Rates - PSAT Math

1. Find the time for combined work: Using the machines from the previous question, how many minutes per part together?

   $2\text{ minutes per part}$. Reciprocal of combined rate: $\frac{1}{\frac{1}{2}} = 2$.

2. A machine makes $120$ parts in $8$ hours. What is the rate in parts per hour?

   $15$ parts per hour. Divide total parts by hours: $\frac{120}{8}=15$.

3. What time is needed to travel $150$ miles at $50\text{ mph}$?

   $3\text{ hours}$. Using $t = \frac{d}{r}$: $150 \div 50 = 3$ hours.

4. What is the conversion from $\text{mi/h}$ to $\text{ft/s}$ for a speed of $1\ \text{mi/h}$?

   $1\ \text{mi/h}=\frac{22}{15}\ \text{ft/s}$. Use $\frac{5280\text{ ft}}{3600\text{ s}}=\frac{22}{15}\text{ ft/s}$.

5. What is the slope (rate) of the line through $(2,3)$ and $(6,11)$?

   $2$. Use $m=\frac{y_2-y_1}{x_2-x_1}$: $\frac{11-3}{6-2}=\frac{8}{4}=2$.

6. If a pipe fills a tank in $6$ hours, what fraction of the tank is filled per hour?

   $\frac{1}{6}$ of the tank per hour. Rate is reciprocal of time to complete one job.

7. What is the slope-intercept form of a line, showing constant rate of change?

   $y=mx+b$. Standard form where $m$ is the rate and $b$ is initial value.

8. What time is needed to travel $90$ miles at $45\ \text{mi/h}$?

   $2\ \text{hours}$. Divide distance by rate: $\frac{90}{45}=2$.

9. What is the percent rate formula for $p\%$ written as a decimal?

   $\frac{p}{100}$. Convert percent to decimal by dividing by 100.

10. What distance is traveled at $45\text{ mi/hr}$ for $2$ hours?

    $90\text{ mi}$. Apply $d = rt = 45 \times 2$.

11. What is the percent increase from $50$ to $65$?

    $30\%$. $\frac{65-50}{50} \times 100\% = 30\%$ increase.

12. If $y=kx$ with $k=2.5$, what is $y$ when $x=12$?

    $y=30$. Substitute into direct variation: $y=2.5\times 12=30$.

13. What is the density if mass is $36$ g and volume is $12$ cm$^3$?

    $3$ g per cm$^3$. $\rho = \frac{36}{12} = 3$ g/cm³.

14. What is the conversion factor to change hours to minutes?

    $1\text{ hr}=60\text{ min}$. Standard time conversion for hour-to-minute calculations.

15. State the formula for distance traveled at constant speed.

    $d = rt$. Distance equals rate times time for constant motion.

16. State the formula for distance in terms of speed and time.

    $\text{distance}=\text{speed}\times\text{time}$. Rearranged speed formula by multiplying both sides by time.

17. What is the formula for distance in terms of speed $v$ and time $t$?

    $d=vt$. Multiply speed by time to find distance.

18. What is the unit rate if $6$ pounds of apples cost $15?

    $2.50 per pound. Divide total cost by quantity: $\frac{15}{6}=2.50$.

19. What is the combined rate if machine A makes $1$ item in $6$ min and B makes $1$ item in $3$ min?

    $\frac{1}{2}$ item per minute. Add individual rates: $\frac{1}{6} + \frac{1}{3} = \frac{1}{2}$ per minute.

20. What is the combined work rate formula for two workers with rates $r_1$ and $r_2$?

    $r_{\text{total}}=r_1+r_2$. Add individual rates when working together.

21. What is the reciprocal rate of $a$ miles per hour?

    $\frac{1}{a}$ hours per mile. Flip the fraction to get reciprocal rate.

22. State the formula for unit price as a rate.

    $\text{unit price}=\frac{\text{cost}}{\text{items}}$. Total cost divided by quantity gives cost per item.

23. What is the formula for price per unit if total cost is $C$ for $n$ items, with $n \ne 0$?

    $\frac{C}{n}$ dollars per item. Unit price equals total cost divided by quantity.

24. Convert $3\text{ m/s}$ to $\text{m/min}$.

    $180\text{ m/min}$. Multiply by $\frac{60\text{ s}}{1\text{ min}}$ to convert units.

25. What is the constant of proportionality $k$ in the direct variation equation $y=kx$ if $y=5x$?

    $k=5$. In $y=kx$, the coefficient of $x$ is $k$.

26. What is the formula for density in terms of mass $m$ and volume $V$?

    $\rho=\frac{m}{V}$. Density equals mass divided by volume.

27. A printer makes $180$ pages in $6$ minutes. What is the rate in $\text{pages/min}$?

    $30\text{ pages/min}$. Divide total pages by time: $\frac{180}{6}$.

