Free Quantitative Reasoning MCQs with Answers

461. Circle A has a radius of 3n and circle B has a radius of 129n, where n is a positive constant. The area of circle B is how many times the area of circle A?

• A. 43
• B. 86
• C. 129
• D. 1,849

Explanation: Choice D is correct. The area of a circle can be found by using the formula A = πr2 , where A is the area and r is the radius of the circle. It's given that the radius of circle A is 3n. Substituting this value for r into the formula A = πr2 gives A =π(3n2) , or 9πn2. It's also given that the radius of circle B is 129n. Substituting this value for r into the formula A = πr2 gives A = π(129n)2 or 16,641πn2. Dividing the area of circle B by the area of circle A gives 16,641πn2 / 9πn2, which simplifies to 1,849. Therefore, the area of circle B is 1,849 times the area of circle A.

462. The measure of angle R is 2π / 3 radians. The measure of angle T is 5π / 12 radians greater than the measure ofangle R. What is the measure of angle T, in degrees?

• A. 75
• B. 120
• C. 195
• D. 390

Explanation: Choice C is correct. It's given that the measure of angle R is 2π3 radians, and the measure of angle T is π5 / 12 radians greater than the measure of angle R. Therefore, the measure of angle T is equal to 2 5 π π3 12+ radians. Multiplying π23 by 44 to get a common denominator with 5π12 yields π812. Therefore, π π2 53 12+ is equivalent to π π8 512 12+ , or 1312π . Therefore, the measure of angle T is π1312 radians. The measure of angle T , in degrees, can be found by multiplying its measure, in radians, by π180. This yields π π13 18012´, which is equivalent to 195 degrees. Therefore, the measure of angle T is 195 degrees.

463. The perimeter of an equilateral triangle is 624 centimeters. The height of this triangle is k 3 centimeters, where k is a constant. What is the value of k?

• A. 40
• B. 100
• C. 104
• D. 204

Explanation: The correct answer is 104. An equilateral triangle is a triangle in which all three sides have the same length and all three angles have a measure of 60°. The height of the triangle, k 3, is the length of the altitude from one vertex. The altitude divides the equilateral triangle into two congruent 30-60-90 right triangles, where the altitude is the side across from the 60o angle in each 30-60-90 right triangle. Since the altitude has a length of k 3, it follows from the properties of 30-60-90 right triangles that the side across from each o30 angle has a length of k and each hypotenuse has a length of 2k. In this case, the hypotenuse of each 30-60-90 right triangle is a side of the equilateral triangle; therefore, each side length of the equilateral triangle is 2k. The perimeter of a triangle is the sum of the lengths of each side. It's given that the perimeter of the equilateral triangle is 624; therefore, 2k + 2k + 2k = 624 or 6k = 624. Dividing both sides of this equation by 6 yields k =104.

464. A researcher surveyed a random sample of students from a large university about how often they see movies. Using the sample data, the researcher estimated that 23% of the students in the population saw a movie at least once per month. The margin of error for this estimation is 4%. Which of the following is the most appropriate conclusion about all students at the university, based on the given estimate and margin of error?

• A. It is unlikely that less than 23% of the students see a movie atleast once per month.
• B. At least 23%, but no more than 25%, of the students see a movie at least once per month.
• C. The researcher is between 19% and 27% sure that most students see a movie at least once per month.
• D. It is plausible that the percentage of students who see a movie at least once per month is between 19% and 27%.

Explanation: Choice D is correct. The margin of error is applied to the sample statistic to create an interval in which the population statistic most likely falls. An estimate of 23% with a margin of error of 4% creates an interval of 23% ± 4%, or between 19% and 27%. Thus, it's plausible that the percentage of students in the population who see a movie at least once a month is between 19% and 27%.

465. A book was on sale for 40% off its original price. If the sale price of the book was $18.00, what was the original price of the book? (Assume there is no sales tax.)

• A. $7.20
• B. $10.80
• C. $30.00
• D. $45.00

Explanation: Choice C is correct. Let x represent the original price of the book. Then, 40% off of x is (1 − 0.40)x, or 0.60x. Since the sale price is $18.00, then 0.60x = 18. Dividing both sides of this equation by 0.60 yields x = 30. Therefore, the original price of the book was $30.

