Free Quantitative Reasoning MCQs with Answers

221. The table shown summarizes the number of employees at each of the 17 restaurants in a town. Which of the following could be the median number of employees for the restaurants in this town?

• A. 2
• B. 9
• C. 15
• D. 21

Explanation: Choice D is correct. If a data set contains an odd number of data values, the median is represented by the middle data value in the list when the data values are listed in ascending or descending order. Since the numbers of employees are given as ranges of values rather than specific values, it's only possible to determine the range in which the median falls, rather than the exact median. Since there are 17 restaurants included in the data set and the numbers of employees are listed in ascending order, it follows that the median number of employees will be represented by the ninth restaurant in the list. Since the first 2 restaurants each have 2 to 7 employees, the numbers of employees in the 2 to 7 range would be represented by the first and second restaurants in the list. The next 4 restaurants each have 8 to 13 employees. Therefore, the number of employees in the 8 to 13 range will be represented by the third through sixth restaurants in the list. The next 2 restaurants each have 14 to 19 employees. Therefore, the number of employees in the 14 to 19 range will be represented by the seventh and eighth restaurants in the list. Since the next 7 restaurants each have 20 to 25 employees, the number of employees in the 20 to 25 range will be represented by the ninth through fifteenth restaurants in the list. This means that the ninth restaurant in the list, which has the median number of employees for the restaurants in this town, has several employees in the 20 to 25 range. Of the given choices, the only number of employees in the 20 to 25 range is 21.

222. The right triangle ABC is shown. What is the value of tan A?

• A. √3/54
• B. 1/√3
• C. √3
• D. 27√3

Explanation: Choice C is correct. In the triangle shown, the measure of angle B is 30° and angle C is a right angle, which means that it has a measure of 90°. Since the sum of the angles in a triangle is equal to 180°, the measure of angle A is equal to 180° - (30 + 90)o, or 60o. In a right triangle whose acute angles have measures 30° and 60°, the lengths of the legs can be represented by the expressions x, x √3, and 2x, where x is the length of the leg opposite the angle with measure 30°, x √3 is the length of the leg opposite the angle with measure 60°, and 2x is the length of the hypotenuse. In the triangle shown, the hypotenuse has a length of 54. It follows that 2x = 54, or x = 27. Therefore, the length of the leg opposite angle B is 27, and the length of the leg opposite angle A is 27 √3. The tangent of an acute angle in a right triangle is defined as the ratio of the length of the leg opposite the angle to the length of the leg adjacent to the angle. The length of the leg opposite angle A is 27 √3 and the length of the leg adjacent to angle A is 27. Therefore, the value of tan A is (27√3/27), or √3.

223. At the time that an article was first featured on the home page of a news website, there were 40 comments on the article. An exponential model estimates that at the end of each hour after the article was first featured on the home page, the number of comments on the article had increased by 190% of the number of comments on the article at the end of the previous hour. Which of the following equations best represents this model, where C is the estimated number of comments on the article t hours after the article was first featured on the home page and t ≤ 4?

• A. C = 40(1.19)t
• B. C = 40(1.9)t
• C. C = 40(19)t
• D. C = 40(2.9)t

Explanation: Choice D is correct. It's given that an exponential model estimates that the number of comments on an article increases by a fixed percentage at the end of each hour. Therefore, the model can be represented by an exponential equation of the form C = Kat, where C is the estimated number of comments on the article t hours after the article was first featured on the home page and K and a are constants. It's also given that when the article was first featured on the home page of the news website, there were 40 comments on the article. This means that when t = 0, C=40. Substituting 0 for t and 40 for C in the equation C = Kat yields 40 = Kao, or 40=K. It's also given that the number of comments on the article at the end of an hour had increased by 190% from the number of comments on the article at the end of the previous hour. Multiplying the percent increase by the number of comments on the article at the end of the previous hour yields the number of estimated additional comments the article has on its home page: (40)(190/100), or 76 comments. Thus, the estimated number of comments for the following hour is the sum of the comments from the end of the previous hour and the number of additional comments, which is 40+76, or 116. This means that when t=1, C=116. Substituting 1 for t, 116 for C, and 40 for K in equation C = Kat yields 116 = 40a1, or 116 = 40a. Dividing both sides of this equation by 40 yields 2.9 = a. Substituting 40 for K and 2 9. for a in the equation C = Kat yields C = 40(2.9)t. Thus, the equation that best represents this model is C = 40(2.9)t.

224. In right triangle ABC, angle C is the right angle, and BC = 162. Point D on side AB is connected by a line segment with point E on side AC such that line segment DE is parallel to side BC and CE = 2AE. What is the length of line segment DE?