28. What is the time needed to travel $90$ miles at $45$ miles per hour?

    $2$ hours. $t = \frac{d}{r} = \frac{90}{45} = 2$ hours.

29. What conversion factor changes $\text{hours}$ to $\text{minutes}$ in a rate problem?

    $1\text{ hour} = 60\text{ minutes}$. Standard time conversion for rate calculations.

30. Find the slope (rate of change) between $(2,5)$ and $(6,17)$.

    $3$. Use slope formula: $\frac{17-5}{6-2}=\frac{12}{4}=3$.

31. What is the relationship between rate $r$, time $t$, and work $W$ for constant productivity?

    $W=rt$. Work equals rate times time for constant productivity.

32. Identify the unit rate: A car travels $150$ miles in $3$ hours. What is the speed?

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

33. What is the unit price if $5$ notebooks cost $12.50$?

    $2.50$ per notebook. Divide total cost by number of items: $\frac{12.50}{5}=2.50$.

34. State the direct variation equation relating $y$, $k$, and $x$.

    $y=kx$. Shows $y$ is a constant multiple of $x$.

35. What is the time formula for one job when the combined work rate is $r$ jobs per hour?

    $t=\frac{1}{r}$ hours. Time equals 1 divided by the rate.

36. A car goes $60$ miles in $1$ hr, then $60$ miles in $2$ hr. What is the average speed?

    $40\text{ mi/hr}$. Average speed: $\frac{\text{total distance}}{\text{total time}} = \frac{120}{3}$.

37. Identify the unit price: $12 for $8$ pounds equals how many dollars per pound?

    $\$1.50\text{ per lb}$. $12 \div 8 = 1.50$ dollars per pound.

38. How long to make $1$ item together if A makes $1$ item in $6$ min and B makes $1$ item in $3$ min?

    $2\text{ minutes}$. Reciprocal of combined rate: $\frac{1}{\frac{1}{2}} = 2$ minutes.

39. What is the conversion from $1$ mile to feet?

    $1\text{ mi} = 5280\text{ ft}$. Standard distance conversion in US customary units.

40. What is the combined work rate if one person works at $r_1$ job per hour and another at $r_2$ job per hour?

    $r_1+r_2$ job per hour. Add individual rates when working simultaneously.

41. What is the relative speed when two objects move in the same direction at $v_1>v_2$?

    $v_1-v_2$. When moving same direction, subtract slower from faster speed.

42. Two people start $120$ miles apart and drive toward each other at $50$ and $30\text{ mi/hr}$. What is the closing speed?

    $80\text{ mi/hr}$. Add speeds when moving toward each other: $50+30=80$.

43. Identify the average speed: travel $60$ miles in $1$ hour, then $60$ miles in $2$ hours.

    $40$ miles per hour. $\frac{120}{3} = 40$ mph for total distance over total time.

44. What is the formula for time in terms of distance $d$ and speed $v$?

    $t=\frac{d}{v}$. Divide distance by speed to find time.

45. How many hours for two workers together if their combined work rate is $\frac{1}{4}$ job per hour?

    $4$ hours. $t = \frac{1}{\frac{1}{4}} = 4$ hours.

46. State the formula for distance in terms of speed and time.

    $\text{distance}=\text{speed}\times\text{time}$. Rearranges speed formula by multiplying both sides by time.

47. Compute a percent increase: A price rises from $50$ to $60$. What is the percent increase?

    $20\%$. Use $\frac{60-50}{50}\times 100$.

48. Identify the unit rate: $18 for $6$ tickets.

    $3 per ticket. Divide total cost by number of tickets: $\frac{\$18}{6}$.

49. What is the formula for speed in terms of distance $d$ and time $t$?

    $v=\frac{d}{t}$. Speed equals distance divided by time.

50. State the formula for speed in terms of distance and time.

    $\text{speed}=\frac{\text{distance}}{\text{time}}$. Fundamental formula relating motion quantities.

51. What is the better buy per ounce: $12$ oz for $3 or $20$ oz for $4?

    $20$ oz for $4. Compare $\frac{3}{12} = 0.25$ vs $\frac{4}{20} = 0.20$ per oz.

52. What is the constant of proportionality $k$ if $y$ varies directly with $x$?

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

53. What is the formula for percent change from original $A$ to new $B$?

    $\frac{B-A}{A}\times 100\%$. Subtract original from new, divide by original, multiply by $100\%$.

54. What distance is traveled in $2.5$ hours at $40$ miles per hour?

    $100$ miles. $d = 40 \times 2.5 = 100$ miles.

55. State the formula for a constant rate: distance in terms of rate and time.

    $d=rt$. Distance equals rate multiplied by time.

56. State the formula for density.

    $\text{density}=\frac{\text{mass}}{\text{volume}}$. Mass per unit volume measures how compact matter is.

