Word Problems - ACT Math

1. A price increases from $50$ to $60$ dollars; what is the percent increase?

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

2. Identify the translation: "twice a number $x$" becomes what expression?

   $2x$. Multiply the number $x$ by 2.

3. A taxi fare is $3 plus $0.50 per mile; if the fare is $8, how many miles were traveled?

   $10$ miles. $8 = 3 + 0.50m$, so $m = 10$ miles.

4. State the formula to calculate distance in word problems.

   Distance = Speed × Time. Fundamental motion formula relating three key variables.

5. Find the original price if an item costs $36$ after a $10\%$ discount.

   $40$. $36 = 0.90x$, so $x = 40$

6. What equation represents 'Three times a number decreased by 4 equals 11'?

   Equation: $3x - 4 = 11$. Translates 'three times' as $3x$ and 'decreased by 4' as $-4$.

7. Identify the translation: "a number $x$ less than $7$" becomes what expression?

   $7-x$. Subtract $x$ from 7, not 7 from $x$.

8. How do you express 'half of a number' in algebraic terms?

   $\frac{x}{2}$. Division by 2 represents taking half the value.

9. Worker A completes $\frac{1}{4}$ of a job per hour; how many hours for the whole job?

   $4$ hours. Rate is $\frac{1}{4}$ job per hour, so 4 hours total.

10. A $10$ L solution is $30\%$ acid; how many liters of acid does it contain?

    $3$ L. $10 \times 0.30 = 3$ liters of acid

11. How do you express consecutive even numbers algebraically?

    Use $x, x+2, x+4, \text{etc.}$. Even numbers differ by 2, starting from any even $x$.

12. How do you express 'a number divided by 5' in algebraic terms?

    $\frac{x}{5}$. Division operation represented as a fraction.

13. What is the equation for a simple interest word problem?

    Interest = Principal × Rate × Time. Standard formula for calculating simple interest earned.

14. A store charges $5$ fixed plus $2$ per item; what is the cost for $7$ items?

    $19$. $5 + 7(2) = 19$ dollars total.

15. How is 'twice the sum of a number and 4' expressed algebraically?

    2(x + 4). Multiply 2 by the entire sum in parentheses.

16. A right triangle has legs $6$ and $8$; what is the hypotenuse length?

    $10$. Use Pythagorean theorem: $\sqrt{6^2 + 8^2} = 10$.

17. What is $30\%$ of $80$?

    $24$. $0.30 \times 80 = 24$

18. How do you write 'the product of 5 and a number is 45' as an equation?

    5x = 45. Product relationship between 5 and unknown variable.

19. A job takes A $6$ hours alone and B $3$ hours alone; how long together at constant rates?

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

20. What is the total amount formula for simple interest with principal $P$ and interest $I$?

    $A=P+I$. Total amount is principal plus accumulated interest.

21. What equation represents 'the product of 6 and a number is 36'?

    6x = 36. Multiplication equation with variable and constant.

22. Identify the equation for 'The sum of a number and 9 is 15.'

    Equation: $x + 9 = 15$. Direct translation of 'sum' as addition operation.

23. What is the weighted average concentration formula for mixing solutions?

    $\frac{c_1a+c_2b}{a+b}$. Weighted average using concentration and amounts.

24. What equation represents 'Four more than twice a number is 10'?

    Equation: $2x + 4 = 10$. Translates 'twice a number' as $2x$ and 'four more' as $+4$.

25. How do you calculate the selling price with a discount?

    Selling Price = Original Price × (1 - Discount Rate). Multiplies by $(1 - \text{decimal discount})$ to reduce price.

26. At $60$ miles per hour, how far is traveled in $2.5$ hours?

    $150$ miles. $60 \times 2.5 = 150$ miles

27. What is the key step in solving mixture word problems?

    Set up an equation based on total quantity or concentration. Balance total amounts or percentages to find unknowns.

28. How do you express 'three more than a number' algebraically?

    $x + 3$. Addition of 3 to the unknown variable $x$.

