Free Quantitative Reasoning MCQs with Answers

301. The volume of a sphere is given by the formula V = 4/3πr3, where r is the radius of the sphere. Which of the following gives the radius of the sphere in terms of the volume of the sphere?

• A. r = 4π⁄3V
• B. r = 3V⁄4π
• C. r = 3√4π⁄3V
• D. 3√3V/4π

Explanation: To find the radius in terms of the volume, start with the formula for the volume of a sphere: V = 4/3πr3. Rearrange the equation to solve for r3 by dividing both sides by 4/3π, yielding r3 = 3V⁄4π. Finally, take the cube root of both sides to solve for r, giving r = ³√3V⁄4π. Thus, option D is correct. The other options either incorrectly rearrange the formula or misunderstand the mathematical operations needed to isolate r.

302. The maximum value of a data set consisting of 25 positive integers is 84. A new data set consisting of 26 positive integers is created by including 96 in the original data set. Which of the following measures must be 12 greater for the new data set than for the original data set?

• A. The mean
• B. The median
• C. The range
• D. The standard deviation

Explanation: Choice C is correct. Let m be the minimum value of the original data set. The range of a data set is the difference between the maximum value and the minimum value. The range of the original data set is therefore 84 − m. The new data set consists of the original set and the positive integer 96. Thus, the new data set has the same minimum m and a maximum of 96. Therefore, the range of the new data set is 96 − m. The difference in the two ranges can be found by subtracting the ranges: (96 − m) − (84 − m). Using the distributive property, this can be rewritten as 96 − m − 84 + m, which is equal to 12. Therefore, the range of the new data set must be 12 greater than the range of the original data set.

303. The first year Eleanor organized a fund-raising event, she invited 30 people. For each of the next 5 years, she invited double the number of people she had invited the previous year. If f n is the number of people invited to the fund-raiser n years after Eleanor began organizing the event, which of the following statements best describes the function f?

• A. The function f is a decreasing linear function.
• B. The function f is an increasing linear function.
• C. The function f is a decreasing exponential function.
• D. The function f is an increasing exponential function.

Explanation: Choice D is correct. It's given that the number of people Eleanor invited the first year was 30 and that the number of people invited doubles each of the following years, which is the same as increasing by a constant factor of 2. Therefore, the function f can be defined by f (n) = 30(2)n, where n is the number of years after Eleanor began organizing the event. This is an increasing exponential function.

304. Some values of x and their corresponding values of y are shown in the table, where a is a constant. If there is a linear relationship between x and y, which of the following equations represents the relationship?

• A. x + 2y = a
• B. x + 2y = 5a
• C. 2x - y = -5a
• D. 2x - y = 7a

Explanation: Choice A is correct. The slope-intercept form of a linear equation in the xy-plane is y = mx + b, where m is the slope of the graph of the equation and b is the y-coordinate of the y-intercept of the graph. Any two ordered pairs (x1, y1) and (x2, y2) that satisfy a linear equation can be used to compute the slope of the graph of the equation using the formula m = y2 − y1 / x2 − x1. Substituting the two pairs (a, 0) and (3a, −a) from the table into the formula gives m = −a − 0/ 3a − a, which can be rewritten as −a/ 2a, or − 1/2. Substituting this value for m in the slope-intercept form of the equation produces y = -1/2 x + b. Substituting values from the ordered pair (a, 0) in the table into this equation produces 0 = -1/2(a) + b, which simplifies to b = a/2. Substituting this value for b in the slope-intercept form of the equation produces y = −1/2 x + a/2. Rewriting this equation in standard form by adding 1/2x to both sides and then multiplying both sides by 2 gives the equation x + 2y = a.

305. 700, 1200, 1600, 2000, xIf the mean of the five numbers above is 1600, what is the value of x?

