import { type PracticeQuestion } from "../../types/lesson";

export const LINEAR_EQ_ONE_VAR_EASY: PracticeQuestion[] = [
  {
    id: "097e10f5",
    type: "mcq",
    questionHtml:
      "What value of <strong>p</strong> satisfies the equation <strong>5 p + 180 = 250</strong>?",
    choices: [
      { label: "A", text: "<strong>14</strong>" },
      { label: "B", text: "<strong>65</strong>" },
      { label: "C", text: "<strong>86</strong>" },
      { label: "D", text: "<strong>250</strong>" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. Subtracting <strong>180</strong> from both sides of the given equation yields <strong>5 p = 70</strong>. Dividing both sides of this equation by <strong>5</strong> yields <strong>p = 14</strong>. Therefore, the value of <strong>p</strong> that satisfies the equation <strong>5 p + 180 = 250</strong> is <strong>14</strong>.<br>Choice B is incorrect. This value of <strong>p</strong> satisfies the equation <strong>5 p + 180 = 505</strong>.<br>Choice C is incorrect. This value of <strong>p</strong> satisfies the equation <strong>5 p + 180 = 610</strong>.<br>Choice D is incorrect. This value of <strong>p</strong> satisfies the equation <strong>5 p + 180 = 1, 430</strong>.",
    hasFigure: false,
  },
  {
    id: "0d685865",
    type: "mcq",
    questionHtml:
      "If <strong>x = 7</strong>, what is the value of <strong>x + 20</strong>?",
    choices: [
      { label: "A", text: "<strong>13</strong>" },
      { label: "B", text: "<strong>20</strong>" },
      { label: "C", text: "<strong>27</strong>" },
      { label: "D", text: "<strong>34</strong>" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. It’s given that <strong>x = 7</strong>. Substituting <strong>7</strong> for <strong>x</strong> into the given expression <strong>x + 20</strong> yields <strong>7 + 20</strong>, which is equivalent to <strong>27</strong>.<br>Choice A is incorrect. This is the value of <strong>x + 6</strong>.<br>Choice B is incorrect. This is the value of <strong>x + 13</strong>.<br>Choice D is incorrect. This is the value of <strong>x + 27</strong>.",
    hasFigure: false,
  },
  {
    id: "12255364",
    type: "mcq",
    questionHtml:
      "A gym charges its members a onetime <strong>dollar sign 36</strong> enrollment fee and a membership fee of <strong>dollar sign 19</strong> per month. If there are no charges other than the enrollment fee and the membership fee, after how many months will a member have been charged a total of <strong>dollar sign 188</strong> at the gym?",
    choices: [
      { label: "A", text: "<strong>4</strong>" },
      { label: "B", text: "<strong>5</strong>" },
      { label: "C", text: "<strong>8</strong>" },
      { label: "D", text: "<strong>10</strong>" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. It’s given that a gym charges its members a onetime <strong>dollar sign 36</strong> enrollment fee and a membership fee of <strong>dollar sign 19</strong> per month. Let <strong>x</strong> represent the number of months at the gym after which a member will have been charged a total of <strong>dollar sign 188</strong>. If there are no charges other than the enrollment fee and the membership fee, the equation <strong>36 + 19 x = 188</strong> can be used to represent this situation. Subtracting <strong>36</strong> from both sides of this equation yields <strong>19 x = 152</strong>. Dividing both sides of this equation by <strong>19</strong> yields <strong>x = 8</strong>. Therefore, a member will have been charged a total of <strong>dollar sign 188</strong> after <strong>8</strong> months at the gym.<br>Choice A is incorrect. A member will have been charged a total of <strong>dollar sign (36 + 19 · 4)</strong>, or <strong>dollar sign 112</strong>, after <strong>4</strong> months at the gym.<br>Choice B is incorrect. A member will have been charged a total of <strong>dollar sign (36 + 19 · 5)</strong>, or <strong>dollar sign 131</strong>, after <strong>5</strong> months at the gym.<br>Choice D is incorrect. A member will have been charged a total of <strong>dollar sign (36 + 19 · 10)</strong>, or <strong>dollar sign 226</strong>, after <strong>10</strong> months at the gym.",
    hasFigure: false,
  },
  {
    id: "2f0a43b2",
    type: "spr",
    questionHtml:
      "If <strong>(x) / (8) = 5</strong>, what is the value of <strong>(8) / (x)</strong>?",
    choices: [],
    correctAnswer: ".2, 1/5",
    explanation:
      "The correct answer is <strong>one fifth</strong>. Since the number <strong>5</strong> can also be written as <strong>five oneths</strong>, the given equation can also be written as <strong>(x) / (8) = five oneths</strong>. This equation is equivalent to <strong>(8) / (x) = one fifth</strong>. Therefore, the value of <strong>(8) / (x)</strong> is <strong>one fifth</strong>. Note that 1/5 and .2 are examples of ways to enter a correct answer.<br>Alternate approach: Multiplying both sides of the equation <strong>(x) / (8) = 5</strong> by <strong>8</strong> yields <strong>x = 40</strong>. Substituting <strong>40</strong> for <strong>x</strong> into the expression <strong>(8) / (x)</strong> yields <strong>eight fortieths</strong>, or <strong>one fifth</strong>.",
    hasFigure: false,
  },
  {
    id: "349a5bc1",
    type: "spr",
    questionHtml:
      "<strong>4 x + 5 = 165</strong><br>What is the solution to the given equation?",
    choices: [],
    correctAnswer: "40",
    explanation:
      "The correct answer is <strong>40</strong>. Subtracting <strong>5</strong> from both sides of the given equation yields <strong>4 x = 160</strong>. Dividing both sides of this equation by <strong>4</strong> yields <strong>x = 40</strong>. Therefore, the solution to the given equation is <strong>40</strong>.",
    hasFigure: false,
  },
  {
    id: "3c4ce699",
    type: "spr",
    questionHtml:
      "If <strong>6 + x = 9</strong>, what is the value of <strong>18 + 3 x</strong>?",
    choices: [],
    correctAnswer: "27",
    explanation:
      "The correct answer is <strong>27</strong>. Multiplying both sides of the given equation by <strong>3</strong> yields <strong>3 (6 + x) = 3 (9)</strong>, or <strong>18 + 3 x = 27</strong>. Therefore, the value of <strong>18 + 3 x</strong> is <strong>27</strong>.",
    hasFigure: false,
  },
  {
    id: "3d04de9c",
    type: "mcq",
    questionHtml:
      "A principal used a total of <strong>25</strong> flags that were either blue or yellow for field day. The principal used <strong>20</strong> blue flags. How many yellow flags were used?",
    choices: [
      { label: "A", text: "<strong>5</strong>" },
      { label: "B", text: "<strong>20</strong>" },
      { label: "C", text: "<strong>25</strong>" },
      { label: "D", text: "<strong>30</strong>" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. It's given that a principal used a total of <strong>25</strong> blue flags and yellow flags. It's also given that of the <strong>25</strong> flags used, <strong>20</strong> flags were blue. Subtracting the number of blue flags used from the total number of flags used results in the number of yellow flags used. It follows that the number of yellow flags used is <strong>25 − 20</strong>, or <strong>5</strong>. <br>Choice B is incorrect. This is the number of blue flags used.<br>Choice C is incorrect. This is the total number of flags used.<br>Choice D is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "46f68129",
    type: "mcq",
    questionHtml:
      "A librarian has 43 books to distribute to a group of children. If he gives each child 2 books, he will have 7 books left over. How many children are in the group?",
    choices: [
      { label: "A", text: "15" },
      { label: "B", text: "18" },
      { label: "C", text: "25" },
      { label: "D", text: "29" },
    ],
    correctAnswer: "",
    explanation:
      "Choice B is correct. Subtracting the number of books left over from the total number of books results in <strong>43 − 7 = 36</strong>, which is the number of books distributed. Dividing the number of books distributed by the number of books given to each child results in <strong>the fraction 36 over 2 = 18</strong>.Choice A is incorrect and results from dividing the total number of books by the number of books given to each child, <strong>the fraction 43 over 2, is ≈ 22</strong>, then subtracting the number of books left over from the result, <strong>22 − 7 = 15</strong>. Choice C is incorrect and results from adding the number of books left over to the total number of books, <strong>43 + 7 = 50</strong>, then dividing the result by the number of books given to each child, <strong>the fraction 50 over 2 = 25</strong>. Choice D is incorrect and results from dividing the total number of books by the number of books given to each child, <strong>the fraction 43 over 2, is ≈ 22</strong>, then adding the number of books left over, <strong>22 + 7 = 29</strong>.",
    hasFigure: false,
  },
  {
    id: "4de87c9a",
    type: "mcq",
    questionHtml:
      "<strong>3</strong> more than <strong>8</strong> times a number <strong>x</strong> is equal to <strong>83</strong>. Which equation represents this situation?",
    choices: [
      { label: "A", text: "<strong>(3) (8) x = 83</strong>" },
      { label: "B", text: "<strong>8 x = 83 + 3</strong>" },
      { label: "C", text: "<strong>3 x + 8 = 83</strong>" },
