Free Quantitative Reasoning MCQs with Answers

231. The table shows the linear relationship between the number of cars, c, on a commuter train and the maximum number of passengers and crew, p, that the train can carry. Which equation represents the linear relationship between c and p?

• A. 55c − p = −9
• B. 55c − p = 9
• C. 55p - c = -9
• D. 55p - c = 9

Explanation: Choice A is correct. It's given that there is a linear relationship between the number of cars, c, on a commuter train and the maximum number of passengers and crew, p, that the train can carry. It follows that this relationship can be represented by an equation of the form p = mc + b, where m is the rate of change of p in this relationship and b is a constant. The rate of change of p in this relationship can be calculated by dividing the difference in any two values of p by the difference in the corresponding values of c. Using two pairs of values given in the table, the rate of change of p in this relationship is 284 - 174 / 5 - 3, or 55. Substituting 55 for m in the equation p = mc + b yields p = 55c + b. The value of b can be found by substituting any value of c and its corresponding value of p for c and p, respectively, in this equation. Substituting 10 for c and 559 for p yields 559 = 55(10) + b, or 559 = 550 + b. Subtracting 550 from both sides of this equation yields 9=b. Substituting 9 for b in the equation p = 55c + b yields p = 55c + 9. Subtracting 9 from both sides of this equation yields p - 9 = 55c. Subtracting p from both sides of this equation yields -9 = 55c - p, or 55c - p = -9.

232. If 48c = 3√47, what is the value of c?

• A. 7/20
• B. 7/10
• C. 7/15
• D. 7/24

Explanation: The correct answer is 7/24. Begin by expressing the cube root as a fractional exponent: 3√47 becomes 47/3. Equate this to the left side of the equation, 48c. This gives us the equation 8c = 7/3. Solve for c by dividing both sides by 8, resulting in c = 7/24.Other options are incorrect due to errors in converting the root to a fractional exponent or errors in dividing the equation.

233. (x - 2) - 4 (y + 7) = 117(x - 2) + 4 (y + 7) = 442The solution to the given system of equations is (x, y). What is the value of 6(x - 2)?

• A. 1,660
• B. 1,665
• C. 1,677
• D. 1,680

Explanation: To find the value of 6(x - 2), first, notice that adding the two equations together helps eliminate the y terms. The equations are:(x - 2) - 4(y + 7) = 117(x - 2) + 4(y + 7) = 442Adding these equations results in: 2(x - 2) = 559Divide both sides by 2 to find x - 2: x - 2 = 279.5Then multiply by 6 to find 6(x - 2): 6(x - 2) = 6 * 279.5 = 1,677This confirms that the correct answer is 1,677. The other options do not account for the appropriate simplification and multiplication steps.

234. In triangle ABC, angle B is a right angle. The length of side AB is 10√37 and the length of side BC is 24√37. What is the length of the side AC?

• A. 14√37
• B. 26√37
• C. 34√37
• D. √(34.37)

Explanation: Choice B is correct. The Pythagorean theorem states that for a right triangle, c2 = a2 + b2, where c represents the length of the hypotenuse and a and b represent the lengths of the legs. It's given that in triangle ABC, angle B is a right angle. Therefore, triangle ABC is a right triangle, where the hypotenuse is side AC and the legs are sides AB and BC. It's given that the lengths of sides AB and BC are 10√37 and 24√37, respectively. Substituting these values for a and b in the formula c2 = a2 + b2 yields c2 = (10√37)2 + (24√37)2, which is equivalent to c2 = 100(37) + 576(37), or c2 = 676(37). Taking the square root of both sides of this equation yields c = +,- 26√37. Since c represents the length of the hypotenuse, side AC, c must be positive. Therefore, the length of side AC is 26√37.

235. x = y - 3x/2 + 2y = 6Which ordered pair (x, y) satisfies the system of equations shown above?

• A. (-3, 0)
• B. (0, 3)
• C. (6, -3)
• D. (36, -6)

Explanation: Choice B is correct. The first equation can be rewritten as y - x = 3 and the second as x/4 + y = 3, which implies that -x = x/4 and so x = 0. The ordered pair (0, 3) satisfies the first equation and also the second, since 0 + 2(3) = 6 is a true equality.Alternatively, the first equation can be rewritten as y = x + 3. Substituting x + 3 for y in the second equation gives x/2 + 2(x+3) = 6. This can be rewritten using the distributive property as x/2 + 2x + 6 = 6.It follows that 2x + x/2 must be 0. Thus, x = 0. Substituting 0 for x in the equation y = x + 3 gives y = 3. Therefore, the ordered pair (0, 3) is the solution to the system of equations shown.