29. Which operation is essential in work-related word problems?

    Addition of rates. Combined work rates determine completion time.

30. What equation represents 'a number decreased by 8 is 12'?

    x - 8 = 12. Subtraction translates to the minus operation algebraically.

31. Identify the equation to solve: 'A number is 5 more than twice another number.'

    Equation: $x = 2y + 5$. Translates 'twice another' as $2y$ and 'is 5 more than' as $+5$.

32. How is 'ten more than twice a number' expressed algebraically?

    2x + 10. Multiply by 2 first, then add 10.

33. What is the formula for simple interest on principal $P$ at rate $r$ for time $t$ (years)?

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

34. What is the first step in solving a word problem involving percentages?

    Identify the given and unknown quantities. Essential first step to organize the problem structure.

35. A fair coin is flipped twice; what is the probability of exactly one head?

    $\frac{1}{2}$. Two outcomes (HT, TH) out of four possible.

36. What equation represents 'Four times a number is reduced by 8 to get 16'?

    Equation: $4x - 8 = 16$. Translates 'reduced by 8' as subtraction from $4x$.

37. How do you solve a percent increase word problem?

    $\text{New Value} = \text{Original Value} \times (1 + \text{Percent Increase})$ . Multiplies original by $(1 + \text{decimal percent})$ for increase.

38. A store charges $5 fixed plus $2 per item; what is the cost for $7$ items?

    $19. $5 + 7(2) = 19$ dollars total.

39. Identify the equation for 'Three times a number equals 18.'

    Equation: $3x = 18$. Direct translation of 'three times' as multiplication.

40. Identify the translation: "three times the sum of $x$ and $4$" becomes what expression?

    $3(x+4)$. First add $x$ and 4, then multiply by 3.

41. How do you express 'three-fifths of a number' in algebraic terms?

    $\frac{3}{5}x$. Fraction multiplied by variable shows partial amount.

42. A die is rolled once; what is the probability of rolling a multiple of $3$?

    $\frac{1}{3}$. Multiples of 3 are 3 and 6, so 2 out of 6.

43. Identify the translation: "$5$ more than a number $x$" becomes what expression?

    $x+5$. Add 5 to the number $x$.

44. How do you express 'the sum of twice a number and 8' algebraically?

    2x + 8. Addition of doubled variable and constant term.

45. Identify the equation for 'The quotient of a number and 5 is 3.'

    Equation: $\frac{x}{5} = 3$. Translates 'quotient' as division operation.

46. A $5$ lb nut mix costs $4/lb and $6/lb nuts are combined; if total cost is $26, how many pounds at $6/lb?

    $3$ lb. Let $x$ be pounds at $6: $2(4) + x(6) = 26$, so $x = 3$.

47. How do you translate 'twice a number' into an algebraic expression?

    2x. Multiply by 2 to represent doubling the value.

48. What is the key step in solving rate problems involving two objects?

    Set equations based on relative rates. Compare speeds to determine relative motion.

49. A price decreases from $80$ to $68$ dollars; what is the percent decrease?

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

50. How do you set up a system of equations for a word problem?

    Translate conditions into algebraic equations. Convert word relationships into mathematical expressions.

51. State the method to find an unknown in a ratio problem.

    Use cross-multiplication to solve the proportion. Cross-multiplication solves proportional equations efficiently.

52. What is the approach to solve a rate of work problem with two workers?

    Add their rates to find total rate. Combined rates equal sum of individual work rates.

53. What is the constant speed relationship among distance $d$, rate $r$, and time $t$?

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

54. A rectangle has area $48$ and width $6$; what is its length?

    $8$. Area is length times width, so $l = \frac{48}{6} = 8$.

55. A $5$ lb nut mix costs $4$/lb and $6$/lb nuts are combined; if total cost is $26$, how many pounds at $6$/lb?

    $3$ lb. Let $x$ be pounds at $6$: $2(4) + x(6) = 26$, so $x = 3$.

56. What is the key step in solving age-related word problems?

    Set up equations based on age relations. Relationships between ages form the equation foundation.