      { label: "D", text: "<strong>8 x + 3 = 83</strong>" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. The given phrase “<strong>8</strong> times a number <strong>x</strong>” can be represented by the expression <strong>8 x</strong>. The given phrase “<strong>3</strong> more than” indicates an increase of <strong>3</strong> to a quantity. Therefore “<strong>3</strong> more than <strong>8</strong> times a number <strong>x</strong>” can be represented by the expression <strong>8 x + 3</strong>. Since it’s given that <strong>3</strong> more than <strong>8</strong> times a number <strong>x</strong> is equal to <strong>83</strong>, it follows that <strong>8 x + 3</strong> is equal to <strong>83</strong>, or <strong>8 x + 3 = 83</strong>. Therefore, the equation that represents this situation is <strong>8 x + 3 = 83</strong>.<br>Choice A is incorrect. This equation represents <strong>3</strong> times the quantity <strong>8</strong> times a number <strong>x</strong> is equal to <strong>83</strong>. <br>Choice B is incorrect. This equation represents <strong>8</strong> times a number <strong>x</strong> is equal to <strong>3</strong> more than <strong>83</strong>.<br>Choice C is incorrect. This equation represents <strong>8</strong> more than <strong>3</strong> times a number <strong>x</strong> is equal to <strong>83</strong>.",
    hasFigure: false,
  },
  {
    id: "4e77195b",
    type: "spr",
    questionHtml:
      "If <strong>2 + x = 60</strong>, what is the value of <strong>16 + 8 x</strong>?",
    choices: [],
    correctAnswer: "480",
    explanation:
      "The correct answer is <strong>480</strong>. Multiplying both sides of the given equation by <strong>8</strong> yields <strong>8 (2 + x) = 8 (60)</strong>, or <strong>16 + 8 x = 480</strong>. Therefore, if <strong>2 + x = 60</strong>, the value of <strong>16 + 8 x</strong> is <strong>480</strong>.",
    hasFigure: false,
  },
  {
    id: "4f7981a0",
    type: "spr",
    questionHtml:
      "If <strong>3 x + 2 = 8</strong>, what is the value of <strong>9 x + 6</strong> ?",
    choices: [],
    correctAnswer: "",
    explanation:
      "The correct answer is 24. Multiplying both sides of the given equation by 3 yields <strong>3 · (3 x + 2, ) = 24</strong>. Using the distributive property to rewrite the left-hand side of this equation yields <strong>9 x + 6 = 24</strong>.",
    hasFigure: false,
  },
  {
    id: "550b352c",
    type: "mcq",
    questionHtml: "How many solutions exist to the equation shown above?",
    choices: [
      { label: "A", text: "None" },
      { label: "B", text: "Exactly 1" },
      { label: "C", text: "Exactly 3" },
      { label: "D", text: "Infinitely many" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. Subtracting 4 from each side of the given equation yields <strong>6 = 2 x</strong>, or <strong>x = 3</strong>, so the equation has a unique solution of <strong>x = 3</strong>.Choice A is incorrect. Since 3 is a value of x that satisfies the given equation, the equation has at least 1 solution. Choice C is incorrect. Linear equations can have 0, 1, or infinitely many solutions; no linear equation has exactly 3 solutions. Choice D is incorrect. If a linear equation has infinitely many solutions, it can be reduced to <strong>0 = 0</strong>. This equation reduces to <strong>x = 3</strong>, so there is only 1 solution.",
    hasFigure: false,
  },
  {
    id: "5c94e6fa",
    type: "spr",
    questionHtml:
      "<strong>3 x + 21 = 3 x + k</strong><br>In the given equation, <strong>k</strong> is a constant. The equation has infinitely many solutions. What is the value of <strong>k</strong>?",
    choices: [],
    correctAnswer: "21",
    explanation:
      "The correct answer is <strong>21</strong>. It's given that the equation <strong>3 x + 21 = 3 x + k</strong> has infinitely many solutions. If an equation in one variable has infinitely many solutions, then the equation is true for any value of the variable. Subtracting <strong>3 x</strong> from both sides of the given equation yields <strong>k = 21</strong>. Since this equation must be true for any value of <strong>x</strong>, the value of <strong>k</strong> is <strong>21</strong>.",
    hasFigure: false,
  },
  {
    id: "60f71697",
    type: "mcq",
    questionHtml:
      "<strong>8 x = 88</strong><br>What value of <strong>x</strong> is the solution to the given equation?",
    choices: [
      { label: "A", text: "<strong>11</strong>" },
      { label: "B", text: "<strong>80</strong>" },
      { label: "C", text: "<strong>96</strong>" },
      { label: "D", text: "<strong>704</strong>" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. Dividing both sides of the given equation by <strong>8</strong> yields <strong>x = 11</strong>. Therefore, <strong>11</strong> is the solution to the given equation.<br>Choice B is incorrect. This is the solution to the equation <strong>x + 8 = 88</strong>.<br>Choice C is incorrect. This is the solution to the equation <strong>x − 8 = 88</strong>.<br>Choice D is incorrect. This is the solution to the equation <strong>(x) / (8) = 88</strong>.",
    hasFigure: false,
  },
  {
    id: "6105234d",
    type: "mcq",
    questionHtml:
      "John paid a total of <strong>dollar sign 165</strong> for a microscope by making a down payment of <strong>dollar sign 37</strong> plus <strong>p</strong> monthly payments of <strong>dollar sign 16</strong> each. Which of the following equations represents this situation?",
    choices: [
      { label: "A", text: "<strong>16 p − 37 = 165</strong>" },
      { label: "B", text: "<strong>37 p − 16 = 165</strong>" },
      { label: "C", text: "<strong>16 p + 37 = 165</strong>" },
      { label: "D", text: "<strong>37 p + 16 = 165</strong>" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. It’s given that John made a <strong>dollar sign 16</strong> payment each month for <strong>p</strong> months. The total amount of these payments can be represented by the expression <strong>16 p</strong>. The down payment can be added to that amount to find the total amount John paid, yielding the expression <strong>16 p + 37</strong>. It’s given that John paid a total of <strong>dollar sign 165</strong>. Therefore, the expression for the total amount John paid can be set equal to that amount, yielding the equation <strong>16 p + 37 = 165</strong>.<br>Choice A is incorrect and may result from conceptual or calculation errors.<br>Choice B is incorrect and may result from conceptual or calculation errors.<br>Choice D is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "6ac23de7",
    type: "mcq",
    questionHtml:
      "<strong>the fraction 4 x over 5 = 20</strong>In the equation above, what is the value of x ?",
    choices: [
      { label: "A", text: "25" },
      { label: "B", text: "24" },
      { label: "C", text: "16" },
      { label: "D", text: "15" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. Multiplying both sides of the equation by 5 results in <strong>4 x = 100</strong>. Dividing both sides of the resulting equation by 4 results in <strong>x = 25</strong>.Choice B is incorrect and may result from adding 20 and 4. Choice C is incorrect and may result from dividing 20 by 5 and then multiplying the result by 4. Choice D is incorrect and may result from subtracting 5 from 20.",
    hasFigure: false,
  },
  {
    id: "6fa593f1",
    type: "mcq",
    questionHtml:
      "If <strong>x = 40</strong>, what is the value of <strong>x + 6</strong>?",
    choices: [
      { label: "A", text: "<strong>34</strong>" },
      { label: "B", text: "<strong>40</strong>" },
      { label: "C", text: "<strong>46</strong>" },
      { label: "D", text: "<strong>64</strong>" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. It's given that <strong>x = 40</strong>. Adding <strong>6</strong> to both sides of this equation yields <strong>x + 6 = 40 + 6</strong>, or <strong>x + 6 = 46</strong>. Therefore, the value of <strong>x + 6</strong> is <strong>46</strong>.<br>Choice A is incorrect. This is the value of <strong>x − 6</strong>, not <strong>x + 6</strong>.<br>Choice B is incorrect. This is the value of <strong>x</strong>, not <strong>x + 6</strong>.<br>Choice D is incorrect. This is the value of <strong>x + 24</strong>, not <strong>x + 6</strong>.",
    hasFigure: false,
  },
  {
    id: "70774aa4",
    type: "mcq",
    questionHtml:
      "If <strong>5 x = 20</strong>, what is the value of <strong>15 x</strong>?",
    choices: [
      { label: "A", text: "<strong>7</strong>" },
      { label: "B", text: "<strong>12</strong>" },
      { label: "C", text: "<strong>23</strong>" },
      { label: "D", text: "<strong>60</strong>" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. It’s given that <strong>5 x = 20</strong>. Multiplying both sides of this equation by <strong>3</strong> yields <strong>15 x = 60</strong>. Therefore, the value of <strong>15 x</strong> is <strong>60</strong>.<br>Choice A is incorrect and may result from conceptual errors.<br>Choice B is incorrect and may result from conceptual errors.<br>Choice C is incorrect and may result from conceptual errors.",
    hasFigure: false,
  },
  {
    id: "7392dfc1",
    type: "mcq",
    questionHtml:
      "Which of the following is equivalent to <strong>4 x + 6 = 12</strong> ?",
    choices: [
      { label: "A", text: "<strong>2 x + 4 = 6</strong>" },
      { label: "B", text: "<strong>x + 3 = 3</strong>" },
      { label: "C", text: "<strong>3 x + 2 = 4</strong>" },
      { label: "D", text: "<strong>2 x + 3 = 6</strong>" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. Dividing each side of the original equation by [] yields <strong>the fraction with numerator 4 x + 6, and denominator 2 = twelve halves</strong>, which simplifies to <strong>2 x + 3 = 6</strong>.Choice A is incorrect. Dividing each side of the original equation by [] gives <strong>2 x + 3 = 6</strong>, which is not equivalent to <strong>2 x + 4 = 6</strong>. Choice B is incorrect. Dividing each side of the original equation by [] gives <strong>x + three halves = 3</strong>, which is not equivalent to <strong>x + 3 = 3</strong>. Choice C is incorrect. Dividing each side of the original equation by [] gives <strong>four thirds x + 2 = 4</strong>, which is not equivalent to <strong>3 x + 2 = 4</strong>.",