• A. 50
• B. 48
• C. 54
• D. 60

Explanation: The correct answer is 54. To solve this problem, recognize that triangle ADE is similar to triangle ABC because DE is parallel to BC, making corresponding angles congruent. Given CE = 2AE, we can express AC in terms of AE: AC = AE + CE = 3AE. Since DE/BC = AE/AC, substituting the known values gives DE/162 = AE/3AE = 1/3. Solving this proportion by multiplying both sides by 162 gives DE = 54. Other options result from misapplying the proportional relationships or incorrect calculations of the side lengths.

225. The graph of function f is shown, where y = f(x).Which of the following describes function f?

• A. Increasing linear
• B. Decreasing linear
• C. Increasing exponential
• D. Decreasing exponential

Explanation: Choice A is correct. The graph of function f shows that as x increases, f(x) also increases, which means f(x) is an increasing function. The graph of f is a line, which indicates a constant rate of change. A function that has a constant rate of change is a linear function. Therefore, function f can be described as increasing linearly.

226. The graph of the function f is shown, where y = f(x) What is the y-intercept of the graph?

• A. (0, −1)
• B. (0, −4)
• C. (0, 1)
• D. (0, 4)

Explanation: Choice B is correct. The y-intercept of a graph is the point where the graph intersects the y-axis. This occurs when the x-coordinate is 0. In the given graph, the function f intersects the y-axis at the point (0, −4), which means the y-intercept is (0, −4). Choices A, C, and D are incorrect as they either misidentify the intersection point or confuse positive and negative y-values.

227. The function f defined by f (t) = 14t + 9 gives the estimated length, in inches, of a vine plant t months after Tavon purchased it. Which of the following is the best interpretation of 9 in this context?

• A. Tavon will keep the vine plant for 9 months.
• B. The vine plant is expected to grow 9 inches each month.
• C. The vine plant is expected to grow to a maximum length of 9 inches.
• D. The estimated length of the vine plant was 9 inches when Tavon purchased it.

Explanation: Choice D is correct. It's given that the function f defined by f (t) = 14t + 9 gives the estimated length, in inches, of a vine plant t months after Tavon purchased it. For a function defined by an equation of the form f (t) = mt + b, , where m and b are constants, b represents the value of f (0), or the value of f (t) when the value of t is 0. Therefore, for the function defined by f (t) = 14t + 9, 9 represents the value of f (t) when the value of t is 0. This means that 0 months after the vine plant was purchased, the estimated length of the vine plant was 9 inches. Therefore, the best interpretation of 9 in this context is the estimated length of the vine plant was 9 inches when Tavon purchased it.

228. A rectangle has an area of 155 square inches. The length of the rectangle is 4 inches less than 7 times the width of the rectangle. What is the width of the rectangle, in inches?

• A. 5
• B. 8
• C. 10
• D. 15

Explanation: The correct answer is 5. Let x represent the width of the rectangle. The length is 7x - 4. The area is calculated as x(7x - 4) = 155. Solving this equation gives x = 5 as the width. Option B, C, and D are incorrect as they result in areas that do not match the given area of 155 square inches.

229. 4, 10, 18, 4, 4, 5, 6, 5 What is the median of the data set shown?

• A. 4
• B. 5
• C. 7
• D. 14

Explanation: Choice B is correct. If a data set contains an even number of data values, when the data values are listed in ascending or descending order, the median is between the two middle values. The given data set contains 8 values. When listed in ascending order, the data set is 4, 4, 4, 5, 5, 6, 10, 18 and the two middle values are 5 and 5. Since the two middle values are the same, the median must be 5.

230. A right circular cylinder has a volume of 432 cubic centimeters. The area of the base of the cylinder is 24 square centimeters. What is the height, in centimeters, of the cylinder?

• A. 18
• B. 24
• C. 216
• D. 10,368

Explanation: Choice A is correct. The volume V of a right circular cylinder is calculated using the formula V = πr^2h, where πr^2 is the area of the base of the cylinder and h is the height. Given a volume of 432 cubic centimeters and a base area of 24 square centimeters, substituting these values into the formula yields 432 = 24h. Solving for h, divide both sides by 24 to find h = 18 centimeters. Therefore, the height of the cylinder is 18 centimeters.Choices B, C, and D are incorrect because they result from misunderstandings or miscalculations of either the formula or the given values. Choice B confuses the area of the base with the height. Choices C and D involve incorrect assumptions about the base area, leading to significantly wrong calculations.