57. What is the formula for average rate of change from $(x_1,y_1)$ to $(x_2,y_2)$ with $x_2 \ne x_1$?

    $\frac{y_2-y_1}{x_2-x_1}$. Change in $y$ divided by change in $x$ gives the slope.

58. State the formula for speed as a rate.

    $\text{speed}=\frac{\text{distance}}{\text{time}}$. Fundamental rate formula relating motion quantities.

59. Convert $72\text{ mi/hr}$ to $\text{ft/s}$.

    $105.6\text{ ft/s}$. Multiply by $\frac{5280\text{ ft}}{1\text{ mi}}\times\frac{1\text{ hr}}{3600\text{ s}}$.

60. What is the speed if you travel $45$ miles in $\frac{3}{4}$ hour?

    $60$ miles per hour. Using $r = \frac{d}{t}$: $\frac{45}{\frac{3}{4}} = 45 \times \frac{4}{3} = 60$ mph.

61. Find the distance: A cyclist rides at $12$ miles per hour for $2.5$ hours. How far?

    $30$ miles. Use $d=st$: $12\times 2.5=30$ miles.

62. State the formula for speed in terms of distance and time.

    $\text{speed}=\frac{\text{distance}}{\text{time}}$. Fundamental formula relating motion quantities.

63. What is the average rate of change of $y$ with respect to $x$ from $x=a$ to $x=b$?

    $\frac{y(b)-y(a)}{b-a}$. Measures how much $y$ changes per unit change in $x$.

64. What is the cost of $7$ pounds at $2.40 per pound?

    $16.80. $7 \times 2.40 = 16.80$ dollars.

65. What is the definition of unit rate, and how is it written as a fraction?

    A rate per $1$ unit, written as $\frac{a}{1}$ or $a$ per $1$. Unit rates show the amount per one unit of the denominator.

66. What is the combined work rate if one machine does $\frac{1}{6}$ job/hr and another does $\frac{1}{3}$ job/hr?

    $\frac{1}{2}$ job per hour. $\frac{1}{6} + \frac{1}{3} = \frac{1}{6} + \frac{2}{6} = \frac{3}{6} = \frac{1}{2}$.

67. Find the time to travel $90$ miles at $45\text{ mi/hr}$.

    $2\text{ hr}$. Rearrange $d=rt$ to get $t=\frac{d}{r}=\frac{90}{45}=2$.

68. State the combined work formula for two workers with times $a$ and $b$ (hours per job).

    $\frac{1}{a}+\frac{1}{b}=\frac{1}{t}$. Combined rates add when working together.

69. If one worker finishes a job in $8$ hours and another in $12$ hours, what is $t$ together?

    $t=\frac{24}{5}$ hours. Use $\frac{1}{8}+\frac{1}{12}=\frac{1}{t}$; solve for $t$.

70. Identify the unit rate: $18 for $6$ pounds of apples. What is the cost per pound?

    $3 per pound. $\frac{18}{6} = 3$ dollars per pound.

71. State the formula for distance when traveling at constant speed for a given time.

    $d = rt$. Distance equals rate times time for constant speed motion.

72. Find the distance: Traveling at $60\text{ mph}$ for $2.5$ hours, what distance is covered?

    $150\text{ miles}$. Apply $d = rt$: $60 \times 2.5 = 150$.

73. What is the constant rate of change if a value increases from $20$ to $35$ in $5$ hours?

    $3$ per hour. $\frac{35-20}{5} = \frac{15}{5} = 3$ per hour.

74. What is the formula for speed when distance is $d$ and time is $t$ (with $t \ne 0$)?

    $r = \frac{d}{t}$. Rate equals distance divided by time.

75. What is the time to finish $1$ job at constant work rate $r$ jobs per hour, with $r \ne 0$?

    $\frac{1}{r}$ hours. Time is reciprocal of rate for one job.

76. What is the definition of a rate in math?

    A ratio comparing two quantities with different units. Rates express relationships between different measurement units.

77. Identify the unit rate: $180$ miles in $3$ hours equals how many miles per hour?

    $60$ miles per hour. $\frac{180}{3} = 60$ miles per hour.

78. Identify the unit price: $12$ apples cost $6. What is the cost per apple?

    $0.50 per apple. Divide total cost by quantity: $\frac{\$6}{12}=\$0.50$.

79. Find the unit price: $12 for $8$ notebooks. What is the cost per notebook?

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

80. What is the definition of a rate?

    A ratio comparing two quantities with different units. Rates express relationships between different measurement types.

81. What is the unit price for $6$ items costing $18$ dollars?

    $3$ dollars per item. Divide $18 by $6$ items to get price per item.

82. If $y$ varies directly with $x$ and $y=18$ when $x=6$, what is $k$?

    $k=3$. Use $k=\frac{y}{x}$: $k=\frac{18}{6}=3$.