    hasFigure: false,
  },
  {
    id: "7a987ae4",
    type: "mcq",
    questionHtml:
      "If <strong>2 n over 5 = 10</strong>, what is the value of <strong>2 n − 1</strong> ?",
    choices: [
      { label: "A", text: "24" },
      { label: "B", text: "49" },
      { label: "C", text: "50" },
      { label: "D", text: "99" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. Multiplying both sides of the given equation by 5 yields <strong>2 n = 50</strong>. Substituting 50 for <strong>2 n</strong> in the expression <strong>2 n − 1</strong> yields <strong>50 − 1 = 49</strong>.Alternate approach: Dividing both sides of <strong>2 n = 50</strong> by 2 yields <strong>n = 25</strong>. Evaluating the expression <strong>2 n − 1</strong> for <strong>n = 25</strong> yields <strong>2 · 25 − 1 = 49</strong>.<br>Choice A is incorrect and may result from finding the value of <strong>n − 1</strong> instead of <strong>2 n − 1</strong>. Choice C is incorrect and may result from finding the value of <strong>2 n</strong> instead of <strong>2 n − 1</strong>. Choice D is incorrect and may result from finding the value of <strong>4 n − 1</strong> instead of <strong>2 n − 1</strong>.",
    hasFigure: false,
  },
  {
    id: "8339793c",
    type: "spr",
    questionHtml:
      "Nasir bought <strong>9</strong> storage bins that were each the same price. He used a coupon for <strong>dollar sign 63</strong> off the entire purchase. The cost for the entire purchase after using the coupon was <strong>dollar sign 27</strong>. What was the original price, in dollars, for <strong>1</strong> storage bin?",
    choices: [],
    correctAnswer: "10",
    explanation:
      "The correct answer is <strong>10</strong>. It’s given that the cost for the entire purchase was <strong>dollar sign 27</strong> after a coupon was used for <strong>dollar sign 63</strong> off the entire purchase. Adding the amount of the coupon to the purchase price yields <strong>27 + 63 = 90</strong>. Thus, the cost for the entire purchase before using the coupon was <strong>dollar sign 90</strong>. It’s given that Nasir bought <strong>9</strong> storage bins. The original price for <strong>1</strong> storage bin can be found by dividing the total cost by <strong>9</strong>. Therefore, the original price, in dollars, for <strong>1</strong> storage bin is <strong>(90) / (9)</strong>, or <strong>10</strong>.",
    hasFigure: false,
  },
  {
    id: "87071893",
    type: "spr",
    questionHtml:
      "<strong>x + 40 = 95</strong><br>What value of <strong>x</strong> is the solution to the given equation?",
    choices: [],
    correctAnswer: "55",
    explanation:
      "The correct answer is <strong>55</strong>. Subtracting <strong>40</strong> from both sides of the given equation yields <strong>x = 55</strong>. Therefore, the value of <strong>x</strong> is <strong>55</strong>.",
    hasFigure: false,
  },
  {
    id: "93954cfa",
    type: "mcq",
    questionHtml:
      "One pound of grapes costs $2. At this rate, how many dollars will c pounds of grapes cost?",
    choices: [
      { label: "A", text: "<strong>2 c</strong>" },
      { label: "B", text: "<strong>2 + c</strong>" },
      { label: "C", text: "<strong>the fraction 2 over c</strong>" },
      { label: "D", text: "<strong>the fraction c over 2</strong>" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. If one pound of grapes costs $2, two pounds of grapes will cost 2 times $2, three pounds of grapes will cost 3 times $2, and so on. Therefore, c pounds of grapes will cost c times $2, which is 2c dollars.Choice B is incorrect and may result from incorrectly adding instead of multiplying. Choice C is incorrect and may result from assuming that c pounds cost $2, and then finding the cost per pound. Choice D is incorrect and could result from incorrectly assuming that 2 pounds cost $c, and then finding the cost per pound.",
    hasFigure: false,
  },
  {
    id: "997bec28",
    type: "spr",
    questionHtml:
      "The perimeter of an isosceles triangle is <strong>83</strong> inches. Each of the two congruent sides of the triangle has a length of <strong>24</strong> inches. What is the length, in inches, of the third side?",
    choices: [],
    correctAnswer: "35",
    explanation:
      "The correct answer is <strong>35</strong>. It’s given that the perimeter of an isosceles triangle is <strong>83</strong> inches and that each of the two congruent sides has a length of <strong>24</strong> inches. The perimeter of a triangle is the sum of the lengths of its three sides. The equation <strong>24 + 24 + x = 83</strong> can be used to represent this situation, where <strong>x</strong> is the length, in inches, of the third side. Combining like terms on the left-hand side of this equation yields <strong>48 + x = 83</strong>. Subtracting <strong>48</strong> from both sides of this equation yields <strong>x = 35</strong>. Therefore, the length, in inches, of the third side is <strong>35</strong>.",
    hasFigure: false,
  },
  {
    id: "9ff10b3b",
    type: "mcq",
    questionHtml:
      "If <strong>one half x − one sixth x = 1</strong>, what is the value of x ?",
    choices: [
      { label: "A", text: "<strong>−4</strong>" },
      { label: "B", text: "<strong>one third</strong>" },
      { label: "C", text: "<strong>3</strong>" },
      { label: "D", text: "<strong>6</strong>" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. To make it easier to add like terms on the left-hand side of the given equation, both sides of the equation can be multiplied by 6, which is the lowest common denominator of <strong>one half</strong> and <strong>one sixth</strong>. This yields <strong>3 x − x = 6</strong>, which can be rewritten as <strong>2 x = 6</strong>. Dividing both sides of this equation by 2 yields <strong>x = 3</strong>.Choice A is incorrect and may result from subtracting the denominators instead of numerators with common denominators to get <strong>−one fourth x</strong>, rather than <strong>one third x</strong>, on the left-hand side of the equation. Choice B is incorrect and may result from rewriting the given equation as <strong>one half x = one sixth</strong> instead of <strong>2 x = 6</strong>. Choice D is incorrect and may result from conceptual or computational errors.",
    hasFigure: false,
  },
  {
    id: "a9c04a21",
    type: "mcq",
    questionHtml:
      "What is the solution to the equation <strong>2 x + 3 = 7</strong>?",
    choices: [
      { label: "A", text: "1" },
      { label: "B", text: "1.5" },
      { label: "C", text: "2" },
      { label: "D", text: "4" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. Subtracting 3 from both sides of the given equation yields <strong>2 x = 4</strong>. Dividing both sides by 2 results in <strong>x = 2</strong>.Choices A and B are incorrect and may result from computational errors. Choice D is incorrect. This is the value of <strong>2 x</strong>.",
    hasFigure: false,
  },
  {
    id: "b4553284",
    type: "spr",
    questionHtml:
      "If <strong>2 x = 12</strong>, what is the value of <strong>9 x</strong>?",
    choices: [],
    correctAnswer: "54",
    explanation:
      "The correct answer is <strong>54</strong>. Dividing both sides of the given equation by <strong>2</strong> yields <strong>x = 6</strong>. Multiplying both sides of this equation by <strong>9</strong> yields <strong>9 x = 54</strong>. Thus, the value of <strong>9 x</strong> is <strong>54</strong>.",
    hasFigure: false,
  },
  {
    id: "b82a943c",
    type: "mcq",
    questionHtml:
      "If <strong>7 x = 28</strong>, what is the value of <strong>8 x</strong>?",
    choices: [
      { label: "A", text: "<strong>21</strong>" },
      { label: "B", text: "<strong>32</strong>" },
      { label: "C", text: "<strong>168</strong>" },
      { label: "D", text: "<strong>224</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. Dividing both sides of the given equation <strong>7 x = 28</strong> by <strong>7</strong> yields <strong>x = 4</strong>. Substituting <strong>4</strong> for <strong>x</strong> in the expression <strong>8 x</strong> yields <strong>8 (4)</strong>, which is equivalent to <strong>32</strong>.<br>Choice A is incorrect. This is the value of <strong>(21) / (4) x</strong>.<br>Choice C is incorrect. This is the value of <strong>42 x</strong>.<br>Choice D is incorrect. This is the value of <strong>56 x</strong>.",
    hasFigure: false,
  },
  {
    id: "c3989ef8",
    type: "mcq",
    questionHtml:
      "Henry receives a <strong>dollar sign 60.00</strong> gift card to pay for movies online. He uses his gift card to buy <strong>3</strong> movies for <strong>dollar sign 7.50</strong> each. If he spends the rest of his gift card balance on renting movies for <strong>dollar sign 1.50</strong> each, how many movies can Henry rent?",
    choices: [
      { label: "A", text: "<strong>10</strong>" },
      { label: "B", text: "<strong>25</strong>" },
      { label: "C", text: "<strong>35</strong>" },
      { label: "D", text: "<strong>40</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. It's given that Henry uses his <strong>dollar sign 60.00</strong> gift card to buy <strong>3</strong> movies for <strong>dollar sign 7.50</strong> each. Therefore, Henry spends <strong>3 (dollar sign 7.50)</strong>, or <strong>dollar sign 22.50</strong>, of his <strong>dollar sign 60.00</strong> gift card to buy <strong>3</strong> movies. After buying <strong>3</strong> movies with his <strong>dollar sign 60.00</strong> gift card, Henry has a gift card balance of <strong>dollar sign 60.00 − dollar sign 22.50</strong>, or <strong>dollar sign 37.50</strong>. It's also given that Henry spends the rest of his gift card balance on renting movies for <strong>dollar sign 1.50</strong> each. Therefore, Henry can rent <strong>(dollar sign 37.50) / (dollar sign 1.50)</strong>, or <strong>25</strong>, movies.<br>Choice A is incorrect and may result from conceptual or calculation errors.<br>Choice C is incorrect and may result from conceptual or calculation errors.<br>Choice D is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "c841e8e8",
