$. $3$ cm $\times 50{,}000 = 150{,}000$ cm $= 1.5$ km.

115. Find $x$ if $5:8=x:40$.

     $x=25$. Cross-multiply: $8x=200$, so $x=25$.

116. What is the defining constant relationship for inverse variation between $x$ and $y$?

     $xy=k$. The product of inversely varying quantities is constant.

117. What is the definition of a scale factor $s$ from an original to a scaled copy?

     $s=\frac{\text{new}}{\text{original}}$. Scale factor is the ratio of new size to original size.

118. What is the percent form of the ratio $\frac{a}{b}$?

     $\frac{a}{b}\times 100\%$. Multiply the fraction by 100 to convert to percentage form.

119. What percent is equivalent to the ratio $3:5$ (as a part-to-whole percent)?

     $60\%$. Convert ratio to fraction: $\frac{3}{5}=0.6=60\%$.

120. Identify the missing value $x$ in the proportion $\frac{3}{4}=\frac{x}{12}$.

     $x=9$. Cross-multiply: $3 \cdot 12 = 4 \cdot x$, so $36 = 4x$ and $x = 9$.

121. What is $y$ if $y$ is directly proportional to $x$ with $y=8$ when $x=2$, and $x=7$?

     $y=28$. Find $k=\frac{8}{2}=4$, then $y=4×7=28$.

122. Identify $x$ if $\frac{x}{12}=\frac{5}{8}$.

     $x=\frac{15}{2}$. Cross-multiply: $8x=12×5=60$, so $x=\frac{60}{8}=\frac{15}{2}$.

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

     $ad=bc$. Cross-multiply: the product of means equals the product of extremes.

124. Find the missing term $x$ in the scale ratio $1:50$ if the actual length is $200$ units.

     $x=4$ units. Set up proportion: $\frac{1}{50} = \frac{x}{200}$, so $x = 4$.

125. What is the proportion equation for "$a$ is to $b$ as $c$ is to $d$"?

     $\frac{a}{b}=\frac{c}{d}$. This proportion states that the ratio of $a$ to $b$ equals the ratio of $c$ to $d$.

126. If $y=kx$ with $k=\frac{3}{5}$, what is $y$ when $x=20$?

     $y=12$. Substitute: $y = \frac{3}{5} \cdot 20 = \frac{60}{5} = 12$.

127. A recipe uses flour:sugar $=3:2$. How much sugar for $12$ cups of flour?

     $8$ cups. Set up proportion: $\frac{3}{2} = \frac{12}{x}$, so $x = 8$ cups sugar.

128. What is the formula for direct variation of $y$ with $x$ using constant $k$?

     $y=kx$. Direct variation means $y$ is proportional to $x$ with constant $k$.

129. What is the ratio of part to whole if the whole is $P+Q$ and the part is $P$?

     $\frac{P}{P+Q}$. Part-to-whole ratio divides the part by the total.

130. Find $x$ if $\frac{x}{12}=\frac{3}{4}$.

     $x=9$. Cross-multiply: $4x=36$, so $x=9$.

131. What is the unit rate for $150$ miles in $3$ hours?

     $50$ miles per hour. Divide total distance by total time: $\frac{150}{3}=50$.

132. A map scale is $1$ inch $=5$ miles. What distance is shown by $3.2$ inches?

     $16$ miles. Multiply: $3.2 \times 5 = 16$ miles.

133. Two quantities are in the ratio $5:8$. If the smaller quantity is $25$, what is the larger quantity?

     $40$. Set up proportion: $\frac{5}{8} = \frac{25}{x}$, cross-multiply to get $x = 40$.

134. A store has boys:girls $=5:7$. If there are $84$ students total, how many are girls?

     $49$. Total ratio parts: $5+7=12$; girls: $\frac{7}{12} \times 84 = 49$.

135. What is the definition of the ratio $a:b$ written as a fraction?

     $\frac{a}{b}$. A ratio $a:b$ is expressed as the fraction $\frac{a}{b}$.

136. Identify the extremes in the proportion $\frac{a}{b}=\frac{c}{d}$.

     $a$ and $d$. The extremes are the outer terms when written as $a:b::c:d$.

137. What is the unit rate for $\frac{45\ \text{miles}}{3\ \text{hours}}$?

     $15\ \text{miles per hour}$. Divide total miles by total hours: $\frac{45}{3} = 15$.