83. What is the combined work time if one worker takes $6$ hours and another takes $3$ hours?

    $2$ hours. Combined rate $\frac{1}{6} + \frac{1}{3} = \frac{1}{2}$, so time is $2$ hours.

84. Find the constant of proportionality: If $y=18$ when $x=6$ in a proportional relationship, what is $k$?

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

85. A printer makes $120$ pages in $3$ minutes. How many pages per minute is this rate?

    $40$ pages per minute. $\frac{120}{3} = 40$ pages per minute.

86. What is the formula for distance in terms of speed $v$ and time $t$?

    $d=vt$. Rearrange $v=\frac{d}{t}$ to solve for distance.

87. What distance is traveled at $60\ \text{mi/h}$ for $2.5$ hours?

    $150\ \text{miles}$. Multiply rate by time: $60 \times 2.5=150$.

88. What is the formula for time when $d$ is distance and $r$ is rate?

    $t=\frac{d}{r}$. Time equals distance divided by rate.

89. What is the formula for a population after $t$ hours with initial $P_0$ and constant rate $r$ per hour?

    $P=P_0+rt$. Linear growth: initial value plus rate times time.

90. What is the formula for rate when $d$ is distance and $t$ is time?

    $r=\frac{d}{t}$. Rate equals distance divided by time.

91. Identify the equation that represents a proportional relationship with unit rate $k$.

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

92. Two pipes fill a tank in $6$ hours and $3$ hours. How long to fill $1$ tank together?

    $2$ hours. $\frac{1}{\frac{1}{2}} = 2$ hours to fill one tank.

93. What is the formula for constant speed in terms of distance $d$ and time $t$?

    $v=\frac{d}{t}$. Speed equals distance divided by time.

94. What is the unit rate if a car travels $180$ miles in $3$ hours?

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

95. What is the time to finish $1$ job with combined work rate $r_1+r_2$ job per hour?

    $\frac{1}{r_1+r_2}$ hours. Time equals work divided by combined rate.

96. What is the percent decrease from $80$ to $60$?

    $25\%$. Use $\frac{60-80}{80}\times 100\%=\frac{-20}{80}\times 100\%=25\%$.

97. State the formula for speed when distance traveled and time are known.

    $r = \frac{d}{t}$. Rate equals distance divided by time.

98. Two pipes fill a tank in $6$ hours and $3$ hours. What is the combined fill rate in tanks per hour?

    $\frac{1}{2}$ tank per hour. $\frac{1}{6} + \frac{1}{3} = \frac{1}{6} + \frac{2}{6} = \frac{3}{6} = \frac{1}{2}$.

99. One pipe fills a tank in $6$ hours. What is its fill rate in tanks per hour?

    $\frac{1}{6}$ tank per hour. Rate is reciprocal of time: $\frac{1}{6}$ tank per hour.

100. What is the speed in $\text{mi/hr}$ if you travel $150$ miles in $3$ hours?

     $50\text{ mi/hr}$. Apply $r = \frac{d}{t} = \frac{150}{3}$.

101. Solve a work-rate problem: One machine makes $30$ parts per hour. How many parts in $20$ minutes?

     $10$ parts. $20$ min $= \frac{1}{3}$ hr, so $30\times\frac{1}{3}=10$.

102. What is the formula for density given mass $m$ and volume $V$, with $V \ne 0$?

     $\rho = \frac{m}{V}$. Density equals mass divided by volume.

103. Identify the constant of proportionality $k$ if $y$ varies directly with $x$ and $y=18$ when $x=6$.

     $k=3$. Use $k=\frac{y}{x}$: $\frac{18}{6}=3$.

104. What is the slope of the line given by $3x+2y=10$?

     $-\frac{3}{2}$. Rewrite as $y=-\frac{3}{2}x+5$; slope is $-\frac{3}{2}$.

105. State the formula for average rate of change from $(x_1,y_1)$ to $(x_2,y_2)$.

     $\frac{y_2-y_1}{x_2-x_1}$. Calculates slope between two points on a graph.

106. A tank fills at $3\ \text{gal/min}$ and drains at $1\ \text{gal/min}$. What is the net rate?

     $2\ \text{gal/min}$. Net rate is fill rate minus drain rate: $3-1=2$.

107. At a constant rate of $4$ gallons per minute, how many gallons in $7$ minutes?

     $28$ gallons. Multiply rate by time: $4\times 7=28$ gallons.

108. What does the slope $m$ represent in $y=mx+b$ when $x$ is time and $y$ is distance?

     Speed (distance per time). Slope equals velocity when graphing position vs. time.

109. What is the conversion factor to change feet to inches?

     $1\text{ ft}=12\text{ in}$. Standard length conversion in imperial system.

110. What is the average speed formula for total distance $D$ over total time $T$, with $T \ne 0$?

     $\text{avg speed} = \frac{D}{T}$. Average speed is total distance over total time.