    type: "mcq",
    questionHtml:
      "<strong>k + 12 = 336</strong><br>What is the solution to the given equation?",
    choices: [
      { label: "A", text: "<strong>28</strong>" },
      { label: "B", text: "<strong>324</strong>" },
      { label: "C", text: "<strong>348</strong>" },
      { label: "D", text: "<strong>4, 032</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. Subtracting <strong>12</strong> from both sides of the given equation yields <strong>k = 324</strong>. Therefore, the solution to the given equation is <strong>324</strong>.<br>Choice A is incorrect and may result from conceptual or calculation errors.<br>Choice C is incorrect and may result from conceptual or calculation errors.<br>Choice D is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "d9d83c02",
    type: "mcq",
    questionHtml:
      "For what value of w does <strong>w − 10 = 2 · (w + 5, )</strong> ?",
    choices: [
      { label: "A", text: "[]" },
      { label: "B", text: "<strong>0</strong>" },
      { label: "C", text: "<strong>−15</strong>" },
      { label: "D", text: "<strong>−20</strong>" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. To solve the equation, use the distributive property to multiply on the right-hand side of the equation which gives w – 10 = 2w + 10. Subtract w from both sides of the equation, which gives –10 = w + 10. Finally, subtract 10 from both sides of the equation, which gives –20 = w.Choices A and B are incorrect and may result from making sign errors. Choice C is incorrect and may result from incompletely distributing the 2 in the expression 2(w + 5).",
    hasFigure: false,
  },
  {
    id: "e53870b6",
    type: "spr",
    questionHtml:
      "In the given equation, k is a constant. If the equation has infinitely many solutions, what is the value of k ?",
    choices: [],
    correctAnswer: "",
    explanation:
      "The correct answer is 5. Subtracting <strong>6 x</strong> from both sides of the given equation gives <strong>k = 5</strong>, so for any value of x, <strong>6 x + k = 6 x + 5</strong> if and only if <strong>k = 5</strong>. Therefore, if the given equation has infinitely many solutions, the value of k is 5.",
    hasFigure: false,
  },
  {
    id: "e7b6f0d1",
    type: "mcq",
    questionHtml:
      "<strong>4 x + 6 = 18</strong><br>Which equation has the same solution as the given equation?",
    choices: [
      { label: "A", text: "<strong>4 x = 108</strong>" },
      { label: "B", text: "<strong>4 x = 24</strong>" },
      { label: "C", text: "<strong>4 x = 12</strong>" },
      { label: "D", text: "<strong>4 x = 3</strong>" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. Subtracting <strong>6</strong> from both sides of the given equation yields <strong>4 x = 12</strong>, which is the equation given in choice C. Since this equation is equivalent to the given equation, it has the same solution as the given equation.<br>Choice A is incorrect and may result from conceptual or calculation errors.<br>Choice B is incorrect and may result from conceptual or calculation errors.<br>Choice D is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "eac739b2",
    type: "mcq",
    questionHtml:
      "If <strong>4 x + 2 = 12</strong>, what is the value of <strong>16 x + 8</strong>?",
    choices: [
      { label: "A", text: "<strong>40</strong>" },
      { label: "B", text: "<strong>48</strong>" },
      { label: "C", text: "<strong>56</strong>" },
      { label: "D", text: "<strong>60</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. Multiplying both sides of the given equation by <strong>4</strong> yields <strong>(4) (4 x + 2) = (4) (12)</strong>, or <strong>16 x + 8 = 48</strong>. Therefore, the value of <strong>16 x + 8</strong> is <strong>48</strong>.<br>Choice A is incorrect and may result from conceptual or calculation errors.<br>Choice C is incorrect and may result from conceptual or calculation errors.<br>Choice D is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "ed18c4f7",
    type: "mcq",
    questionHtml:
      "Cathy has n CDs. Gerry has 3 more than twice the number of CDs that Cathy has. In terms of n, how many CDs does Gerry have?",
    choices: [
      { label: "A", text: "<strong>3 n − 2</strong>" },
      { label: "B", text: "<strong>3 n + 2</strong>" },
      { label: "C", text: "<strong>2 n − 3</strong>" },
      { label: "D", text: "<strong>2 n + 3</strong>" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. The term 2n represents twice the number of CDs that Cathy has, and adding 3 represents 3 more than that amount.Choices A and B are incorrect. The expression 3n represents three times the number of CDs that Cathy has. Choice C is incorrect. Subtracting 3 represents 3 fewer than twice the number of CDs that Cathy has.",
    hasFigure: false,
  },
  {
    id: "fbb0ea7f",
    type: "mcq",
    questionHtml:
      "A rocket contained <strong>467, 000</strong> kilograms (kg) of propellant before launch. Exactly <strong>21</strong> seconds after launch, <strong>362, 105</strong> kg of this propellant remained. On average, approximately how much propellant, in kg, did the rocket burn each second after launch?",
    choices: [
      { label: "A", text: "<strong>4, 995</strong>" },
      { label: "B", text: "<strong>17, 243</strong>" },
      { label: "C", text: "<strong>39, 481</strong>" },
      { label: "D", text: "<strong>104, 895</strong>" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. It’s given that the rocket contained <strong>467, 000 kilograms (kg)</strong> of propellant before launch and had <strong>362, 105 kg</strong> remaining exactly <strong>21</strong> seconds after launch. Finding the difference between the amount, in <strong>kg</strong>, of propellant before launch and the remaining amount, in <strong>kg</strong>, of propellant after launch gives the amount, in <strong>kg</strong>, of propellant burned during the <strong>21</strong> seconds: <strong>467, 000 − 362, 105 = 104, 895</strong>. Dividing the amount of propellant burned by the number of seconds yields <strong>(104, 895) / (21) = 4, 995</strong>. Thus, an average of <strong>4, 995 kg</strong> of propellant burned each second after launch.<br>Choice B is incorrect and may result from conceptual or calculation errors.<br>Choice C is incorrect and may result from conceptual or calculation errors.<br>Choice D is incorrect and may result from finding the amount of propellant burned, rather than the amount of propellant burned each second.",
    hasFigure: false,
  },
];

export const LINEAR_EQ_ONE_VAR_MEDIUM: PracticeQuestion[] = [
  {
    id: "12ee1edc",
    type: "mcq",
    questionHtml:
      "In the given equation, b is a constant. If the equation has no solution, what is the value of b ?",
    choices: [
      { label: "A", text: "2" },
      { label: "B", text: "4" },
      { label: "C", text: "6" },
      { label: "D", text: "10" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. This equation has no solution when there is no value of x that produces a true statement. Solving the given equation for x by dividing both sides by <strong>(b − 2, )</strong> gives <strong>x = the fraction with numerator 8, and denominator, (b − 2, )</strong>. When <strong>(b − 2, ) = 0</strong>, the right-hand side of this equation will be undefined, and the equation will have no solution. Therefore, when <strong>b = 2</strong>, there is no value of x that satisfies the given equation.Choices B, C, and D are incorrect. Substituting 4, 6, and 10 for b in the given equation yields exactly one solution, rather than no solution, for x. For example, substituting 4 for b in the given equation yields <strong>(4 − 2, ) · x = 8</strong>, or <strong>2 x = 8</strong>. Dividing both sides of <strong>2 x = 8</strong> by 2 yields <strong>x = 4</strong>. Similarly, if <strong>b = 6</strong> or <strong>b = 10</strong>, <strong>x = 2</strong> and <strong>x = 1</strong>, respectively.",
    hasFigure: false,
  },
  {
    id: "15daa8d6",
    type: "spr",
    questionHtml:
      "<strong>2 x + 16 = a (x + 8)</strong><br>In the given equation, <strong>a</strong> is a constant. If the equation has infinitely many solutions, what is the value of <strong>a</strong>?",
    choices: [],
    correctAnswer: "2",
    explanation:
      "The correct answer is <strong>2</strong>. An equation with one variable, <strong>x</strong>, has infinitely many solutions only when both sides of the equation are equal for any defined value of <strong>x</strong>. It's given that <strong>2 x + 16 = a (x + 8)</strong>, where <strong>a</strong> is a constant. This equation can be rewritten as <strong>2 (x + 8) = a (x + 8)</strong>. If this equation has infinitely many solutions, then both sides of this equation are equal for any defined value of <strong>x</strong>. Both sides of this equation are equal for any defined value of <strong>x</strong> when <strong>2 = a</strong>. Therefore, if the equation has infinitely many solutions, the value of <strong>a</strong> is <strong>2</strong>.<br>Alternate approach: If the given equation, <strong>2 x + 16 = a (x + 8)</strong>, has infinitely many solutions, then both sides of this equation are equal for any value of <strong>x</strong>. If <strong>x = 0</strong>, then substituting <strong>0</strong> for <strong>x</strong> in <strong>2 x + 16 = a (x + 8)</strong> yields <strong>2 (0) + 16 = a (0 + 8)</strong>, or <strong>16 = 8 a</strong>. Dividing both sides of this equation by <strong>8</strong> yields <strong>2 = a</strong>.",