• A. 2300
• B. 2400
• C. 2500
• D. 2600

Explanation: The correct answer is 2500. The mean number of the list is found by dividing the sum of all the numbers in the list by the count of numbers in the list. It's given that the mean of the five numbers in this list is 1600; therefore, 700 + 1200 + 1600 + 2000 + x / 5 = 1600. Multiplying both sides of this equation by 5 gives 700 + 1200 + 1600 + 2000 + x = 8000. The left-hand side of this equation can be rewritten as 5500 + x = 8000. Subtracting 5500 from both sides of this equation gives x = 2500.

306. Tilly earns p dollars for every w hours of work. Which expression represents the amount of money, in dollars, Tilly earns for 39w hours of work?

• A. 39p
• B. p/39
• C. p + 39
• D. p - 39

Explanation: Choice A is correct. It's given that Tilly earns p dollars for every w hours of work. This can be represented by the proportion p w . The amount of money, x , Tilly earns for 39w hours of work can be found by setting up the proportion p/w = x/39w. This can be rewritten as 39pw = xw . Dividing both sides by w results in x p =39 .

307. For the right triangle shown, a = 4 and b = 5. Which expression represents the value of c ?

• A. 4 + 5
• B. √(4)(5)
• C. √4 + 5
• D. √42 + 52

Explanation: Choice D is correct. By the Pythagorean theorem, if a right triangle has a hypotenuse with length c and legs with lengths a and b, then c2 = a2 + b2 . In the right triangle shown, the hypotenuse has length c and the legs have lengths a and b. It's given that a = 4 and b =5. Substituting 4 for a and 5 for b in the Pythagorean theorem yields c2 = 42 + 52 . Taking the square root of both sides of this equation yields c =± √42 + 52 . Since the length of a side of a triangle must be positive, the value of c is √42 + 52 .

308. The equation E(t) = 5(1.8)t gives the estimated number of employees at a restaurant, where t is the number of years since the restaurant opened. Which of the following is the best interpretation of the number 5 in this context?

• A. The estimated number of employees when the restaurant opened
• B. The increase in the estimated number of employees each year
• C. The number of years the restaurant has been open
• D. The percent increase in the estimated number of employees each year

Explanation: The correct answer is that the number 5 in the function E(t) = 5(1.8)t represents the estimated number of employees when the restaurant opened. In an exponential function like E(t) = a(b)t, 'a' denotes the initial value at t = 0. Thus, 5 indicates the starting number of employees. The other options are incorrect because they misinterpret the roles of the constants and variables in the exponential function. The base of the exponent, 1.8, determines the growth factor, not the initial or yearly increase in employees.

309. Each year, the value of an investment increases by 0.49% of its value the previous year. Which of the following functions best models how the value of the investment changes over time?

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

Explanation: Choice C is correct. Because the value of the investment increases each year, the function that best models how the value of the investment changes over time is an increasing function. It′s given that each year, the value of the investment increases by 0.49% of its value the previous year. Since the value of the investment changes by a fixed percentage each year, the function that best models how the value of the investment changes over time is an exponential function. Therefore, the function that best models how the value of the investment changes over time is an increasing exponential function.

310. The population of Greenville increased by 7% from 2015 to 2016. If the 2016 population is k times the 2015 population, what is the value of k ?

• A. 0.07
• B. 0.7
• C. 1.07
• D. 1.7

Explanation: The correct answer is Choice C: 1.07. To find the value of k, consider that the population increased by 7%. This means that the 2016 population is the original 2015 population plus 7% of the original. Mathematically, this can be expressed as x + 0.07x = 1.07x, where x is the 2015 population. Therefore, the 2016 population is 1.07 times the 2015 population, making k = 1.07.Choice A (0.07) is incorrect because it represents the percentage increase as a decimal, not the actual multiplier for the population. Choice B (0.7) is a misunderstanding of the percentage increase concept and would imply a significant decrease rather than an increase. Choice D (1.7) incorrectly assumes a 70% increase rather than the actual 7%.