57. Find the final price after a $25\%$ discount on $40$ dollars.

    $30$. $40 - (0.25 \times 40) = 30$

58. What is the first step in solving ratio problems?

    Set up a proportion. Creates equivalent fractions to solve for unknowns.

59. What is the mixture formula for total amount when combining $a$ units and $b$ units?

    $\text{total amount}=a+b$. Total quantity is sum of individual amounts.

60. What is the primary step in solving proportion word problems?

    Set up a proportion equation. Ratios create equivalent fraction relationships to solve.

61. Mix $2$ L of $20\%$ with $3$ L of $50\%$; what is the resulting concentration?

    $38\%$. $\frac{2(0.20) + 3(0.50)}{5} = 0.38 = 38\%$

62. What equation represents 'the sum of a number and 11 is 25'?

    x + 11 = 25. Addition equation with variable and constants.

63. How do you translate 'the square of a number' into an expression?

    $x^2$. Exponent of 2 represents multiplying by itself.

64. How do you write 'the ratio of a number to 7' as an expression?

    $\frac{x}{7}$. Division creates the fractional relationship to 7.

65. How long does it take to travel $180$ miles at $45$ miles per hour?

    $4$ hours. $\frac{180}{45} = 4$ hours

66. How do you express 'four times a number' in algebraic terms?

    4x. Multiplication by 4 represents the repeated addition.

67. If $15$ is $25\%$ of $x$, what is $x$?

    $60$. $15 = 0.25x$, so $x = 60$

68. What is the first step in solving mixture problems?

    Identify the components and their quantities. Organize what's being mixed and in what amounts.

69. What is the key concept in solving profit and loss problems?

    Profit = Selling Price - Cost Price. Revenue minus expenses determines financial gain.

70. Which operation is used to find the total in consecutive number problems?

    Addition. Sums consecutive integers to find their total.

71. How do you solve a problem involving proportional reasoning?

    Set up an equation with equivalent ratios. Proportions express equal ratios for solving unknowns.

72. Identify the translation: "the difference of $x$ and $4$" becomes what expression?

    $x-4$. Subtract the second number from the first.

73. What is the first step to solve a proportion word problem?

    Set up a proportion using equivalent ratios. Cross-multiplying equivalent ratios solves for unknown values.

74. A rectangle has perimeter $30$ and length $10$; what is its width?

    $5$. Perimeter is $2l + 2w = 30$, so $w = 5$.

75. Identify the translation: "$7$ less than a number $x$" becomes what expression?

    $x-7$. Subtract 7 from the number $x$.

76. Identify the translation: "$7$ less than a number" means what expression in $x$?

    $x-7$. Subtract 7 from the number $x$.

77. What is the primary concept in solving interest problems?

    Interest = Principal × Rate × Time. Standard formula for calculating earned interest.

78. What is the first step in setting up equations for word problems?

    Define variables for unknowns. Assigns letters to represent unknown quantities.

79. Identify the translation: "the quotient of $x$ and $5$" becomes what expression?

    $\frac{x}{5}$. Divide the first number by the second.

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

    $\frac{f(b)-f(a)}{b-a}$. Change in function value divided by change in input.

81. What is the key formula for solving average problems?

    Average = Total Sum / Number of Items. Divides total by count to find mean value.

82. What is the first step in solving a time, rate, and distance problem?

    Identify given values for time, rate, or distance. Organize known information before applying $d = rt$ formula.

83. Translate 'the difference between a number and 5 is 10' into an equation.

    x - 5 = 10. Subtraction relationship between two quantities equals 10.

84. What equation represents 'Seven less than a number is 12'?

    Equation: $x - 7 = 12$. Translates 'seven less than' as $x - 7$.

85. What is the equation for 'a number plus 9 equals 17'?

    x + 9 = 17. Direct translation of addition equals relationship.

86. Express 'The product of a number and 7 is 21' as an equation.

    Equation: $7x = 21$. Translates 'product of number and 7' directly as multiplication.

