Free Quantitative Reasoning MCQs with Answers

261. Two types of tickets were sold for a concert held at an amphitheater. Tickets to sit on a bench during the concert cost $75 each, and tickets to sit on the lawn during the concert cost $40 each. Organizers of the concert announced that 350 tickets had been sold and that $19,250 had been raised through ticket sales alone. Which of the following systems of equations could be used to find the number of tickets for bench seats, B, and the number of tickets for lawn seats, L, that were sold for the concert?

• A. (75B) + (40L) = 1,950B + L = 350
• B. 40B + 75L = 19,250B + L = 350
• C. 75B + 40L = 350B + L = 19,250
• D. 75B + 40L = 19,250B + L = 350

Explanation: Choice D is correct. Since only two types of tickets were sold and a total of 350 tickets were sold, the sum of the numbers of both types of ticket sold must be 350. Therefore, B + L = 350. Since the bench tickets were $75 each, the income from B bench tickets was 75B. Similarly, since the lawn tickets were $40 each, the income from L lawn tickets sold was 40L. The total income from all tickets was $19,250. So the sum of the income from bench tickets and lawn tickets sold must equal 19,250. Therefore, 75B + 40L = 19,250. Only choice D has both correct equations.

262. The glass pictured can hold a maximum volume of 473 cubic centimeters, which is approximately 16 fluid ounces.What is the value of k, in centimeters?

• A. 2.52
• B. 7.67
• C. 7.79
• D. 10.11

Explanation: The glass can hold a maximum volume of 473 cubic centimeters. By substituting this volume into the formula V = 7πk3/48 and solving for k, we find k ≈ 10.11. This value matches Option D, which is correct. The other options lead to volumes that do not align with the given information, making them incorrect choices.

263. Water pours into the glass slowly and at a constant rate. Which of the following graphs best illustrates the height of the water level in the glass as it fills?

• A. A
• B. B
• C. C
• D. D

Explanation: Option C is the correct choice. The initial rapid increase in water height corresponds to the glass's narrow base, which causes the water level to rise quickly. As the glass widens, the rate of increase in water height slows down, resulting in a graph that is steeper at the start and less steep later. Option A incorrectly suggests a uniform rate of increase, Option B implies an accelerating increase which is not aligned with the glass's shape, and Option D indicates no increase at all, which is not possible as water is being added.

264. Roberto is an insurance agent who sells two types of policies: a $50,000 policy and a $100,000 policy. Last month, his goal was to sell at least 57 insurance policies. While he did not meet his goal, the total value of the policies he sold was over $3,000,000. Which of the following systems of inequalities describes x, the possible number of $50,000 policies, and y, the possible number of $100,000 policies, that Roberto sold last month?

• A. x + y < 5750,000 x + 100,000 y < 3,000,000
• B. x + y > 5750,000 x + 100,000 y > 3,000,000
• C. x + y < 5750,000 x + 100,000 y > 3,000,000
• D. x + y > 5750,000 x + 100,000 y < 3,000,000

Explanation: Choice C is correct. Since Roberto sells only two types of policies and he didn't meet his goal of selling at least 57 policies, the sum of x, the number of $50,000 policies, and y, the number of $100,000 policies, must be less than 57. Symbolically, that is x + y < 57. The total value, in dollars, from selling x number of $50,000 policies is 50,000x. The total value, in dollars, from selling y number of $100,000 policies is 100,000y. Since the total value of the policies he sold was over $3,000,000, it follows that 50,000x + 100,000y > 3,000,000. Only choice C has both correct inequalities.

265. The table shows the results of a research study that investigated the therapeutic value of vitamin C in preventing colds. A random sample of 300 adults received either a vitamin C pill or a sugar pill each day during 2 weeks, and the adults reported whether they contracted a cold during that period. What proportion of adults who received a sugar pill reported contracting a cold?

• A. 11/18
• B. 11/50
• C. 9/50
• D. 11/100

Explanation: Choice B is correct. A total of 150 adults received the sugar pill. Of those, 33 reported contracting a cold. Therefore, 33/150 , or the equivalent of 11/50 , is the proportion of adults receiving a sugar pill who reported contracting a cold.