111. What is the combined work rate if one machine does a job in $6$ hours and another in $3$ hours?

     $\frac{1}{2}$ job per hour. Add individual rates: $\frac{1}{6}+\frac{1}{3}=\frac{1}{6}+\frac{2}{6}=\frac{3}{6}=\frac{1}{2}$.

112. What distance is traveled at $60$ mph for $2.5$ hours?

     $150$ miles. Use $d = rt$: $60 \times 2.5 = 150$ miles.

113. State the formula for time in terms of distance and speed.

     $\text{time}=\frac{\text{distance}}{\text{speed}}$. Rearranges distance formula by dividing both sides by speed.

114. What is the distance traveled in $2.5$ hours at $40$ miles per hour?

     $100$ miles. Use $d=vt$: $40 \times 2.5=100$.

115. What is the miles-per-gallon formula if you drive $m$ miles using $g$ gallons, with $g \ne 0$?

     $\frac{m}{g}$ miles per gallon. Fuel efficiency equals miles divided by gallons.

116. What is the time needed to travel $150$ miles at $50$ miles per hour?

     $3$ hours. Using $t = \frac{d}{r}$: $\frac{150}{50} = 3$ hours.

117. What is the time to travel $150$ miles at a constant speed of $50$ miles per hour?

     $3$ hours. Use $t=\frac{d}{v}$: $\frac{150}{50}=3$.

118. A tap fills $12$ liters in $3$ minutes. What is the rate in $\text{L/min}$?

     $4\text{ L/min}$. Divide volume by time: $\frac{12\text{ L}}{3\text{ min}}=4\text{ L/min}$.

119. What is the distance traveled at $55\text{ mph}$ for $2$ hours?

     $110\text{ miles}$. Using $d = rt$: $55 \times 2 = 110$ miles.

120. What is the unit price per ounce for $24$ ounces costing $6?

     $0.25 per ounce. Divide total price by ounces: $\frac{\$6}{24}$.

121. Find the time: A runner goes $9$ miles at $6$ miles per hour. How long does it take?

     $1.5$ hours. Use $t=\frac{d}{s}$: $\frac{9}{6}=1.5$ hours.

122. What is the formula for time when distance is $d$ and speed is $r$ (with $r \ne 0$)?

     $t = \frac{d}{r}$. Time equals distance divided by rate.

123. What is the flow rate in gallons per minute if $8$ gallons fill in $40$ seconds?

     $12$ gallons per minute. $8$ gallons in $\frac{40}{60}$ minutes = $8 \div \frac{2}{3} = 12$ gal/min.

124. What is the average speed formula for total distance $D$ traveled in total time $T$?

     $\frac{D}{T}$. Average speed is total distance over total time.

125. What is $36$ miles per hour in feet per second?

     $52.8$ ft/s. Multiply by $\frac{22}{15}$: $36 \times \frac{22}{15}=52.8$.

126. What is the conversion factor to change centimeters to meters?

     $100\text{ cm}=1\text{ m}$. Standard metric conversion for length measurements.

127. State the formula for average speed.

     $\text{average speed}=\frac{\text{total distance}}{\text{total time}}$. Total distance divided by total time gives overall rate.

128. Identify the unit rate: $18$ miles in $3$ hours.

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

129. What is the formula for average speed over a trip with total distance $D$ and total time $T$?

     $\text{avg speed}=\frac{D}{T}$. Total distance divided by total time for entire trip.

130. What is the constant of proportionality $k$ in $y=kx$?

     $k=\frac{y}{x}$. The ratio that relates $x$ and $y$ in direct variation.

131. Identify the unit rate: $18 for $6$ pounds of apples.

     $3 per pound. Divide total cost by quantity: $\frac{\$18}{6}=\$3$.

132. What is the formula for time given distance $d$ at constant speed $r$, with $r \ne 0$?

     $t = \frac{d}{r}$. Time equals distance divided by rate.

133. What is the total amount formula for simple interest with principal $P$, rate $r$, time $t$?

     $A=P+Prt$. Total equals principal plus interest earned.

134. What is the formula for work rate when $1$ job is completed in $t$ hours?

     $\text{rate}=\frac{1}{t}\text{ job/hour}$. Reciprocal relationship: slower time means faster rate.

135. What is the percent change formula from an original value $A$ to a new value $B$?

     \frac{B-A}{A}\times 100\text{%}. Measures relative change as a percentage of original.

136. What is the formula for average rate of change from $x=a$ to $x=b$ for $f(x)$?

     $\frac{f(b)-f(a)}{b-a}$. Measures the change in $f$ per unit change in $x$ over an interval.

137. What is the definition of a unit rate?

     A rate with denominator $1$ unit (for example, per $1$). Simplifies comparison by standardizing to one unit.