138. A quantity increases from $40$ to $50$. What is the percent increase?

     $25\%$. Percent increase: $\frac{50-40}{40} \times 100\% = 25\%$.

139. Identify whether $y=4x+1$ is proportional, and state the reason in one phrase.

     Not proportional; it does not pass through $(0,0)$. The $+1$ term means the line doesn't pass through the origin.

140. What is the formula for a part-to-whole fraction if parts are $a$ and $b$?

     $\frac{a}{a+b}$. Part $a$ over the total $(a+b)$ gives the part-to-whole fraction.

141. If $y$ varies inversely with $x$ and $x$ doubles, what happens to $y$?

     $y$ is multiplied by $\frac{1}{2}$. In inverse variation, when one variable doubles, the other halves.

142. Find $y$ if $y$ is proportional to $x$, $k=4$, and $x=7$.

     $y=28$. Use $y = kx$: $y = 4 \cdot 7 = 28$.

143. A quantity decreases from $80$ to $68$. What is the percent decrease?

     $15\%$. Percent decrease: $\frac{80-68}{80} \times 100\% = 15\%$.

144. What is the decimal form of $r\%$?

     $\frac{r}{100}$. To convert percent to decimal, divide by 100.

145. What is the percent form of a ratio $\frac{a}{b}$?

     $\frac{a}{b}\times 100\%$. Convert a ratio to percent by multiplying by $100\%$.

146. Identify whether the table is proportional: $(x,y)=(2,6),(5,15),(8,24)$.

     Yes, because $\frac{y}{x}=3$ each time. Check each ratio: $\frac{6}{2} = \frac{15}{5} = \frac{24}{8} = 3$.

147. Which expression gives the constant of proportionality when $y$ is proportional to $x$?

     $k=\frac{y}{x}$. The constant ratio between proportional quantities.

148. What is the constant of proportionality $k$ if $y$ is proportional to $x$ and $y=18$ when $x=6$?

     $k=3$. Find $k$ using $y=kx$: $18=k(6)$, so $k=3$.

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

     $ad=bc$. Cross-multiply: the product of extremes equals the product of means.

150. State the proportion equation that represents $a:b=c:d$.

     $\frac{a}{b}=\frac{c}{d}$. A proportion equates two ratios as fractions.

151. What is the ratio of $x$ to $y$ if $x=27$ and $y=36$, in simplest form?

     $3:4$. Simplify $\frac{27}{36}$ by dividing by GCD of 9.

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

     $60$ miles per hour. Divide total distance by time: $180÷3=60$.

153. What is $15\%$ of $80$?

     $12$. Calculate $0.15 \times 80 = 12$.

154. If lengths scale by $k$, what is the area scale factor in terms of $k$?

     Area scales by $k^2$. Area is 2-dimensional, so it scales by the square of the linear scale factor.

155. Convert the ratio $3:8$ to a percent (nearest tenth is not needed).

     $37.5\%$. $\frac{3}{8} \times 100\% = 37.5\%$.

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

     $\frac{a}{b}=\frac{c}{d}$ (with $b,d\ne 0$). Two ratios are equal when their cross-products are equal.

157. What is the equation of a proportional relationship with constant $k$ using variables $x$ and $y$?

     $y=kx$. Direct variation means $y$ varies directly with $x$.

158. A recipe uses flour:sugar $=3:2$. What is the sugar amount for $12$ cups of flour?

     $8$ cups. Set up proportion: $\frac{3}{2} = \frac{12}{x}$, so $x = 8$.

159. What is the equation for direct variation if $y$ varies directly with $x$ with constant $k$?

     $y=kx$. Direct variation means $y$ is a constant multiple of $x$.

160. What is the percent formula relating part, whole, and percent $p$?

     $\text{part}=\frac{p}{100}\cdot\text{whole}$. Convert percent to decimal by dividing by $100$, then multiply by the whole.

161. What is $x$ if $\frac{x+2}{10}=\frac{3}{5}$?

     $x=4$. Cross-multiply: $5(x+2)=30$, so $x+2=6$ and $x=4$.

162. What is the unit rate for $\frac{a}{b}$ (with $b\ne 0$) expressed "per $1$"?

     $\frac{a}{b}$ per $1$. Unit rate divides the numerator by the denominator to get the rate per one unit.

163. What is the slope of the line $y=kx$ in a proportional relationship graph?

     $k$. In $y = kx$, the coefficient $k$ is the slope.