    hasFigure: false,
  },
  {
    id: "36ab4122",
    type: "mcq",
    questionHtml:
      "Megan’s regular wage at her job is p dollars per hour for the first 8 hours of work in a day plus 1.5 times her regular hourly wage for work in excess of 8 hours that day. On a given day, Megan worked for 10 hours, and her total earnings for that day were $137.50. What is Megan’s regular hourly wage?",
    choices: [
      { label: "A", text: "$11.75" },
      { label: "B", text: "$12.50" },
      { label: "C", text: "$13.25" },
      { label: "D", text: "$13.75" },
    ],
    correctAnswer: "",
    explanation:
      "Choice B is correct. Since p represents Megan’s regular pay per hour, 1.5p represents the pay per hour in excess of 8 hours. Since Megan worked for 10 hours, she must have been paid p dollars per hour for 8 of the hours plus 1.5p dollars per hour for the remaining 2 hours. Therefore, since Megan earned $137.50 for the 10 hours, the situation can be represented by the equation 137.5 = 8p + 2(1.5)p. Distributing the 2 in the equation gives 137.5 = 8p + 3p, and combining like terms gives 137.5 = 11p. Dividing both sides by 11 gives p = 12.5. Therefore, Megan’s regular wage is $12.50.Choices A and C are incorrect and may be the result of calculation errors. Choice D is incorrect and may result from finding the average hourly wage that Megan earned for the 10 hours of work.",
    hasFigure: false,
  },
  {
    id: "45bba652",
    type: "mcq",
    questionHtml:
      "If <strong>2 · (x − 5, ) + 3 · (x − 5, ) = 10</strong>, what is the value of <strong>x − 5</strong> ?",
    choices: [
      { label: "A", text: "2" },
      { label: "B", text: "5" },
      { label: "C", text: "7" },
      { label: "D", text: "12" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. Adding the like terms on the left-hand side of the given equation yields <strong>5 · (x − 5, ) = 10</strong>. Dividing both sides of this equation by 5 yields <strong>x − 5 = 2</strong>.Choice B is incorrect and may result from subtracting 5, not dividing by 5, on both sides of the equation <strong>5 · (x − 5, ) = 10</strong>. Choice C is incorrect. This is the value of x, not the value of <strong>x − 5</strong>. Choice D is incorrect. This is the value of <strong>x + 5</strong>, not the value of <strong>x − 5</strong>.",
    hasFigure: false,
  },
  {
    id: "4f669597",
    type: "spr",
    questionHtml: "What value of p is the solution of the equation above?",
    choices: [],
    correctAnswer: "",
    explanation:
      "The correct answer is 1.2. One way to solve the equation <strong>2 · (p + 1, ) + 8 · (p − 1, ) = 5 p</strong> is to first distribute the terms outside the parentheses to the terms inside the parentheses: <strong>2 p + 2 + 8 p − 8 = 5 p</strong>. Next, combine like terms on the left side of the equal sign: <strong>10 p − 6 = 5 p</strong> . Subtracting 10p from both sides yields <strong>−6 = −5 p</strong> . Finally, dividing both sides by <strong>−5</strong> gives <strong>p = six fifths</strong>, which is equivalent to <strong>p = 1 . 2</strong>. Note that 1.2 and 6/5 are examples of ways to enter a correct answer.",
    hasFigure: false,
  },
  {
    id: "5a7ab8e8",
    type: "mcq",
    questionHtml:
      "<strong>66 x = 66 x</strong><br>How many solutions does the given equation have?",
    choices: [
      { label: "A", text: "Exactly one" },
      { label: "B", text: "Exactly two" },
      { label: "C", text: "Infinitely many" },
      { label: "D", text: "Zero" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. If the two sides of a linear equation are equivalent, then the equation is true for any value. If an equation is true for any value, it has infinitely many solutions. Since the two sides of the given linear equation <strong>66 x = 66 x</strong> are equivalent, the given equation has infinitely many solutions.<br>Choice A is incorrect and may result from conceptual or calculation errors.<br>Choice B is incorrect and may result from conceptual or calculation errors.<br>Choice D is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "5ad9eff0",
    type: "mcq",
    questionHtml:
      "The width of a rectangular dance floor is w feet. The length of the floor is 6 feet longer than its width. Which of the following expresses the perimeter, in feet, of the dance floor in terms of w ?",
    choices: [
      { label: "A", text: "<strong>2 w + 6</strong>" },
      { label: "B", text: "<strong>4 w + 12</strong>" },
      { label: "C", text: "<strong>w² + 6</strong>" },
      { label: "D", text: "<strong>w² + 6 w</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. It is given that the width of the dance floor is w feet. The length is 6 feet longer than the width; therefore, the length of the dance floor is <strong>w + 6</strong>. So the perimeter is <strong>w + w + (w + 6, ) + (w + 6, ) = 4 w + 12</strong>.Choice A is incorrect because it is the sum of one length and one width, which is only half the perimeter. Choice C is incorrect and may result from using the formula for the area instead of the formula for the perimeter and making a calculation error. Choice D is incorrect because this is the area, not the perimeter, of the dance floor.",
    hasFigure: false,
  },
  {
    id: "620abf36",
    type: "mcq",
    questionHtml:
      "If <strong>5 (x + 4) = 4 (x + 4) + 29</strong>, what is the value of <strong>x + 4</strong>?",
    choices: [
      { label: "A", text: "<strong>−4</strong>" },
      { label: "B", text: "<strong>25</strong>" },
      { label: "C", text: "<strong>29</strong>" },
      { label: "D", text: "<strong>33</strong>" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. Subtracting <strong>4 (x + 4)</strong> from both sides of the given equation yields <strong>1 (x + 4) = 29</strong>, or <strong>x + 4 = 29</strong>. Therefore, the value of <strong>x + 4</strong> is <strong>29</strong>.<br>Choice A is incorrect and may result from conceptual or calculation errors.<br>Choice B is incorrect. This is the value of <strong>x</strong>, not <strong>x + 4</strong>.<br>Choice D is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "70e29454",
    type: "mcq",
    questionHtml:
      "In the equation above, a and b are constants. If the equation has infinitely many solutions, what are the values of a and b ?",
    choices: [
      { label: "A", text: "<strong>a = 2</strong> and <strong>b = 1</strong>" },
      { label: "B", text: "<strong>a = 2</strong> and <strong>b = 7</strong>" },
      {
        label: "C",
        text: "<strong>a = −2</strong> and <strong>b = 5</strong>",
      },
      {
        label: "D",
        text: "<strong>a = −2</strong> and <strong>b = −5</strong>",
      },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. Distributing the a on the left-hand side of the equation gives 3a – b – ax = –1 – 2x. Rearranging the terms in each side of the equation yields –ax + 3a – b = –2x –1. Since the equation has infinitely many solutions, it follows that the coefficients of x and the free terms on both sides must be equal. That is,  –a = –2, or a = 2, and 3a – b = –1. Substituting 2 for a in the equation 3a – b = –1 gives 3(2) – b = –1, so  b = 7.Choice A is incorrect and may be the result of a conceptual error when finding the value of b. Choices C and D are incorrect and may result from making a sign error when simplifying.",
    hasFigure: false,
  },
  {
    id: "7a5a74a6",
    type: "mcq",
    questionHtml:
      "If x is the solution to the equation above, what is the value of <strong>x − 3</strong> ?",
    choices: [
      { label: "A", text: "<strong>23 over 2</strong>" },
      { label: "B", text: "<strong>17 over 2</strong>" },
      { label: "C", text: "<strong>15 over 2</strong>" },
      { label: "D", text: "<strong>negative, 15 over 2</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. Because 2 is a factor of both <strong>2 x</strong> and 6, the expression <strong>2 x − 6</strong> can be rewritten as <strong>2 · (x − 3, )</strong>. Substituting <strong>2 · (x − 3, )</strong> for <strong>2 x − 6</strong> on the left-hand side of the given equation yields <strong>3 · 2 · (x − 3, ) − 11 = 4 · (x − 3, ) + 6</strong>, or <strong>6 · (x − 3, ) − 11 = 4 · (x − 3, ) + 6</strong>. Subtracting <strong>4 · (x − 3, )</strong> from both sides of this equation yields <strong>2 · (x − 3, ) − 11 = 6</strong>. Adding 11 to both sides of this equation yields <strong>2 · (x − 3, ) = 17</strong>. Dividing both sides of this equation by 2 yields <strong>x − 3 = the fraction 17 over 2</strong>.Alternate approach: Distributing 3 to the quantity <strong>2 x − 6</strong> on the left-hand side of the given equation and distributing 4 to the quantity <strong>x − 3</strong> on the right-hand side yields <strong>6 x − 18 − 11 = 4 x − 12 + 6</strong>, or <strong>6 x − 29 = 4 x − 6</strong>. Subtracting <strong>4 x</strong> from both sides of this equation yields <strong>2 x − 29 = −6</strong>. Adding 29 to both sides of this equation yields <strong>2 x = 23</strong>. Dividing both sides of this equation by 2 yields <strong>x = the fraction 23 over 2</strong>. Therefore, the value of <strong>x − 3</strong> is <strong>the fraction 23 over 2, end fraction − 3</strong>, or <strong>the fraction 17 over 2</strong>.<br>Choice A is incorrect. This is the value of x, not <strong>x − 3</strong>. Choices C and D are incorrect. If the value of <strong>x − 3</strong> is <strong>the fraction 15 over 2</strong> or <strong>−of the fraction 15 over 2</strong>, it follows that the value of x is <strong>the fraction 21 over 2</strong> or <strong>−of the fraction 9 over 2</strong>, respectively. However, solving the given equation for x yields <strong>x = the fraction 23 over 2</strong>. Therefore, the value of <strong>x − 3</strong> can’t be <strong>the fraction 15 over 2</strong> or <strong>−of the fraction 15 over 2</strong>.",
