Free Quantitative Reasoning MCQs with Answers

341. 3 more than 8 times a number x is equal to 83. Which equation represents this situation?

• A. (3)(8)x = 83
• B. 8x = 83 + 3
• C. 3x + 8 = 83
• D. 8x + 3 = 83

Explanation: The correct equation is 8x + 3 = 83. The phrase '8 times a number x' translates to 8x. '3 more than' indicates that 3 is added to this product, forming 8x + 3. The equation states that this expression is equal to 83. Other options misinterpret the order or meaning of the phrases.

342. Hana deposited a fixed amount into her bank account each month. The function f (t) = 100 + 25t gives the amount, in dollars, in Hana's bank account after t monthly deposits. What is the best interpretation of 25 in this context?

• A. With each monthly deposit, the amount in Hana's bank account increased by $25.
• B. Before Hana made any monthly deposits, the amount in her bank account was $25.
• C. After 1 monthly deposit, the amount in Hana's bank account was $25.
• D. Hana made a total of 25 monthly deposits.

Explanation: Choice A is correct. It's given that t represents the number of monthly deposits. In the given function f (t) = 100 + 25t , the coefficient of t is 25. This means that for every increase in the value of t by 1, the value of f increases by 25. It follows that with each monthly deposit, the amount in Hana's bank account increased by $25.

343. Right triangles PQR and STU are similar, where P corresponds to S. If the measure of angle Q is 18°,what is the measure of angle S ?

• A. 18°
• B. 72°
• C. 82°
• D. 162°

Explanation: Choice B is correct. In similar triangles, corresponding angles are congruent. It's given that right triangles PQR and STU are similar, where angle P corresponds to angle S. It follows that angle P is congruent to angle S. In the triangles shown, angle R and angie U are both marked as right angles, so angle R and angle U are corresponding angles. It follows that angle Q and angle T are corresponding angles, and thus, angle Q is congruent to angle T. It's given that the measure of angle Q is 18°, so the measure of angle T is also 18°. Angle U is a right angle, so the measure of angle U is 90°. The sum of the measures of the interior angles of a triangle is 180°. Thus, the sum of the measures of the interior angles of triangle STU is 180 degrees. Let s represent the measure, in degrees, of angle S. It follows that s + 18 + 90 = 180, or s + 108 =180. Subtracting 108 from both sides of this equation yields s = 72. Therefore, if the measure of angle Q is 18 degrees, then the measure of angle S is 72 degrees.

344. 2.5b + 5r = 80The given equation describes the relationship between the number of birds, b, and the number of reptiles, r, that can be cared for at a pet care business on a given day. If the business cares for 16 reptiles on a given day, how many birds can it care for on this day?

• A. 0
• B. 5
• C. 40
• D. 80

Explanation: Choice A is correct. The number of birds can be found by calculating the value of b when r =16 in the given equation. Substituting 16 for r in the given equation yields 2.5b + 5(16) = 80 , or 2.5b + 80 = 80 . Subtracting 80 from both sides of this equation yields 2.5b = 0. Dividing both sides of this equation by 2.5 yields b = 0. Therefore, if the business cares for 16 reptiles on a given day, it can care for 0 birds on this day.

345. What is an equation of the graph shown?

• A. y = -2x - 8
• B. y = x - 8
• C. y = - x - 8
• D. y = 2x - 8

Explanation: Choice C is correct. The equation of a line in slope-intercept form is y = mx + b, where m is the slope and b is the y-intercept. The graph shows a line with a y-intercept at (0, -8), so b = -8. The line also passes through the point (-8, 0). The slope m can be calculated using the points (0, -8) and (-8, 0) as m = (0 - (-8)) / (-8 - 0) = -1. Substituting m = -1 and b = -8 into the equation form gives y = -x - 8. Therefore, the equation of the line shown is y = -x - 8.Option A is incorrect as it has a slope of -2. Option B is incorrect due to its slope of 1, and Option D is incorrect with a slope of 2.

346. For a certain rectangular region, the ratio of its length to its width is 35 to 10. If the width of the rectangular region increases by 7 units, how must the length change to maintain this ratio?

• A. It must increase by 24.5 units.
• B. It must decrease by 24.5 units.
• C. It must decrease by 7 units.
• D. It must increase by 7 units.