164. A map has scale $1$ inch $=5$ miles. How many miles correspond to $3.5$ inches?

     $17.5$ miles. Multiply: $3.5 \text{ inches} \times 5 \frac{\text{miles}}{\text{inch}} = 17.5$ miles.

165. What is the slope of a proportional relationship graphed by $y=kx$?

     $k$. In $y=kx$, the coefficient $k$ is the slope.

166. A recipe uses flour:sugar $=3:2$. How much sugar for $12$ cups flour?

     $8$ cups. Set up proportion: $\frac{3}{2}=\frac{12}{x}$, so $x=8$.

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

     $\frac{a}{b}=\frac{c}{d}$. A proportion states that two ratios are equal.

168. Identify the missing value $x$ in the proportion $\frac{x}{5}=\frac{14}{10}$.

     $x=7$. Cross-multiply: $x \cdot 10 = 5 \cdot 14$, so $10x = 70$ and $x = 7$.

169. Identify $x$ if $3:5=x:40$.

     $x=24$. Set up proportion $\frac{3}{5}=\frac{x}{40}$; cross-multiply: $5x=120$.

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

     $ad=bc$. Cross-multiply to get the products of opposite terms equal.

171. If a figure is scaled by factor $\frac{3}{2}$, what is the area scale factor?

     $\left(\frac{3}{2}\right)^2=\frac{9}{4}$. Area scales by the square of the linear scale factor.

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

     $\frac{a}{b} = \frac{c}{d}$, with $b \ne 0$ and $d \ne 0$. A proportion states that two ratios are equal.

173. If $y$ is proportional to $x$ and $y=15$ when $x=6$, what is $k$ in $y=kx$?

     $k=\frac{5}{2}$. Substitute values: $15 = k \cdot 6$, so $k = \frac{15}{6} = \frac{5}{2}$.

174. What is the $y$-intercept of a proportional relationship written as $y=kx$?

     $0$. Proportional relationships pass through the origin $(0,0)$.

175. What is the cross-products condition for $\frac{a}{b}=\frac{c}{d}$ with $b\neq 0$ and $d\neq 0$?

     $ad=bc$. Cross-multiply to get the products equal.

176. If similar figures have side ratio $3:5$, what is the ratio of their areas?

     $9:25$. Square the side ratio: $(\frac{3}{5})^2=\frac{9}{25}$.

177. What is the unit rate for $180$ miles in $4$ hours?

     $45$ miles per hour. Divide total distance by time: $\frac{180}{4}=45$ mph.

178. In a mixture with ratio water:juice $=2:5$, what fraction of the mixture is water?

     $\frac{2}{7}$. Water is $\frac{2}{2+5} = \frac{2}{7}$ of the total mixture.

179. What is the percent formula that relates part, whole, and percent $p$?

     $\text{part}=\frac{p}{100}\cdot\text{whole}$. To find a percent of a whole, multiply by the decimal form.

180. Convert the ratio $3:5$ to a fraction representing the first part of the whole.

     $\frac{3}{8}$. The whole is $3+5=8$, so the first part is $\frac{3}{8}$ of the whole.

181. State the formula for converting part-to-part ratio $a:b$ to part-to-whole.

     $a:(a+b)$ and $b:(a+b)$. Each part becomes a ratio with the sum $(a+b)$ as the whole.

182. A figure is enlarged by scale factor $k=\frac{3}{2}$. By what factor does area change?

     $\frac{9}{4}$. Area scales by $k^2 = (\frac{3}{2})^2 = \frac{9}{4}$.

183. What graph feature must a proportional relationship $y=kx$ have?

     It passes through $(0,0)$. Proportional relationships always start at the origin since $y=0$ when $x=0$.

184. Find the missing value $x$ in the proportion $\frac{7}{x}=\frac{21}{30}$.

     $x=10$. Cross-multiply: $21x=210$, so $x=10$.

185. What is the constant of proportionality $k$ in $y=kx$ if $y=15$ when $x=5$?

     $k=3$. Substitute values: $15=k(5)$, so $k=15÷5=3$.

186. Find the unit price if $6$ notebooks cost $9.

     $1.50 per notebook. Divide total cost by number of items: $\frac{\$9}{6} = \$1.50$.

187. What is the equation for a proportional relationship between $y$ and $x$ using constant $k$?

     $y=kx$. Direct variation means $y$ equals a constant times $x$.

188. What is the unit rate (dollars per item) for $18 for $6$ items?

     $3 per item. Divide total cost by number of items: $\frac{\$18}{6} = \$3$ per item.