    hasFigure: false,
  },
  {
    id: "8c515062",
    type: "mcq",
    questionHtml:
      "A candle is made of <strong>17</strong> ounces of wax. When the candle is burning, the amount of wax in the candle decreases by <strong>1</strong> ounce every <strong>4</strong> hours. If <strong>6</strong> ounces of wax remain in this candle, for how many hours has it been burning?",
    choices: [
      { label: "A", text: "<strong>3</strong>" },
      { label: "B", text: "<strong>6</strong>" },
      { label: "C", text: "<strong>24</strong>" },
      { label: "D", text: "<strong>44</strong>" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. It’s given that the candle starts with <strong>17</strong> ounces of wax and has <strong>6</strong> ounces of wax remaining after a period of time has passed. The amount of wax the candle has lost during the time period can be found by subtracting the remaining amount of wax from the amount of wax the candle was made of, which yields <strong>17 − 6</strong> ounces, or <strong>11</strong> ounces. This means the candle loses <strong>11</strong> ounces of wax during that period of time. It’s given that the amount of wax decreases by <strong>1</strong> ounce every <strong>4</strong> hours. If <strong>h</strong> represents the number of hours the candle has been burning, it follows that <strong>one fourth = (11) / (h)</strong>. Multiplying both sides of this equation by <strong>4 h</strong> yields <strong>h = 44</strong>. Therefore, the candle has been burning for <strong>44</strong> hours. <br>Choice A is incorrect and may result from using the equation <strong>one fourth = (h) / (11)</strong> rather than <strong>one fourth = (11) / (h)</strong> to represent the situation, and then rounding to the nearest whole number. <br>Choice B is incorrect. This is the amount of wax, in ounces, remaining in the candle, not the number of hours it has been burning. <br>Choice C is incorrect and may result from using the equation <strong>one fourth = (6) / (h)</strong> rather than <strong>one fourth = (11) / (h)</strong> to represent the situation.",
    hasFigure: false,
  },
  {
    id: "aa85b138",
    type: "mcq",
    questionHtml:
      "A tree had a height of 6 feet when it was planted. The equation above can be used to find how many years n it took the tree to reach a height of 14 feet. Which of the following is the best interpretation of the number 2 in this context?",
    choices: [
      {
        label: "A",
        text: "The number of years it took the tree to double its height",
      },
      {
        label: "B",
        text: "The average number of feet that the tree grew per year",
      },
      {
        label: "C",
        text: "The height, in feet, of the tree when the tree was 1 year old",
      },
      {
        label: "D",
        text: "The average number of years it takes similar trees to grow 14 feet",
      },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. The height of the tree at a given time is equal to its height when it was planted plus the number of feet that the tree grew. In the given equation, 14 represents the height of the tree at the given time, and 6 represents the height of the tree when it was planted. It follows that <strong>2 n</strong> represents the number of feet the tree grew from the time it was planted until the time it reached a height of 14 feet. Since n represents the number of years between the given time and the time the tree was planted, 2 must represent the average number of feet the tree grew each year.Choice A is incorrect and may result from interpreting the coefficient 2 as doubling instead of as increasing by 2 each year. Choice C is incorrect. The height of the tree when it was 1 year old was <strong>2 · 1 + 6 = 8</strong> feet, not 2 feet. Choice D is incorrect. No information is given to connect the growth of one particular tree to the growth of similar trees.",
    hasFigure: false,
  },
  {
    id: "ce314070",
    type: "mcq",
    questionHtml:
      "If <strong>4 x − one half = −5</strong>, what is the value of <strong>8 x − 1</strong>?",
    choices: [
      { label: "A", text: "<strong>2</strong>" },
      { label: "B", text: "<strong>−nine eighths</strong>" },
      { label: "C", text: "<strong>−five halves</strong>" },
      { label: "D", text: "<strong>−10</strong>" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. Multiplying the given equation by 2 on each side yields <strong>2 · (4 x − one half, ) = 2 · −5</strong>. Applying the distributive property, this equation can be rewritten as <strong>2 · 4 x − 2 · one half = 2 · −5</strong>, or <strong>8 x − 1 = −10</strong>.Choices A, B, and C are incorrect and may result from calculation errors in solving the given equation for <strong>x</strong> and then substituting that value of x in the expression <strong>8 x − 1</strong>.",
    hasFigure: false,
  },
  {
    id: "ce6b52d8",
    type: "spr",
    questionHtml:
      "If <strong>2 (3 t − 10) + t = 40 + 4 t</strong>, what is the value of <strong>3 t</strong>?",
    choices: [],
    correctAnswer: "60",
    explanation:
      "The correct answer is <strong>60</strong>. Subtracting <strong>t</strong> from both sides of the given equation yields <strong>2 (3 t − 10) = 40 + 3 t</strong>. Applying the distributive property to the left-hand side of this equation yields <strong>6 t − 20 = 40 + 3 t</strong>. Adding <strong>20</strong> to both sides of this equation yields <strong>6 t = 60 + 3 t</strong>. Subtracting <strong>3 t</strong> from both sides of this equation yields <strong>3 t = 60</strong>. Therefore, the value of <strong>3 t</strong> is <strong>60</strong>.",
    hasFigure: false,
  },
  {
    id: "eafdbbbd",
    type: "mcq",
    questionHtml:
      "<strong>one fourth (x + 5) − one third (x + 5) = −7</strong><br>What value of <strong>x</strong> is the solution to the given equation?",
    choices: [
      { label: "A", text: "<strong>−12</strong>" },
      { label: "B", text: "<strong>−5</strong>" },
      { label: "C", text: "<strong>79</strong>" },
      { label: "D", text: "<strong>204</strong>" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. For the given equation, <strong>(x + 5)</strong> is a factor of both terms on the left-hand side. Therefore, the given equation can be rewritten as <strong>(one fourth − one third) (x + 5) = −7</strong>, or <strong>(three twelfths − four twelfths) (x + 5) = −7</strong>, which is equivalent to <strong>− one twelfth (x + 5) = −7</strong>. Multiplying both sides of this equation by <strong>−12</strong> yields <strong>x + 5 = 84</strong>. Subtracting <strong>5</strong> from both sides of this equation yields <strong>x = 79</strong>. <br>Choice A is incorrect. This is the value of <strong>x</strong> for which the left-hand side of the given equation equals <strong>seven twelfths</strong>, not <strong>−7</strong>.<br>Choice B is incorrect. This is the value of <strong>x</strong> for which the left-hand side of the given equation equals <strong>0</strong>, not <strong>−7</strong>.<br>Choice D is incorrect. This is the value of <strong>x</strong> for which the left-hand side of the given equation equals <strong>−(209) / (12)</strong>, not <strong>−7</strong>.",
    hasFigure: false,
  },
  {
    id: "f09097b1",
    type: "spr",
    questionHtml:
      "An agricultural scientist studying the growth of corn plants recorded the height of a corn plant at the beginning of a study and the height of the plant each day for the next 12 days. The scientist found that the height of the plant increased by an average of 1.20 centimeters per day for the 12 days. If the height of the plant on the last day of the study was 36.8 centimeters, what was the height, in centimeters, of the corn plant at the beginning of the study?",
    choices: [],
    correctAnswer: "",
    explanation:
      "The correct answer is 22.4. If the height of the plant increased by an average of 1.20 centimeters per day for 12 days, then its total growth over the 12 days was <strong>1 . 2 0 · 12 = 14.4</strong> centimeters. The plant was 36.8 centimeters tall after 12 days, so at the beginning of the study its height was <strong>36 . 8 − 14 . 4 = 22 . 4</strong> centimeters. Note that 22.4 and 112/5 are examples of ways to enter a correct answer.Alternate approach: The equation <strong>36 . 8 = 12 · 1 . 2 0 + h</strong> can be used to represent this situation, where h is the height of the plant, in centimeters, at the beginning of the study. Solving this equation for h yields 22.4 centimeters.",
    hasFigure: false,
  },
];

export const LINEAR_EQ_ONE_VAR_HARD: PracticeQuestion[] = [
  {
    id: "0cb57740",
    type: "mcq",
    questionHtml:
      "Each side of a <strong>30</strong>-sided polygon has one of three lengths. The number of sides with length <strong>8 centimeters (cm)</strong> is <strong>5</strong> times the number of sides <strong>n</strong> with length <strong>3 cm</strong>. There are <strong>6</strong> sides with length <strong>4 cm</strong>. Which equation must be true for the value of <strong>n</strong>?",
    choices: [
      { label: "A", text: "<strong>5 n + 6 = 30</strong>" },
      { label: "B", text: "<strong>6 n + 6 = 30</strong>" },
      { label: "C", text: "<strong>8 n + 3 n + 4 n = 30</strong>" },