87. What is the equation for 'the product of a number and 4 is 20'?

    4x = 20. Multiplication relationship between variable and constant.

88. What does the phrase "per" usually indicate in an ACT word problem expression?

    "Per" usually means division, as in $60$ miles per $2$ hours is $\frac{60}{2}$. This indicates division to find rate in word problems.

89. What is the formula to find the distance in a distance-speed-time problem?

    Distance = Speed × Time. Fundamental relationship connecting these three quantities.

90. How is 'the quotient of a number and 3' expressed algebraically?

    $\frac{x}{3}$. Division represents the quotient operation in algebra.

91. What is the equation for: 'The sum of a number and twice another is 20'?

    Equation: $x + 2y = 20$. Translates 'sum' and 'twice another' into algebraic form.

92. A jar has $5$ red and $3$ blue marbles; what is the probability of drawing a red marble?

    $\frac{5}{8}$. 5 red out of 8 total marbles.

93. What is the equation for 'the sum of three times a number and 9 is 24'?

    3x + 9 = 24. Combines multiplication and addition in one equation.

94. Which mathematical operation is used when distributing items evenly?

    Division. Splits quantities into equal parts among recipients.

95. What is the work-rate equation for two workers with rates $r_1$ and $r_2$ working together?

    $r_1+r_2=r_{\text{total}}$. Combined work rates add when working together.

96. How do you express consecutive odd numbers algebraically?

    Use $x, x+2, x+4, \text{etc.}$. Odd numbers differ by 2, starting from any odd $x$.

97. What is the first step in solving a work-rate word problem?

    Determine the rate of work for each individual or machine. Individual rates combine to find total completion time.

98. What is the primary step in solving linear equation word problems?

    Translate the words into an equation. Converts verbal statements into mathematical symbols.

99. What equation represents 'a number minus 6 equals 14'?

    x - 6 = 14. Subtraction equation with variable and constant.

100. What does the phrase "of" usually indicate in an ACT word problem expression?

     "Of" usually means multiplication, as in $\frac{3}{4}$ of $x$ is $\frac{3}{4}x$. This indicates multiplication in word problems.

101. Express 'The difference between a number and 6 is 8' as an equation.

     Equation: $x - 6 = 8$. Translates 'difference' as subtraction operation.

102. State the formula for calculating the average in word problems.

     Average = $\frac{\text{Sum of values}}{\text{Number of values}}$. Divides total sum by count of data points.

103. What equation represents 'the sum of a number and 7 is 15'?

     $x + 7 = 15$. Addition translates to the plus operation in algebra.

104. How do you solve a percent decrease word problem?

     New Value = Original Value × (1 - Percent Decrease). Multiplies by $(1 - \text{decimal percent})$ for decrease.

105. Find the total cost if 5 items cost $x$ each and a $20 fee is added.

     Total Cost = $5x + 20$. Multiplies unit cost by quantity, then adds fixed fee.

106. Find the equation: 'Twice a number added to 3 equals 11.'

     Equation: $2x + 3 = 11$. Translates 'twice a number added to 3' as $2x + 3$.

107. Which operation is typically used for problems involving total cost?

     Addition. Combines individual amounts to find the sum.

108. What is the percent equation relating percent, part, and whole?

     $\text{part}=\frac{\text{percent}}{100}\cdot\text{whole}$. Basic percent formula: part equals percent times whole divided by 100.

109. Two cars start $300$ miles apart and drive toward each other at $60$ mph and $40$ mph; when do they meet?

     $3$ hours. Combined speed is $100$ mph, so $\frac{300}{100} = 3$ hours.

110. How is 'seven more than the product of a number and 3' expressed?

     3x + 7. Multiply by 3 first, then add 7.

111. What is the equation for 'A number is divided by 4 to give 3'?

     Equation: $\frac{x}{4} = 3$. Translates division phrase into fractional equation.

112. Translate 'five less than twice a number' into an expression.

     2x - 5. Multiply by 2 first, then subtract 5.

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