189. What is the cross-multiplication rule for $\frac{a}{b}=\frac{c}{d}$ with $b\ne 0$ and $d\ne 0$?

     $ad=bc$. Cross-multiply to solve proportions: product of extremes equals product of means.

190. Find $y$ if $y$ is proportional to $x$ with $y=18$ when $x=6$.

     $y=3x$. Find $k = \frac{18}{6} = 3$, so $y = 3x$.

191. Which value of $k$ makes $y=kx$ pass through $(4,14)$?

     $k=\frac{7}{2}$. Substitute: $14 = k(4)$, so $k = \frac{14}{4} = \frac{7}{2}$.

192. If a scale is $1$ inch $=5$ miles, what distance (miles) is shown by $3$ inches?

     $15$ miles. Multiply: $3$ inches $×$ $5$ miles/inch $=15$ miles.

193. What is the relationship between side lengths and areas under scale factor $s$?

     Lengths $\times s$; areas $\times s^2$. Linear dimensions scale by $s$; areas scale by $s^2$.

194. What value of $x$ makes the proportion $\frac{x}{12} = \frac{5}{8}$ true?

     $\frac{15}{2}$. Cross-multiply: $8x = 60$, so $x = \frac{60}{8} = \frac{15}{2}$.

195. Identify $k$ if $y$ is proportional to $x$ and $(x,y)=(8,20)$ lies on $y=kx$.

     $k=\frac{5}{2}$. Substitute into $y = kx$: $20 = k(8)$, so $k = \frac{20}{8} = \frac{5}{2}$.

196. Which statement identifies a proportional relationship in a table of $(x,y)$ values?

     $\frac{y}{x}$ is constant for all pairs. In proportional relationships, the ratio $\frac{y}{x}$ remains constant.

197. Find $x$ if $\frac{x}{12}=\frac{5}{8}$.

     $x=\frac{15}{2}$. Cross-multiply: $8x = 60$, so $x = \frac{60}{8} = \frac{15}{2}$.

198. A ratio is simplified to $5:6$. What is an equivalent ratio with the second term $=24$?

     $20:24$. Multiply both terms by $4$: $5 \times 4 : 6 \times 4 = 20:24$.

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

     $\frac{a}{b}=\frac{c}{d}$. A proportion states that two ratios are equal.

200. What is the standard equation for a proportional relationship between $y$ and $x$?

     $y=kx$. Direct proportionality means $y$ equals a constant times $x$.

201. What is the scale factor from a length of $4$ to a length of $10$?

     $\frac{5}{2}$. Scale factor = $\frac{\text{new length}}{\text{original length}}=\frac{10}{4}=\frac{5}{2}$.

202. What is the slope of a proportional relationship graphed by $y=kx$?

     $k$. In $y=kx$, the coefficient $k$ represents the slope of the line.

203. What does the ratio $a:b$ mean in fraction form?

     $\frac{a}{b}$. A ratio $a:b$ is written as a fraction with first term over second term.

204. What is the scale factor formula for similar figures using corresponding sides $a$ and $b$?

     $\text{scale factor}=\frac{a}{b}$. Scale factor is the ratio of corresponding lengths in similar figures.

205. What is the equation of a proportional relationship on a coordinate plane?

     A line through the origin: $y=kx$. Proportional relationships pass through $(0,0)$ with slope $k$.

206. What is the definition of the ratio $a:b$ in fraction form?

     $\frac{a}{b}$. A ratio $a:b$ is written as a fraction with $a$ in the numerator and $b$ in the denominator.

207. Find $y$ if $y$ is proportional to $x$, $k=\frac{3}{4}$, and $x=20$.

     $y=15$. Use $y = kx$: $y = \frac{3}{4} \cdot 20 = 15$.

208. If $y$ is proportional to $x$ and $y=18$ when $x=6$, what is $k$ in $y=kx$?

     $k=3$. Substitute values into $y = kx$: $18 = k(6)$, so $k = 3$.

209. Find $x$ if $7:x=14:20$.

     $x=10$. Cross-multiply: $7 \times 20 = 14x$, so $140 = 14x$ and $x = 10$.

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

     $x=\frac{15}{2}$. Cross-multiply: $8x=60$, so $x=\frac{60}{8}=\frac{15}{2}$.

211. Identify $x$ if $\frac{7}{x}=\frac{21}{30}$.

     $x=10$. Cross-multiply: $210 = 21x$, so $x = 10$.