      { label: "D", text: "<strong>8 (5 n) + 3 n + 4 (6) = 30</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. It’s given that each side of a <strong>30</strong>-sided polygon has one of three lengths. It's also given that the number of sides with length <strong>8</strong> <strong>centimeters (cm)</strong> is <strong>5</strong> times the number of sides <strong>n</strong> with length <strong>3 cm</strong>. Therefore, there are <strong>5 · n</strong>, or <strong>5 n</strong>, sides with length <strong>8 cm</strong>. It’s also given that there are <strong>6</strong> sides with length <strong>4 cm</strong>. Therefore, the number of <strong>3 cm</strong>, <strong>4 cm</strong>, and <strong>8 cm</strong> sides are <strong>n</strong>, <strong>6</strong>, and <strong>5 n</strong>, respectively. Since there are a total of <strong>30</strong> sides, the equation <strong>n + 6 + 5 n = 30</strong> represents this situation. Combining like terms on the left-hand side of this equation yields <strong>6 n + 6 = 30</strong>. Therefore, the equation that must be true for the value of <strong>n</strong> is <strong>6 n + 6 = 30</strong>.<br>Choice A is incorrect and may result from conceptual or calculation errors.<br>Choice C is incorrect and may result from conceptual or calculation errors.<br>Choice D is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "153ee763",
    type: "mcq",
    questionHtml:
      "<strong>− 3 x + 21 p x = 84</strong><br>In the given equation, <strong>p</strong> is a constant. The equation has no solution. What is the value of <strong>p</strong>?",
    choices: [
      { label: "A", text: "<strong>0</strong>" },
      { label: "B", text: "<strong>one seventh</strong>" },
      { label: "C", text: "<strong>four thirds</strong>" },
      { label: "D", text: "<strong>4</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. A linear equation in one variable has no solution if and only if the equation is false; that is, when there is no value of <strong>x</strong> that produces a true statement. It's given that in the equation <strong>− 3 x + 21 p x = 84</strong>, <strong>p</strong> is a constant and the equation has no solution for <strong>x</strong>. Therefore, the value of the constant <strong>p</strong> is one that results in a false equation. Factoring out the common factor of <strong>− 3 x</strong> on the left-hand side of the given equation yields <strong>− 3 x (1 − 7 p) = 84</strong>. Dividing both sides of this equation by <strong>−3</strong> yields <strong>x (1 − 7 p) = −28</strong>. Dividing both sides of this equation by <strong>(1 − 7 p)</strong> yields <strong>x = (−28) / (1 − 7 p)</strong>. This equation is false if and only if <strong>1 − 7 p = 0</strong>. Adding <strong>7 p</strong> to both sides of <strong>1 − 7 p = 0</strong> yields <strong>1 = 7 p</strong>. Dividing both sides of this equation by <strong>7</strong> yields <strong>one seventh = p</strong>. It follows that the equation <strong>x = (−28) / (1 − 7 p)</strong> is false if and only if <strong>p = one seventh</strong>. Therefore, the given equation has no solution if and only if the value of <strong>p</strong> is <strong>one seventh</strong>.<br>Choice A is incorrect and may result from conceptual or calculation errors.<br>Choice C is incorrect and may result from conceptual or calculation errors.<br>Choice D is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "25e1cfed",
    type: "mcq",
    questionHtml:
      "How many solutions does the equation <strong>10 (15 x − 9) = − 15 (6 − 10 x)</strong> have?",
    choices: [
      { label: "A", text: "Exactly one" },
      { label: "B", text: "Exactly two" },
      { label: "C", text: "Infinitely many" },
      { label: "D", text: "Zero" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. Applying the distributive property to each side of the given equation yields <strong>150 x − 90 = −90 + 150 x</strong>. Applying the commutative property of addition to the right-hand side of this equation yields <strong>150 x − 90 = 150 x − 90</strong>. Since the two sides of the equation are equivalent, this equation is true for any value of <strong>x</strong>. Therefore, the given equation has infinitely many solutions.<br>Choice A is incorrect and may result from conceptual or calculation errors.<br>Choice B is incorrect and may result from conceptual or calculation errors.<br>Choice D is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "2937ef4f",
    type: "mcq",
    questionHtml:
      "Hector used a tool called an auger to remove corn from a storage bin at a constant rate. The bin contained 24,000 bushels of corn when Hector began to use the auger. After 5 hours of using the auger, 19,350 bushels of corn remained in the bin. If the auger continues to remove corn at this rate, what is the total number of hours Hector will have been using the auger when 12,840 bushels of corn remain in the bin?",
    choices: [
      { label: "A", text: "3" },
      { label: "B", text: "7" },
      { label: "C", text: "8" },
      { label: "D", text: "12" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. After using the auger for 5 hours, Hector had removed 24,000 – 19,350 = 4,650 bushels of corn from the storage bin. During the 5-hour period, the auger removed corn from the bin at a constant rate of <strong>the fraction 4, 650 over 5 = 930</strong> bushels per hour. Assuming the auger continues to remove corn at this rate, after x hours it will have removed 930x bushels of corn. Because the bin contained 24,000 bushels of corn when Hector started using the auger, the equation 24,000 – 930x = 12,840 can be used to find the number of hours, x, Hector will have been using the auger when 12,840 bushels of corn remain in the bin. Subtracting 12,840 from both sides of this equation and adding 930x to both sides of the equation yields 11,160 = 930x. Dividing both sides of this equation by 930 yields x = 12. Therefore, Hector will have been using the auger for 12 hours when 12,840 bushels of corn remain in the storage bin.<br> Choice A is incorrect. Three hours after Hector began using the auger, 24,000 – 3(930) = 21,210 bushels of corn remained, not 12,840. Choice B is incorrect. Seven hours after Hector began using the auger, 24,000 – 7(930) = 17,490 bushels of corn will remain, not 12,840. Choice C is incorrect. Eight hours after Hector began using the auger, 24,000 – 8(930) = 16,560 bushels of corn will remain, not 12,840.",
    hasFigure: false,
  },
  {
    id: "3f8a701b",
    type: "mcq",
    questionHtml:
      "The equation <strong>9 x + 5 = a · (x + b, )</strong>, where a and b are constants, has no solutions. Which of the following must be true?<br> I. <strong>a = 9</strong><br> <br><br> II. <strong>b = 5</strong><br> <br><br> III. <strong>b ≠ five ninths</strong>",
    choices: [
      { label: "A", text: "None" },
      { label: "B", text: "I only" },
      { label: "C", text: "I and II only" },
      { label: "D", text: "I and III only" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. For a linear equation in a form <strong>a, x + b = c x + d</strong> to have no solutions, the x-terms must have equal coefficients and the remaining terms must not be equal. Expanding the right-hand side of the given equation yields <strong>9 x + 5 = a, x + a, b</strong>. Inspecting the x-terms, 9 must equal a, so statement I must be true. Inspecting the remaining terms, 5 can’t equal <strong>9 b</strong>. Dividing both of these quantities by 9 yields that b can’t equal <strong>five ninths</strong>. Therefore, statement III must be true. Since b can have any value other than <strong>five ninths</strong>, statement II may or may not be true.Choice A is incorrect. For the given equation to have no solution, both <strong>a = 9</strong> and <strong>b ≠ five ninths</strong> must be true. Choice B is incorrect because it must also be true that <strong>b ≠ five ninths</strong>. Choice C is incorrect because when <strong>a = 9</strong>, there are many values of b that lead to an equation having no solution. That is, b might be 5, but b isn’t required to be 5.",
    hasFigure: false,
  },
  {
    id: "628300a9",
    type: "mcq",
    questionHtml:
      "A science teacher is preparing the 5 stations of a science laboratory. Each station will have either Experiment A materials or Experiment B materials, but not both. Experiment A requires 6 teaspoons of salt, and Experiment B requires 4 teaspoons of salt. If x is the number of stations that will be set up for Experiment A and the remaining stations will be set up for Experiment B, which of the following expressions represents the total number of teaspoons of salt required?",
    choices: [
      { label: "A", text: "<strong>5 x</strong>" },
      { label: "B", text: "<strong>10 x</strong>" },
      { label: "C", text: "<strong>2 x + 20</strong>" },
      { label: "D", text: "<strong>10 x + 20</strong>" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. It is given that x represents the number of stations that will be set up for Experiment A and that there will be 5 stations total, so it follows that 5 – x is the number of stations that will be set up for Experiment B. It is also given that Experiment A requires 6 teaspoons of salt and that Experiment B requires 4 teaspoons of salt, so the total number of teaspoons of salt required is 6x + 4(5 – x), which simplifies to 2x + 20.Choices A, B, and D are incorrect and may be the result of not understanding the description of the context.",
    hasFigure: false,
  },
  {
    id: "771bd0ca",
    type: "spr",
    questionHtml:
      "<strong>5 (t + 3) − 7 (t + 3) = 38</strong><br>What value of <strong>t</strong> is the solution to the given equation?",
    choices: [],
    correctAnswer: "-22",
    explanation:
      "The correct answer is <strong>−22</strong>. The given equation can be rewritten as <strong>− 2 (t + 3) = 38</strong>. Dividing both sides of this equation by <strong>−2</strong> yields <strong>t + 3 = −19</strong>. Subtracting <strong>3</strong> from both sides of this equation yields <strong>t = −22</strong>. Therefore, <strong>−22</strong> is the value of <strong>t</strong> that is the solution to the given equation.",
    hasFigure: false,