266. The average annual energy cost for a certain home is $4,334. The homeowner plans to spend $25,000 to install a geothermal heating system. The homeowner estimates that the average annual energy cost will then be $2,712. Which of the following inequalities can be solved to find t, the number of years after installation at which the total amount of energy cost savings will exceed the installation cost?

• A. 25,000 > (4,334 − 2,712)t
• B. 25,000 < (4,334 − 2,712)t
• C. 25,000 − 4,334 > 2,712t
• D. 25,000 > (4,332/2,712)t

Explanation: Choice B is correct. The savings each year from installing the geothermal heating system will be the average annual energy cost for the home before the geothermal heating system installation minus the average annual energy cost after the geothermal heating system installation, which is (4,334 − 2,712) dollars. In t years, the savings will be (4,334 − 2,712)t dollars. Therefore, the inequality that can be solved to find the number of years after installation at which the total amount of energy cost savings will exceed (be greater than) the installation cost, $25,000, is 25,000 < (4,334 − 2,712)t.

267. The 22 students in a health class conducted an experiment in which they each recorded their pulse rates, in beats per minute, before and after completing a light exercise routine. The dot plots below display the results.Let s1 and r1 be the standard deviation and range, respectively, of the data before exercise, and let s2 and r2 be the standard deviation and range, respectively, of the data after exercise. Which of the following is true?

• A. s1 = s2 and r1 = r2
• B. s1 < s2 and r1 < r2
• C. s1 > s2 and r1 > r2
• D. s1 ≠ s2 and r1 = r2

Explanation: Choice D is correct. The two data sets have the same range. The first data set has a range of 88 − 56 = 32, and the second data set has a range of 112 − 80 = 32. Alternatively, it can be seen visually that the ranges are the same because the two dot plots are aligned, the scales of the graphs are the same, and the graphs have the same width. The two data sets have different standard deviations. Both dot plots show distributions that have a mean near the center value of the dot plot. The first dot plot shows most values clustered near the mean, while the second dot plot shows most values farther from the mean. Therefore, the standard deviations of the two data sets are not equal-the data represented by the second dot plot has a greater standard deviation.

268. In triangle RST above, point W (not shown) lies on RT. What is the value of cos(∠RSW ) − sin(∠WST )?

• A. 0
• B. 1
• C. 2
• D. 3

Explanation: The correct answer is 0. Note that no matter where point W is on _ RT, the sum of the measures of ∠RSW and ∠WST is equal to the measure of ∠RST, which is 90°. Thus, ∠RSW and ∠WST are complementary angles. Since the cosine of an angle is equal to the sine of its complementary angle, cos(∠RSW) = sin(∠WST). Therefore, cos(∠RSW) − sin(∠WST) = 0.

269. 2x - y = 8x + 2y = 4For the system of equations above, what is the value of x + y?

• A. -1
• B. 4
• C. 5
• D. 20

Explanation: Choice B is correct. Multiplying both sides of the first equation in the system by 2 yields 4x − 2y = 16. Adding 4x − 2y = 16 to the second equation in the system yields 5x = 20. Dividing both sides of 5x = 20 by 5 yields x = 4. Substituting 4 for x in x + 2y = 4 yields 4 + 2y = 4. Subtracting 4 from both sides of 4 + 2y = 4 yields 2y = 0. Dividing both sides of this equation by 2 yields y = 0. Substituting 4 for x and 0 for y in the expression x + y yields 4 + 0 = 4.

270. The circle has center O, the length of arc ADC is 5π, and x = 100. What is the length of arc ABC?

• A. 9π
• B. 13π
• C. 18π
• D. 13/2π

Explanation: Choice B is correct. To find the length of arc ABC, first determine the measure of the central angle that subtends it. The angle for arc ADC is 100°, so the angle for arc ABC is 360° − 100° = 260°. The ratio of the arc lengths is equal to the ratio of their central angles: s/5π = 260/100. Simplifying 260/100 gives 13/5. Therefore, s/5π = 13/5, leading to s = 13π when both sides are multiplied by 5π.Option A is incorrect because 9π is less than half the circle's circumference. Option C is incorrect because 18π is the total circumference. Option D is incorrect because it represents only half of the arc ABC's length.