212. A map scale is $1:50{,}000$. What actual distance is $3$ cm on the map?

     $150{,}000$ cm. Multiply map distance by scale: $3 \times 50{,}000 = 150{,}000$ cm.

213. What value of $x$ makes the proportion $\frac{3}{4} = \frac{x}{20}$ true?

     $15$. Cross-multiply: $3 × 20 = 4x$, so $60 = 4x$ and $x = 15$.

214. A recipe uses flour:sugar $=2:3$. If flour is $10$ cups, how many cups of sugar?

     $15$ cups. Set up proportion: $\frac{2}{3} = \frac{10}{x}$, cross-multiply to get $x = 15$.

215. A mixture has ratio water:juice $=2:3$. What fraction of the mixture is water?

     $\frac{2}{5}$. Total parts = $2+3=5$, water fraction = $\frac{2}{5}$.

216. What is the percent equivalent of the ratio $3:20$?

     $15\%$. Convert: $\frac{3}{20}=0.15=15\%$.

217. Find the unit rate if $12$ items cost $30 (dollars per item).

     $2.50 per item. Divide total cost by number of items: $\frac{30}{12} = 2.50$.

218. What is the unit rate for $\frac{a}{b}$ when $b\neq 0$?

     $\frac{a}{b}$ per $1$ unit of $b$. Divide $a$ by $b$ to get the amount of $a$ per one unit of $b$.

219. What is the simplified ratio for $24:36$ in lowest terms?

     $2:3$. Divide both terms by their GCD: $24÷12:36÷12=2:3$.

220. What is the graph shape of a proportional relationship $y=kx$ in the coordinate plane?

     A line through $(0,0)$. Proportional relationships always pass through the origin.

221. What is the definition of the ratio $a:b$ in fraction form?

     $\frac{a}{b}$. A ratio $a:b$ is expressed as a fraction with $a$ in the numerator and $b$ in the denominator.

222. Identify the equivalent ratio to $7:9$ with second term $45$.

     $35:45$. Scale up by factor $5$: $7×5:9×5=35:45$.

223. What is the definition of a ratio comparing quantities $a$ and $b$?

     $a:b$ or $ \frac{a}{b}$, with $b \ne 0$. A ratio compares two quantities by division or colon notation.

224. What is the unit rate for a ratio written as $\frac{a}{b}$ (with $b>0$)?

     $\frac{a}{b}$ per $1$. Unit rate shows the amount per one unit of the denominator.

225. Find $x$ if $x:9=4:3$.

     $x=12$. Cross-multiply: $3x = 9 \cdot 4 = 36$, so $x = 12$.

226. A recipe uses flour:sugar $=3:2$. If flour is $12$ cups, how many cups of sugar are needed?

     $8$ cups. Set up proportion: $\frac{3}{2} = \frac{12}{x}$, so $x = 8$.

227. Find $x$ if $\frac{3}{5}=\frac{x}{20}$.

     $x=12$. Cross-multiply: $3 \times 20 = 5x$, so $x = 12$.

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

     $\frac{a}{b}=\frac{c}{d}$. A proportion states that two ratios are equal.

229. What is the simplified form of the ratio $18:24$?

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

230. A recipe uses flour:sugar $=3:2$. If flour is $12$ cups, how many cups of sugar?

     $8$ cups. Set up proportion $\frac{3}{2} = \frac{12}{x}$; cross-multiply to get $x = 8$.

231. What is the constant of proportionality $k$ in $y=kx$ if $(x,y)=(6,15)$?

     $k=\frac{5}{2}$. Substitute into $y=kx$: $15=k(6)$, so $k=\frac{15}{6}=\frac{5}{2}$.

232. Identify the unit rate: $18$ miles in $3$ hours equals how many miles per hour?

     $6$ miles per hour. Divide total miles by total hours: $\frac{18}{3} = 6$ miles per hour.

233. What is the equation for a proportional relationship between $y$ and $x$?

     $y=kx$. Direct proportionality means $y$ equals a constant times $x$.

234. Which option is equivalent to the ratio $7:2$ written as a fraction?

     $\frac{7}{2}$. The ratio $a:b$ converts to fraction $\frac{a}{b}$.

235. What is the relationship between ratios and scale factor in similar figures?

     All corresponding side ratios equal the scale factor. In similar figures, each side ratio equals the scale factor.

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

     $x=10$. Cross-multiply: $210=21x$, so $x=10$.

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