466. 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. 4π/3V
• B. 3V/4π
• C. 3√4π/3V
• D. 3√3V/4π

Explanation: Choice D is correct. Dividing both sides of the equation V = 4/3 πr3 by 4/3π results in 3V/4π = r3. Taking the cube root of both sides produces 3 √3V/4π = r. Therefore, 3 √3V/4π gives the radius of the sphere in terms of the volume of the sphere.

467. In quadrilateral ABCD, AD  BC and CD = 1/2 AB. What is the measure of angle B?

• A. 150°
• B. 135°
• C. 120°
• D. 90°

Explanation: Choice A is correct. A segment can be drawn inside of quadrilateral ABCD from point B to point F (not shown) on segment AD such that segment BF is perpendicular to segment AD. This will create a rectangle FBCD such that FB = CD. This will also create a right triangle ABF such that FB = 1/2 AB. An acute angle in a right triangle measures 30° if and only if the side opposite this angle is half the length of the hypotenuse. (Such a triangle is called a 30°-60°-90° triangle.) Since AB is the hypotenuse of right triangle ABF and FB = 1/2 AB, triangle ABF must be a 30°-60°-90° triangle, and angle ABF must measure 60°. The measure of angle ABC equals the sum of the measures of angles ABF and FBC. Because angle FBC is in rectangle FBCD, it has a measure of 90°. Therefore, the measure of angle ABC, or angle B shown in the original figure, is 60° + 90° = 150°.

468. Lynne has $8.00 to spend on apples and oranges. Apples cost $0.65 each, and oranges cost $0.75 each. If there is no tax on this purchase and she buys 5 apples, what is the maximum number of whole oranges she can buy?

• A. 5
• B. 6
• C. 7
• D. 8

Explanation: The correct answer is 6. It's given that apples cost $0.65 each and oranges cost $0.75 each. If x is the number of apples, the cost of buying x apples is 0.65x dollars. If y is the number of oranges, the cost of buying y oranges is 0.75y dollars. Lynne has $8.00 to spend; therefore, the inequality for the number of apples and oranges Lynne can buy is 0.65x + 0.75y ≤ 8.00. Since Lynne bought 5 apples, x = 5. Substituting 5 for x yields 0.65(5) + 0.75y ≤ 8.00, which can be rewritten as 3.25 + 0.75y ≤ 8.00. Subtracting 3.25 from both sides of the inequality yields 0.75y ≤ 4.75. Dividing both sides of this inequality by 0.75 yields y ≤ 6.33. Therefore, the maximum number of whole oranges Lynne can buy is 6.

469. The peregrine falcon can reach speeds of up to 200 miles per hour while diving to catch prey, making it the fastest animal on the planet when in a dive.What is a peregrine falcon's maximum speed while diving to catch prey, in feet per second? (Round your answer to the nearest whole number. 1 mile = 5280 feet)

• A. 289
• B. 290
• C. 293
• D. 295

Explanation: The correct answer is 293. It's given that a peregrine falcon's maximum speed while diving is 200 miles per hour and that 1 mile = 5280 feet. Therefore, a peregrine falcon's maximum speed while diving is (200 miles/1 hour)(5280 feet/1 mile) = 1,056,000 feet per hour. There are 60 minutes in 1 hour and 60 seconds in each minute, so there are (60)(60) = 3600 seconds in 1 hour. A peregrine falcon's maximum speed while diving is therefore (1,056,000 feet/1 hour)(1 hour/3600 seconds), which is approximately 293.33 feet per second. To the nearest whole number, this is 293 feet per second.

470. The peregrine falcon can reach speeds of up to 200 miles per hour while diving to catch prey, making it the fastest animal on the planet when in a dive.If a peregrine falcon dove at its maximum speed for half a mile to catch prey, how many seconds would the dive take? (Round your answer to the nearest second.)

• A. 6
• B. 7
• C. 8
• D. 9

Explanation: The correct answer is 9. If x is the number of hours it will take the falcon to dive 0.5 miles, then the speed of 200 miles per hour can be used to create the proportion 200 miles/1 hour = 0.5 mile/ x hours.This proportion can be rewritten as x hours = 0.5 mile/ 200 mile hr-1, which gives x = 0.0025. There are 60 minutes in 1 hour and 60 seconds in each minute, so there are (60)(60) = 3600 seconds in one hour. Therefore, 0.0025 hour is equivalent to (0.0025 hour)(3600 seconds/1 hour) = 9 seconds.