Free Quantitative Reasoning MCQs with Answers

251. In the figure, tan B = 3/4. If BC = 15 and DA = 4, what is the length of DE?

• A. 6
• B. 8
• C. 9
• D. 10

Explanation: The correct answer is 6. Since tan B = 3/4 , ΔABC and ΔDBE are both 3-4-5 triangles. This means that they are both similar to the right triangle with sides of lengths 3, 4, and 5. Since BC = 15, which is 3 times as long as the hypotenuse of the 3-4-5 triangle, the similarity ratio of ΔABC to the 3-4-5 triangle is 3:1. Therefore, the length of _ AC (the side opposite to B) is 3 × 3 = 9, and the length of AB (the side adjacent to angle B) is 4 × 3 = 12. It is also given that DA = 4. Since AB = DA + DB and AB = 12, it follows that DB = 8, which means that the similarity ratio of ΔDBE to the 3-4-5 triangle is 2:1 (DB is the side adjacent to angle B). Therefore, the length of DE, which is the side opposite to angle B, is 3 × 2 = 6.

252. The graph above shows the distance traveled d, in feet, by a product on a conveyor belt m minutes after the product is placed on the belt. Which of the following equations correctly relates d and m?

• A. d = 2m
• B. d = 1/2m
• C. d = m + 2
• D. d = 2m + 2

Explanation: Choice A is correct. The line passes through the origin. Therefore, this is a relationship of the form d = km, where k is a constant representing the slope of the graph. To find the value of k, choose a point (m, d) on the graph of the line other than the origin and substitute the values of m and d into the equation. For example, if the point (2, 4) is chosen, then 4 = k(2), and k = 2. Therefore, the equation of the line is d = 2m.

253. The formula below is often used by project managers to compute E, the estimated time to complete a job, where O is the shortest completion time, P is the longest completion time, and M is the most likely completion time.E = 0 + 4M + P / 6Which of the following correctly gives P in terms of E, O, and M?

• A. P = 6E - O - 4M
• B. P = -6E + O + 4M
• C. P = O + 4M + E / 6
• D. P = O + 4M - E / 6

Explanation: Choice A is correct. Multiplying both sides of the equation by 6 results in 6E = O + 4M + P. Then, subtracting O + 4M from both sides of 6E = O + 4M + P gives P = 6E − O - 4M.

254. In the figure, RT = TU. What is the value of x?

• A. 72
• B. 66
• C. 64
• D. 58

Explanation: Choice C is correct. To solve for x, first recognize that ∆RTU is an isosceles triangle since RT = TU. The base angles ∠TRU and ∠TUR are therefore congruent. Using the triangle sum theorem, we find that 114° + 2t° = 180°, solving for t gives 33°. ∠TUR is congruent to ∠SUV, so ∠SUV also measures 33°. According to the triangle exterior angle theorem, the measure of an exterior angle (x°) is equal to the sum of the measures of the two non-adjacent interior angles. Therefore, x = 31° (∠VSU) + 33° (∠SUV) = 64°.Choices A, B, and D are incorrect because they either miscalculate the angle measures or fail to apply the exterior angle theorem correctly.

255. The width of a rectangular dance floor is w feet. The length of the floor is 6 feet longer than its width. Which of the following expresses the perimeter, in feet, of the dance floor in terms of w?

• A. 2w + 6
• B. 4w + 12
• C. w2 + 6
• D. w2 + 6w

Explanation: Choice B is correct. It is given that the width of the dance floor is w feet. The length is 6 feet longer than the width; therefore, the length of the dance floor is w + 6. So the perimeter is w + w + (w + 6) + (w + 6) = 4w + 12.

256. y > 2x - 12x > 5Which of the following consists of the y-coordinates of all the points that satisfy the system of inequalities above?

• A. y > 6
• B. y > 4
• C. y > 5/2
• D. y > 3/2

Explanation: Choice B is correct. Subtracting the same number from each side of an inequality gives an equivalent inequality. Hence, subtracting 1 from each side of the inequality 2x > 5 gives 2x − 1 > 4. So the given system of inequalities is equivalent to the system of inequalities y > 2x − 1 and 2x − 1 > 4, which can be rewritten as y > 2x − 1 > 4. Using the transitive property of inequalities, it follows that y > 4.

257. A group of 202 people went on an overnight camping trip, taking 60 tents with them. Some of the tents held 2 people each, and the rest held 4 people each. Assuming all the tents were filled to capacity and every person got to sleep in a tent, exactly how many of the tents were 2-person tents?

• A. 30
• B. 20
• C. 19
• D. 18

Explanation: Choice C is correct. Let x represent the number of 2-person tents and let y represent the number of 4-person tents. It is given that the total number of tents was 60 and the total number of people in the group was 202. This situation can be expressed as a system of two equations, x + y = 60 and 2x + 4y = 202. The first equation can be rewritten as y = −x + 60. Substituting −x + 60 for y in the equation 2x + 4y = 202 yields 2x + 4(−x + 60) = 202. Distributing and combining like terms gives −2x + 240 = 202. Subtracting 240 from both sides of −2x + 240 = 202 and then dividing both sides by −2 gives x = 19. Therefore, the number of 2-person tents is 19.Alternate approach: If each of the 60 tents held 4 people, the total number of people that could be accommodated in tents would be 240. However, the actual number of people who slept in tents was 202. The difference of 38 accounts for the 2-person tents. Since each of these tents holds 2 people fewer than a 4-person tent, 38/2 = 19 gives the number of 2-person tents.

258. In the circle above, point A is the center and the length of arc BC is 2/5 of the circumference of the circle. What is the value of x?

• A. 140
• B. 142
• C. 144
• D. 148

Explanation: To find the value of x, note that the length of arc BC is 2/5 of the circle's circumference. In a circle, the ratio of an arc's length to the circumference is equal to the ratio of the arc's angle to 360°. Thus, we set up the equation 2/5 = x/360. Solving this, we find x = 144. Other options result from incorrect calculations or setup of the proportion.

259. Tracy collects, sells, and trades figurines, and she tracks the number of figurines in her collection on the graph below.On what interval did the number of figurines decrease the fastest?

• A. Between 1 and 2 months
• B. Between 2 and 3 months
• C. Between 3 and 4 months
• D. Between 4 and 5 months

Explanation: The correct answer is Option C: Between 3 and 4 months. The graph shows that the line segment between 3 and 4 months is the steepest among all the intervals, indicating the fastest rate of decrease in the number of figurines. Option A is incorrect because, although there was a decrease, it was not as rapid as between 3 and 4 months. Options B and D are incorrect because both intervals show an increase in the number of figurines, not a decrease.

260. In the figure, lines l and m are parallel, y = 20, and z = 60. What is the value of x?

• A. 120
• B. 100
• C. 90
• D. 80

Explanation: The correct answer is Choice B: 100. The parallel lines and the transversal create corresponding angles, so a = y = 20°. In the triangle, the angles a, x, and z must sum to 180°. Given a = 20° and z = 60°, we solve for x: 20 + x + 60 = 180, resulting in x = 100°. Other options result from common misconceptions about angle relationships in triangles and parallel line properties.