236. If f(x) = x2 - 6x + 3 / x - 1, what is f (-1)?

• A. -5
• B. -2
• C. 2
• D. 5

Explanation: Choice A is correct. To find f(-1), substitute -1 into the function: f(x) = x2 - 6x + 3 / x - 1. This becomes (-1)2 - 6(-1) + 3 / (-1) - 1. Simplifying step-by-step: the numerator is 1 + 6 + 3 = 10, and the denominator is -2. Therefore, 10 / -2 equals -5. Other choices result from missteps like incorrect sign handling or misapplying the order of operations, leading to incorrect simplification.

237. Marisa needs to hire at least 10 staff members for an upcoming project. The staff members will be made up of junior directors, who will be paid $640 per week, and senior directors, who will be paid $880 per week. Her budget for paying the staff members is no more than $9,700 per week. She must hire at least 3 junior directors and at least 1 senior director. Which of the following systems of inequalities represents the conditions described if x is the number of junior directors and y is the number of senior directors?

• A. 640x + 880y ≥ 9,700x + y ≤ 10x ≥ 3y ≥ 1
• B. 640x + 880y ≤ 9,700x + y ≥ 10x ≥ 3y ≥ 1
• C. 640x + 880y ≥ 9,700x + y ≥ 10x ≤ 3y ≤ 1
• D. 640x + 880y ≤ 9,700x + y ≤ 10x ≤ 3y ≤ 1

Explanation: The correct answer is Option B. It correctly captures all the conditions of the problem. The inequality 640x + 880y ≤ 9,700 ensures the total weekly payment does not exceed the budget. The inequality x + y ≥ 10 ensures Marisa hires at least 10 staff members. The conditions x ≥ 3 and y ≥ 1 ensure Marisa hires at least 3 junior directors and 1 senior director. Options A, C, and D either misrepresent the budget constraint or the minimum hiring requirements.

238. A shipping service restricts the dimensions of the boxes it will ship for a certain type of service. The restriction states that for boxes shaped like rectangular prisms, the sum of the perimeter of the base of the box and the height of the box cannot exceed 130 inches. The perimeter of the base is determined using the width and length of the box. If a box has a height of 60 inches and its length is 2.5 times the width, which inequality shows the allowable width x, in inches, of the box?

• A. 0 < x ≤ 10
• B. 0 < x ≤ 11 (2/3)
• C. 0 < x ≤ 17 (1/2)
• D. 0 < x ≤ 20

Explanation: Choice A is correct because the perimeter of the base is calculated as 7x inches, and the height is 60 inches. By setting up the inequality 7x + 60 ≤ 130 and solving for x, the allowable range for x is 0 < x ≤ 10. This ensures the box dimensions meet the shipping service's restrictions. Choices B, C, and D are incorrect due to miscalculations, misinterpretations, or overlooking key constraints in the problem.

239. The expression 1/3x2 - 2 can be rewritten as 1/3 (x - k)(x + k), where k is a positive constant. What is the value of k?

• A. 2
• B. 6
• C. √2
• D. √6

Explanation: Choice D is correct. Start by factoring the expression: 1/3 (x2 − 6). Recognize that x2 − 6 is a difference of squares, allowing it to be expressed as (x − √6)(x + √6). Therefore, k must be √6. Choice A is incorrect because it leads to a different constant term. Choice B results from an incorrect factorization, and Choice C arises from a misunderstanding of the expression being factored.

240. The number of radians in a 720-degree angle can be written as aπ, where a is a constant. What is the value of a?

• A. 3
• B. 4
• C. 5
• D. 6

Explanation: The correct answer is 4. To convert degrees to radians, use the formula: Radians = Degrees × (π/180). For a 720-degree angle, the calculation is: 720 × (π/180) = 4π. Thus, the value of a is 4. Option A (3) represents 540 degrees, Option C (5) represents 900 degrees, and Option D (6) represents 1080 degrees.