Free Quantitative Reasoning MCQs with Answers

321. A small business owner budgets $2,200 to purchase candles. The owner must purchase a minimum of 200 candles to maintain the discounted pricing. If the owner pays $4.90 per candle to purchase small candles and$11.60 per candle to purchase large candles, what is the maximum number of large candles the owner can purchase to stay within the budget and maintain the discounted pricing?

• A. 182
• B. 4.9
• C. 7
• D. 9

Explanation: The correct answer is 182. Let s represent the number of small candles the owner can purchase, and let  represent the number of large candles the owner can purchase. It's given that the owner pays $4.90 per candle to purchase small candles and $11.60 per candle to purchase large candles. Therefore, the owner pays 4.90 s dollars for s small candles and 11.60 dollars for  large candles,which means the owner pays a total of 4.90 s+11.60 dollars to purchase candles. It's given that the owner budgets $2,200 to purchase candles.Therefore, 4.90 11.60 s+ £ 2,200. It's also given that the owner must purchase a minimum of 200 candles. Therefore, s+ ³ 200. The inequalities 4.90 11.60 s+ £ 2,200 and s+ ³ 200 can be combined into one compound inequality by rewriting the second inequality so that its left-hand side is equivalent to the left-hand side of the first inequality. Subtracting  from both sides of the inequality s+ ³ 200 yields s³ - 200 . Multiplying both sides of this inequality by 4.90 yields 4.90 4.90 s³ - (200 ), or 4.90 980 4.90 s³ - . Adding 11.60 to both sides of this inequality yields 4.90 11.60 s+ ³- +  980 4.90  11.60 , or 4.90 11.60 s+ ³+   980 6.70 . This inequality can be combined with the inequality 4.90 11.60 s+ £ 2,200, which yields the compound inequality

322. Triangle FGH is similar to triangle JKL, where angle F corresponds to angle J and angles G and K are right angles. If sin (F) = 308 / 317, what is the value of sin ( J ) ?

• A. 75/317
• B. 308/317
• C. 317/308
• D. 317/75

Explanation: Choice B is correct. If two triangles are similar, then their corresponding angles are congruent. It's given that right triangle FGH is similar to right triangle JKLand angle F corresponds to angle J. It follows that angle F is congruent to angle J and, therefore, the measure of angle F is equal to the measure of angle J. The sine ratios of angles of equal measure are equal. Since the measure of angle F is equal to the measureof angle J, sin (F) = sin (J). It's given that sin (F) = 308/317. Therefore, sin(J) is 308/317.

323. The circle has center O, circumference 144π,and diameters PR and QS. The length of arc PS is twice the length of arc PQ. What is the length of arc QR ?

• A. 24π
• B. 48π
• C. 72π
• D. 96π

Explanation: Choice B is correct. Since PR and QS are diameters of the circle shown, OS, OR, OP, and OQ are radii of the circle and are therefore congruent. Since ∠ SOP and ∠ ROQ are vertical angles, they are congruent. Therefore, arc PS and arc QR are formed by congruent radii and have the same angle measure, so they are congruent arcs. Similarly, ∠ SOR and ∠ POQ are vertical angles, so they are congruent. Therefore, arc SR and arc PQ are formed by congruent radii and have the same angle measure, so they are congruent arcs. Let x represent the length of arc SR. Since arc SR and arc PQ are congruent arcs, the length of arc PQ can also be represented by x . It's given that the length of arc PS is twice the length of arc PQ. Therefore, the length of arc PS can be represented by the expression 2x. Since arc PS and arc QR are congruent arcs, the length of arc QR can also be represented by 2x. This gives the expression x + x +2x +2x .Since it's given that the circumference is 144π , the expression x + x +2x +2x isequal to 144π . Thus xx x x ++ + = 2 2 144π , or 6 144 x = π. Dividing both sides ofthis equation by 6 yields x =24π . Therefore, the length of arc QR is 2 24 π , or 48.

324. A sample of oak has a density of 807 kilograms per cubic meter. The sample is in the shape of a cube, where each edge has a length of 0.90 meters. To the nearest whole number, what is the mass, in kilograms, of this sample?

• A. 588
• B. 726
• C. 897
• D. 1,107

Explanation: To find the mass of the cube, calculate the volume first by cubing the edge length to get 0.729 cubic meters. Then, multiply this volume by the density of 807 kilograms per cubic meter to obtain a mass of 588.303 kilograms. Rounding this to the nearest whole number gives 588 kilograms, making Option A the correct answer. The incorrect options stem from errors such as inaccurate volume calculation, incorrect multiplication, or improper rounding.

325. The table shows the results of a poll. A total of 803 voters selected at random were asked which candidate they would vote for in the upcoming election. According to the poll, if 6,424 people vote in the election, by how many votes would Angel Cruz be expected to win?