Explanation: The correct answer is Option B. The length must decrease by 24.5 units when the width of the rectangular region increases by 7 units to maintain the ratio of 35 to 10. This can be calculated by setting up a proportion and solving for the change in length. The other options are incorrect because they do not account for the necessary change in length to maintain the given ratio.

347. Square P has a side length of x inches. Square Q has a perimeter that is 176 inches greater than the perimeter of square P. The function f gives the area of square Q, in square inches. Which of the following defines f?

• A. f(x) = (x + 44)2
• B. f(x) = (x + 176)2
• C. f(x) = (176x + 44)2
• D. f(x) = (176x + 176)2

Explanation: Option A is correct because it accurately calculates the side length of square Q based on the given information and correctly determines the area of square Q. The other options are incorrect as they either miscalculate the side length of square Q or misinterpret the effect of the increased perimeter on the square's dimensions.

348. Point O is the center of a circle. The measure of arc RS on this circle is 100°. What is the measure, in degrees, of its associated angle ROS ?

• A. 100
• B. 60
• C. 40
• D. 10

Explanation: The correct answer is 100. It's given that point O is the center of a circle and the measure of arc RS on the circle is o 100 . It follows that points R and S lie on the circle. Therefore, OR and OS are radii of the circle. A central angle is an angle n formed by two radii of a circle, with its vertex at the center of the circle. Therefore,∠ ROS is a central angle. Because the degree measure of an arc is equal to the measure of its associated central angle, it follows that the measure, in degrees, of ∠ ROS is 100.

349. The expression 65 √35x45. 8√28x is equivalent to axb, where a and b are positive constants and x > 1. What is the value of a + b?

• A. 361/8
• B. 60
• C. 40
• D. 10

Explanation: The given expression is 65√35x45. 8√28x. To simplify, apply the properties of exponents and roots:Recognize that √35 is the same as (35)^(1/2) = 35/2.The expression becomes 65 * 35/2 * 45 * 28/2 * x1. Simplify further:Combine the exponents for common bases, and simplify the expression using properties of exponents: (6 * 4)5 * 35/2 * 24 * x.Calculate the coefficients: (24)5 = 65 * 45.Combine all terms to get a = 361/8 and b = 1. Thus, a + b = 361/8 + 1 = 369/8.Hence, the answer is 361/8. Other options result from incorrect simplification steps or arithmetic errors.

350. In the xy-plane, a parabola has vertex (9, −14) and intersects the x-axis at two points. If the equation ofthe parabola is written in the formy = ax2 + bx + c, where a, b, and c are constants, which of the following could be the value of a + b + c ?

• A. -23
• B. -19
• C. -14
• D. -12

Explanation: Choice D is correct. The equation of a parabola in the xy-plane can be written in the form y=a (x - h)2 + k, where a is a constant and (h. k) is the vertex of the parabola. If a is positive, the parabola will open upward, and if a is negative, the parabola will open downward. It's given that the parabola has vertex (9 - 14). Substituting 9 for h and -14 for k in the equation y = a(x - h)2 + k gives y = a (x - 9)2 - 14, which can be rewritten as y= (ax - 9) (x - 9) - 14, or y = a(x² - 18x + 81) - 14. Distributing the factor of a on the right-hand side of this equation yields y = ax - 18ax + 81a - 14. Therefore, the equation of the parabola, y = ax - 18ax + 81a - 14, can be written in the form y = ax+bx+c, where a = a, b= -18a, and c = 81a - 14. Substituting -18a for b and 81a - 14 for c in the expression a+b+c yields (a)+(-18a)+(81a - 14), or 64a - 14. Since the vertex of the parabola, (9, -14), is below the x-axis, and it's given that the parabola intersects the x-axis at two points, the parabola must open upward. Therefore, the constant a must have a positive value. Setting the expression. 64a - 14 equal to the value in choice D yields 64a - 14 = -12. Adding 14 to both sides of this equation yields 64a = 2. Dividing both sides of this equation by 64 yields a = 2/64 which is a positive value. Therefore, if the equation of the parabola is written in the form y = ax + bx + c, where a, b, and c are constants, the value of a + b + c could be -12.