138. What speed is required to travel $120$ miles in $3$ hours?

     $40$ miles per hour. Use $r = \frac{d}{t}$: $\frac{120}{3} = 40$ mph.

139. Identify the better buy: $16$ oz for $4$ or $24$ oz for $5$.

     $24$ oz for $5$. Compare unit prices: $\frac{4}{16}=0.25$ vs $\frac{5}{24}≈0.21$.

140. What is the formula for time in terms of distance $d$ and speed $v$?

     $t=\frac{d}{v}$. Rearrange $v=\frac{d}{t}$ to solve for time.

141. State the formula for speed in terms of distance and time.

     $r = \frac{d}{t}$. Rate equals distance divided by time elapsed.

142. What is the formula for speed given distance $d$ traveled in time $t$, with $t \ne 0$?

     $r = \frac{d}{t}$. Rate equals distance divided by time.

143. State the formula for time when distance traveled and constant speed are known.

     $t = \frac{d}{r}$. Time equals distance divided by rate.

144. Identify the unit rate: $15$ miles in $\frac{1}{2}$ hour equals how many miles per hour?

     $30$ miles per hour. $15 \div \frac{1}{2} = 15 \times 2 = 30$ miles per hour.

145. Find the distance traveled at $12\text{ m/s}$ for $15$ seconds.

     $180\text{ m}$. Apply $d=rt$: $12\text{ m/s}\times 15\text{ s}=180\text{ m}$.

146. What distance is traveled at $12\text{ m/s}$ for $15\text{ s}$?

     $180\text{ m}$. Apply distance formula: $12\times15=180$.

147. What is the cost per ounce if $12$ ounces cost $3?

     $0.25 per ounce. $\$3 \div 12 = \$0.25$ per ounce.

148. What is the average speed for $120$ miles in $2$ hours?

     $60\ \text{mi/h}$. Divide distance by time: $\frac{120}{2}=60$.

149. A faucet fills $12$ gallons in $3$ minutes. What is the rate in $\text{gal/min}$?

     $4\text{ gal/min}$. Divide total gallons by time: $\frac{12}{3}$.

150. What is the formula for speed when distance is $d$ and time is $t$?

     $r=\frac{d}{t}$. Rate equals distance divided by time.

151. What is the relative speed when two objects move toward each other at $v_1$ and $v_2$?

     $v_1+v_2$. When moving toward each other, speeds add.

152. What time is required to complete $1$ job at a constant rate of $\frac{1}{4}$ job/hr?

     $4\text{ hr}$. Time is reciprocal of rate: $\frac{1}{\frac{1}{4}}$.

153. What is the average speed for a $120$ mile trip that takes $3$ hours total?

     $40$ miles per hour. Average speed = total distance ÷ total time.

154. What is the unit rate for a rate of $\frac{a}{b}$ units per unit, with $b \ne 0$?

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

155. What is the speed in $\text{mph}$ if a car travels $150$ miles in $3$ hours?

     $50\text{ mph}$. Apply speed formula: $\frac{150}{3}=50$.

156. A car uses $12$ gallons to go $360$ miles. What is the fuel efficiency in miles per gallon?

     $30$ miles per gallon. Divide miles by gallons: $\frac{360}{12} = 30$ mpg.

157. Identify the speed in mph for traveling $150$ miles in $3$ hours.

     $50$ mph. Apply speed formula: $\frac{150\text{ miles}}{3\text{ hours}}=50$ mph.

158. Find the unit rate: A car travels $150$ miles in $3$ hours. What is the speed in $\text{mph}$?

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

159. What is the conversion factor to change miles to feet?

     $1\text{ mi}=5280\text{ ft}$. Standard conversion between miles and feet.

160. What is the combined work rate if one job takes $4$ hours and another takes $12$ hours?

     $\frac{1}{3}$ jobs per hour. $\frac{1}{4} + \frac{1}{12} = \frac{3}{12} + \frac{1}{12} = \frac{4}{12} = \frac{1}{3}$.

161. What is the combined work rate if one job takes $a$ hours and another takes $b$ hours?

     $\frac{1}{a} + \frac{1}{b}$ jobs per hour. Combined rate is sum of individual rates $\frac{1}{a}$ and $\frac{1}{b}$.

162. State the formula for density as a rate.

     $\text{density}=\frac{\text{mass}}{\text{volume}}$. Measures how much mass occupies a given space.

163. What is the definition of unit rate in a ratio $\frac{a}{b}$?

     A rate with denominator $1$ unit. The denominator is reduced to one unit.

164. A car goes $60$ miles at $30\text{ mi/hr}$ and $60$ miles at $60\text{ mi/hr}$. What is the average speed?

     $40\text{ mi/hr}$. Total distance $120$ mi, total time $2+1=3$ hr, so $\frac{120}{3}=40$.