• A. 163
• B. 1,304
• C. 3,864
• D. 5,621

Explanation: Choice B is correct. It's given that 483 out of 803 voters responded that they would vote for Angel Cruz. Therefore, the proportion of voters from the poll who responded they would vote for Angel Cruz is 483/803. It's also given that there are a total of 6,424 voters in the election. Therefore, the total number of people who would be expected to vote for Angel Cruz is 6,424(483/803) , or 3,864. Since 3,864 of the 6,424 total voters would be expected to vote for Angel Cruz, it follows that 6,424 - 3,864, or 2,560 voters would be expected not to vote for Angel Cruz. The difference in the number of votes for and against Angel Cruz is 3,864 - 2,560, or 1,304 votes. Therefore, if 6,424 people vote in the election, Angel Cruz would be expected to win by 1,304 votes.

326. A company that provides whale-watching tours takes groups of 21 people at a time. The company'srevenue is 80 dollars per adult and 60 dollars perchild. If the company's revenue for one group consisting of adults and children was 1,440 dollars,how many people in the group were children?

• A. 3
• B. 9
• C. 12
• D. 18

Explanation: To solve this problem, let x be the number of adults and y be the number of children. The total number of people is 21, so x + y = 21. The revenue equation is 80x + 60y = 1440. Substitute x = 21 - y into the revenue equation: 80(21 - y) + 60y = 1440. Simplify to get: 1680 - 80y + 60y = 1440. Combine like terms to get -20y = -240, then divide both sides by -20 to find y = 12. Therefore, there are 12 children.Other options are incorrect as they do not satisfy both the total number of people and the revenue equations simultaneously.

327. One of the factors of 2x3 + 42x 2 + 208x is x + b, where b is a positive constant. What is the smallest possible value of b ?

• A. 2
• B. 4
• C. 6
• D. 8

Explanation: The given expression is 2x3 + 42x2 + 208x. First, factor out the common factor of 2x: this gives 2x(x2 + 21x + 104). Next, factor the quadratic x2 + 21x + 104. To do this, find two numbers that add up to 21 and multiply to 104. These numbers are 8 and 13. Therefore, x2 + 21x + 104 can be factored as (x + 8)(x + 13). Thus, the original expression factors as 2x(x + 8)(x + 13). The factors of the form x + b are x + 8 and x + 13, with the smallest b being 8. Therefore, the correct answer is 8. Options 2, 4, and 6 do not align with the factors obtained from the correct factoring process.

328. y = − 1.5y = x2 + 8x + aIn the given system of equations, a is a positive constant. The system has exactly one distinct real solution. What is the value of a ?

• A. 29/2
• B. 4
• C. 6
• D. 8

Explanation: The correct answer is 29 /2 . According to the first equation in the given system, the value of y is -1.5. Substituting -1.5 for y in the second equation in the given system yields 2 - =++ 1.5 8 x x a. Adding 1.5 to both sides of this equation yields 2 0 8 1.5 = + ++ x x a . If the given system has exactly one distinct real solution, it follows that 2 0 8 1.5 = + ++ x xa has exactly one distinct real solution. A quadratic equation in the form 2 0= ++ px qx r, where p, q, and r are constants, has exactly one distinct real solution if and only if the discriminant, 2 q pr -4 , is equal to 0. The equation 2 0 8 1.5 = + ++ x xa is in this form, where p =1, q =8, andr = a+1.5. Therefore, the discriminant of the equation 2 0 8 1.5 = + ++ x xa is 28 4 1 1.5 - +a , or 58 4 - a. Setting the discriminant equal to 0 to solve for a yields 58 4 0 - =a . Adding 4a to both sides of this equation yields 58= 4 a.Dividing both sides of this equation by 4 yields 584 = a, or 292 = a. Therefore, if the given system of equations has exactly one distinct real solution, the value of a is 29 /2 . Note that 29/2 and 14.5 are examples of ways to enter a correct answer.

329. RS = 20ST = 48TR = 52The side lengths of right triangle RST are given. Triangle RST is similar to triangle UVW, where S corresponds to V and T corresponds to W. What is the value of tan W ?

• A. 5/13
• B. 5/12
• C. 12/13
• D. 12/5

Explanation: Choice B is correct. Given that the triangles RST and UVW are similar, angle T corresponds to angle W. In triangle RST, angle S is the right angle, making TR the hypotenuse. The side RS is opposite to angle T, and ST is adjacent to angle T. The tangent of an angle in a right triangle is the ratio of the length of the opposite side to the length of the adjacent side. Therefore, tan T = RS/ST = 20/48 = 5/12. Since angle T is equal to angle W, it follows that tan W = tan T = 5/12.Choices A and C are incorrect because they represent the sine and cosine of angle W, respectively, which are not relevant for finding the tangent. Choice D is incorrect as it represents the reciprocal of the tangent, which is cotangent.

330. One gallon of paint will cover 220 square feet of a surface. A room has a total wall area of w square feet. Which equation represents the total amount of paint P, in gallons, needed to paint the walls of the room twice?

• A. P = w/110
• B. P = 440w
• C. P = w/220
• D. P = 220w

Explanation: Choice A is correct. The problem specifies that the walls need to be painted twice, covering 2w square feet. By dividing the total area to be painted by the coverage of one gallon (220 square feet), the total gallons needed is calculated as P = 2w/220, which simplifies to P = w/110. Choices B and D are incorrect as they incorrectly multiply the area by the coverage rate instead of dividing. Choice C is inaccurate as it only considers painting the walls once, not twice.