  },
  {
    id: "7d5d1b32",
    type: "spr",
    questionHtml:
      "<strong>2 (k x − n) = −(28) / (15) x − (36) / (19)</strong><br>In the given equation, <strong>k</strong> and <strong>n</strong> are constants and <strong>n > 1</strong>. The equation has no solution. What is the value of <strong>k</strong>?",
    choices: [],
    correctAnswer: "-.9333, -14/15",
    explanation:
      "The correct answer is <strong>−(14) / (15)</strong>. A linear equation in the form <strong>a x + b = c x + d</strong> has no solution only when the coefficients of <strong>x</strong> on each side of the equation are equal and the constant terms are not equal. Dividing both sides of the given equation by <strong>2</strong> yields <strong>k x − n = − (28) / (30) x − (36) / (38)</strong>, or <strong>k x − n = − (14) / (15) x − (18) / (19)</strong>. Since it’s given that the equation has no solution, the coefficient of <strong>x</strong> on both sides of this equation must be equal, and the constant terms on both sides of this equation must not be equal. Since <strong>(18) / (19) < 1</strong>, and it's given that <strong>n > 1</strong>, the second condition is true. Thus, <strong>k</strong> must be equal to <strong>−(14) / (15)</strong>. Note that -14/15, -.9333, and -0.933 are examples of ways to enter a correct answer.",
    hasFigure: false,
  },
  {
    id: "90095507",
    type: "mcq",
    questionHtml:
      "The Townsend Realty Group invested in the five different properties listed in the table above. The table shows the amount, in dollars, the company paid for each property and the corresponding monthly rental price, in dollars, the company charges for the property at each of the five locations. Townsend Realty purchased the Glenview Street property and received a 40% discount off the original price along with an additional 20% off the discounted price for purchasing the property in cash. Which of the following best approximates the original price, in dollars, of the Glenview Street property?",
    choices: [
      { label: "A", text: "$350,000" },
      { label: "B", text: "$291,700" },
      { label: "C", text: "$233,300" },
      { label: "D", text: "$175,000" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. Let x be the original price, in dollars, of the Glenview Street property. After the 40% discount, the price of the property became <strong>0 . 6 x</strong> dollars, and after the additional 20% off the discounted price, the price of the property became <strong>0 . 8 · 0 . 6 x</strong>. Thus, in terms of the original price of the property, x, the purchase price of the property is <strong>0 . 4 8 x</strong> . It follows that <strong>0 . 4 8 x = 140, 000</strong>. Solving this equation for x gives <strong>x = 291, 666 . 6, with the 6 repeating after the decimal .</strong>. Therefore, of the given choices, $291,700 best approximates the original price of the Glenview Street property.Choice A is incorrect because it is the result of dividing the purchase price of the property by 0.4, as though the purchase price were 40% of the original price. Choice C is incorrect because it is the closest to dividing the purchase price of the property by 0.6, as though the purchase price were 60% of the original price. Choice D is incorrect because it is the result of dividing the purchase price of the property by 0.8, as though the purchase price were 80% of the original price.",
    hasFigure: false,
  },
  {
    id: "ae2287e2",
    type: "mcq",
    questionHtml:
      "A certain product costs a company $65 to make. The product is sold by a salesperson who earns a commission that is equal to 20% of the sales price of the product. The profit the company makes for each unit is equal to the sales price minus the combined cost of making the product and the commission. If the sales price of the product is $100, which of the following equations gives the number of units, u, of the product the company sold to make a profit of $6,840 ?",
    choices: [
      { label: "A", text: "[]" },
      { label: "B", text: "[]" },
      { label: "C", text: "<strong>0 . 8 · 100 − 65 u = 6, 840</strong>" },
      { label: "D", text: "[]" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. The sales price of one unit of the product is given as $100. Because the salesperson is awarded a commission equal to 20% of the sales price, the expression 100(1 – 0.2) gives the sales price of one unit after the commission is deducted. It is also given that the profit is equal to the sales price minus the combined cost of making the product, or $65, and the commission: 100(1 – 0.2) – 65. Multiplying this expression by u gives the profit of u units: (100(1 – 0.2) – 65)u. Finally, it is given that the profit for u units is $6,840; therefore (100(1 – 0.2) – 65)u = $6,840.Choice B is incorrect. In this equation, cost is subtracted before commission and the equation gives the commission, not what the company retains after commission. Choice C is incorrect because the number of units is multiplied only by the cost but not by the sale price. Choice D is incorrect because the value 0.2 shows the commission, not what the company retains after commission.",
    hasFigure: false,
  },
  {
    id: "aee9fd2d",
    type: "mcq",
    questionHtml:
      "If <strong>(x + 6) / (3) = (x + 6) / (13)</strong>, the value of <strong>x + 6</strong> is between which of the following pairs of values?",
    choices: [
      { label: "A", text: "<strong>−7</strong> and <strong>−3</strong>" },
      { label: "B", text: "<strong>−2</strong> and <strong>2</strong>" },
      { label: "C", text: "<strong>2</strong> and <strong>7</strong>" },
      { label: "D", text: "<strong>8</strong> and <strong>13</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. Multiplying both sides of the given equation by <strong>(3) (13)</strong>, or <strong>39</strong>, yields <strong>(39) ((x + 6) / (3)) = (39) ((x + 6) / (13))</strong>, or <strong>13 (x + 6) = 3 (x + 6)</strong>. Subtracting <strong>3 (x + 6)</strong> from both sides of this equation yields <strong>10 (x + 6) = 0</strong>. Dividing both sides of this equation by <strong>10</strong> yields <strong>x + 6 = 0</strong>. Therefore, if <strong>(x + 6) / (3) = (x + 6) / (13)</strong>, then the value of <strong>x + 6</strong> is <strong>0</strong>. It follows that of the given choices, the value of <strong>x + 6</strong> is between <strong>−2</strong> and <strong>2</strong>.<br>Choice A is incorrect and may result from conceptual or calculation errors.<br>Choice C is incorrect and may result from conceptual or calculation errors.<br>Choice D is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "b7e6394d",
    type: "mcq",
    questionHtml:
      "Alan drives an average of 100 miles each week. His car can travel an average of 25 miles per gallon of gasoline. Alan would like to reduce his weekly expenditure on gasoline by $5. Assuming gasoline costs $4 per gallon, which equation can Alan use to determine how many fewer average miles, m, he should drive each week?",
    choices: [
      {
        label: "A",
        text: "<strong>the fraction 25 over 4, end fraction · m = 95</strong>",
      },
      {
        label: "B",
        text: "<strong>the fraction 25 over 4, end fraction · m = 5</strong>",
      },
      {
        label: "C",
        text: "<strong>the fraction 4 over 25, end fraction · m = 95</strong>",
      },
      {
        label: "D",
        text: "<strong>the fraction 4 over 25, end fraction · m = 5</strong>",
      },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. Since gasoline costs $4 per gallon, and since Alan’s car travels an average of 25 miles per gallon, the expression <strong>4 over 25</strong> gives the cost, in dollars per mile, to drive the car. Multiplying <strong>4 over 25</strong> by m gives the cost for Alan to drive m miles in his car. Alan wants to reduce his weekly spending by $5, so setting <strong>4 over 25</strong>m equal to 5 gives the number of miles, m, by which he must reduce his driving.Choices A, B, and C are incorrect. Choices A and B transpose the numerator and the denominator in the fraction. The fraction <strong>25 over 4</strong> would result in the unit miles per dollar, but the question requires a unit of dollars per mile. Choices A and C set the expression equal to 95 instead of 5, a mistake that may result from a misconception that Alan wants to reduce his driving by 5 miles each week; instead, the question says he wants to reduce his weekly expenditure by $5.",
    hasFigure: false,
  },
  {
    id: "e6cb2402",
    type: "spr",
    questionHtml:
      "<strong>3 (k x + 13) = (48) / (17) x + 36</strong><br>In the given equation, <strong>k</strong> is a constant. The equation has no solution. What is the value of <strong>k</strong>?",
    choices: [],
    correctAnswer: ".9411, .9412, 16/17",
    explanation:
      "The correct answer is <strong>(16) / (17)</strong>. It's given that the equation <strong>3 (k x + 13) = (48) / (17) x + 36</strong> has no solution. A linear equation in the form <strong>a x + b = c x + d</strong>, where <strong>a</strong>, <strong>b</strong>, <strong>c</strong>, and <strong>d</strong> are constants, has no solution only when the coefficients of <strong>x</strong> on each side of the equation are equal and the constant terms aren't equal. Dividing both sides of the given equation by <strong>3</strong> yields <strong>k x + 13 = (48) / (51) x + (36) / (3)</strong>, or <strong>k x + 13 = (16) / (17) x + 12</strong>. Since the coefficients of <strong>x</strong> on each side of the equation must be equal, it follows that the value of <strong>k</strong> is <strong>(16) / (17)</strong>. Note that 16/17, .9411, .9412, and 0.941 are examples of ways to enter a correct answer.",
    hasFigure: false,
  },
];