165. What is the unit rate if $12$ items cost $18?

     $1.50 per item. Divide total cost by number of items: $\frac{\$18}{12}=\$1.50$.

166. What is the conversion from $1$ hour to minutes?

     $1\text{ hr} = 60\text{ min}$. Standard time conversion.

167. How long does it take to travel $90$ miles at $45$ mph?

     $2$ hours. Use $t = \frac{d}{r}$: $\frac{90}{45} = 2$ hours.

168. What is the formula for distance when speed is $r$ and time is $t$?

     $d = rt$. Distance equals rate multiplied by time.

169. What is the time to travel $150$ miles at $50$ miles per hour?

     $3$ hours. $t = \frac{150}{50} = 3$ hours.

170. What is the combined work rate if one job takes $6$ hr and another takes $3$ hr?

     $\frac{1}{2}\text{ job/hr}$. Combined rate: $\frac{1}{6} + \frac{1}{3} = \frac{1}{2}$.

171. What is the conversion from $1$ foot to inches?

     $1\text{ ft} = 12\text{ in}$. Standard length conversion in US customary units.

172. A printer makes $45$ pages in $3$ minutes. What is the unit rate in $\text{pages/min}$?

     $15\text{ pages/min}$. Divide pages by time: $\frac{45\text{ pages}}{3\text{ min}}=15$.

173. What is the percent decrease from $80$ to $68$?

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

174. Identify the better buy by unit rate: $12$ oz for $3 or $20$ oz for $4?

     $20$ oz for $4. $20$ oz at $0.20/oz beats $12$ oz at $0.25/oz.

175. Convert a rate: If a car travels at $60$ miles per hour, what is the speed in miles per minute?

     $1$ mile per minute. Divide by $60$ since $1$ hour $= 60$ minutes.

176. How long does it take to travel $84$ miles at $28$ mph?

     $3$ hours. Divide distance by speed: $\frac{84}{28}=3$ hours.

177. Identify the unit rate: $120$ miles in $3$ hours equals how many miles per hour?

     $40\text{ mph}$. $120 \div 3 = 40$ miles per hour.

178. What is the conversion from miles per hour to feet per second?

     $1\text{ mph}=\frac{22}{15}\text{ ft/s}$. Multiply by $\frac{5280}{3600}$ and simplify to $\frac{22}{15}$.

179. What is the formula for average speed when traveling distances $d_1,d_2$ in times $t_1,t_2$?

     $\frac{d_1+d_2}{t_1+t_2}$. Total distance divided by total time.

180. A printer makes $240$ pages in $6$ minutes. What is the rate in pages per minute?

     $40$ pages per minute. Divide total pages by time: $\frac{240}{6} = 40$ pages/min.

181. What is the meaning of "per" in a rate such as $12$ dollars per hour?

     It means "for each," representing division. "Per" indicates the denominator in a rate fraction.

182. What is the formula for distance when $r$ is rate and $t$ is time?

     $d=rt$. Distance equals rate times time.

183. What conversion factor changes $\text{minutes}$ to $\text{seconds}$ in a rate problem?

     $1\text{ minute} = 60\text{ seconds}$. Standard time conversion for rate calculations.

184. What is the speed in $\text{mi/hr}$ for $150$ miles in $3$ hours?

     $50\text{ mi/hr}$. Apply speed formula: $\frac{150\text{ mi}}{3\text{ hr}}=50\text{ mi/hr}$.

185. What is the average speed for $120$ miles in $2$ hours, then $60$ miles in $1$ hour?

     $60$ miles per hour. Total: $180$ miles in $3$ hours = $60$ mph average.

186. How long does it take to travel $120$ miles at $60\text{ mi/hr}$?

     $2\text{ hr}$. Apply $t = \frac{d}{r} = \frac{120}{60}$.

187. What is the definition of a rate as used in PSAT Math problems?

     A ratio comparing two quantities with different units. Rates express relationships between different measurement types.

188. Convert $3$ meters per second to meters per minute.

     $180$ meters per minute. Multiply by $60$: $3 \times 60 = 180$ m/min.

189. Convert $72$ miles per hour to feet per second using $1$ mile $=5280$ ft, $1$ hr $=3600$ s.

     $105.6$ ft/s. Multiply: $72\times\frac{5280}{3600}=72\times\frac{22}{15}=105.6$.

190. What is the formula for distance when traveling at constant speed $r$ for time $t$?

     $d = rt$. Distance equals rate times time for constant motion.

191. What is the speed in $\text{mi/h}$ for $150$ miles in $3$ hours?

     $50\ \text{mi/h}$. Divide distance by time: $\frac{150}{3}=50$.

192. What is the conversion factor to change minutes to seconds?

     $1\text{ min}=60\text{ s}$. Standard time conversion for minute-to-second calculations.

193. Identify the unit rate for $\frac{45\text{ miles}}{3\text{ hours}}$.

     $15$ miles per hour. Divide $45$ miles by $3$ hours to get miles per hour.

194. What is the formula for density in terms of mass $m$ and volume $V$?

     $\rho=\frac{m}{V}$. Density equals mass divided by volume.

195. What is the slope interpretation of a unit rate on a graph of $y$ versus $x$?

     Slope $m$ equals unit rate $\frac{\Delta y}{\Delta x}$. On a graph, unit rate appears as the slope of the line.

196. What is the distance traveled at $60$ miles per hour for $\frac{1}{2}$ hour?

     $30$ miles. $d = rt = 60 \times \frac{1}{2} = 30$ miles.

197. What conversion factor changes $\text{inches}$ to $\text{feet}$ in a rate problem?

     $1\text{ foot} = 12\text{ inches}$. Standard distance conversion for rate calculations.

198. What is the unit price per pound if $\frac{3}{2}$ pounds cost $6?

     $4 per pound. $\$6 \div \frac{3}{2} = \$6 \times \frac{2}{3} = \$4$ per pound.

199. What is the interest formula for simple interest with principal $P$, rate $r$, time $t$?

     $I=Prt$. Interest equals principal times rate times time.

200. What time is needed to travel $240$ miles at $60\text{ mph}$?

     $4$ hours. Apply time formula: $\frac{240}{60}=4$.

201. What speed is required to travel $180$ miles in $4$ hours?

     $45\text{ mph}$. Using $r = \frac{d}{t}$: $180 \div 4 = 45$ mph.

202. A job pays $18$ dollars per hour. How much is earned in $7$ hours?

     $126$ dollars. Multiply rate by time: $18 \times 7 = 126$ dollars.

203. What is the formula for time when distance is $d$ and speed is $r$?

     $t=\frac{d}{r}$. Time equals distance divided by rate.

204. What is the conversion rate from miles per hour to feet per second?

     $1$ mph $=\frac{22}{15}$ ft/s. $1$ mile = $5280$ ft, $1$ hour = $3600$ s; $\frac{5280}{3600}=\frac{22}{15}$.

205. What is the slope formula as a rate of change from $(x_1,y_1)$ to $(x_2,y_2)$?

     $m=\frac{y_2-y_1}{x_2-x_1}$. Change in $y$ divided by change in $x$.

206. What is the unit rate for a ratio of $a$ units in $b$ seconds (with $b \ne 0$)?

     $\frac{a}{b}$ units per second. Divide the quantity by time to get the rate per unit time.

207. What is the definition of rate as a ratio of two quantities with different units?

     A ratio comparing quantities in different units, such as $\frac{\text{miles}}{\text{hour}}$. Rates always compare two different measurement units.

208. Compute the average rate of change of $f(x)$ from $x=2$ to $x=5$ if $f(2)=7$ and $f(5)=19$.

     $4$. Apply formula: $\frac{19-7}{5-2}=\frac{12}{3}=4$.

209. State the formula for time in terms of distance and speed.

     $\text{time}=\frac{\text{distance}}{\text{speed}}$. Rearranged distance formula by dividing both sides by speed.

210. A cyclist rides at $12$ mph. What is the time in hours to ride $30$ miles?

     $2.5$ hours. Use $t = \frac{d}{r}$: $\frac{30}{12} = 2.5$ hours.

211. What conversion factor changes $\text{feet}$ to $\text{miles}$ in a rate problem?

     $1\text{ mile} = 5280\text{ feet}$. Standard distance conversion for rate calculations.

212. What is the constant of proportionality $k$ if $y$ varies directly with $x$?

     $k = \frac{y}{x}$. In direct variation, $k$ is the constant ratio $\frac{y}{x}$.

213. What is the formula for distance when traveling at constant speed $r$ for time $t$?

     $d=rt$. Distance equals rate times time for constant speed.

214. State the formula for constant rate problems using $d$, $r$, and $t$.

     $d=rt$. Distance equals rate times time for constant motion.

215. What is the conversion from $1$ minute to seconds?

     $1\text{ min} = 60\text{ s}$. Standard time conversion.

216. What is the slope of the line through $(2,5)$ and $(6,13)$?

     $2$. $m = \frac{13-5}{6-2} = \frac{8}{4} = 2$.

217. Find the time: If you travel $90$ miles at $45\text{ mph}$, how many hours does it take?

     $2\text{ hours}$. Apply $t = \frac{d}{r}$: $\frac{90}{45} = 2$.

218. What is the average speed formula if total distance is $D$ and total time is $T$ (with $T \ne 0$)?

     $\frac{D}{T}$. Total distance divided by total time gives average speed.

219. What is the average speed for a $30$ mile trip out and $30$ miles back in $2$ hours total?

     $30\ \text{mi/h}$. Total distance is $60$ miles, time is $2$ hours: $\frac{60}{2}=30$.

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