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Quantitative comparisons practice This file contains 621 quantitative comparison questions with full answers and explanations! The questions were written and answered by professional test prep teachers and designed to give you the best practice possible. Good luck on your test. http://ThiNganHang.com Broaden your mind, the shortest key to reach your success ThiNganHang.com
1. A: Distance from runner A to Runner B after 9 hours of running at a constant rate B: 45 kilometers AI: Runner A and runner B start running from the same spot. The first runs south at a rate of 3 kilometers per hour and the second runs west at a rate of 4 kilometers per hour. The best answer is C. The runners actually form a triangle, where the hypotenuse is the distance between them and the sides of the triangle are (9 x 3 = 27) and (9 x 4 = 36). The distance between them will be 272 +362 = 45kilometers. 2. A: The distance between car X and car Y after 3 minutes. B: The distance between car X and car Z after 2 minutes. AI: Three cars leave from the same place at different directions. Car X travels north at 1 a rate of 1 Km per minute, car Y travels west at a rate of 1 km per minute and car Z 3 travels south at a rate of 2 Km per minute. The best answer is B. Calculate the distances of A and B. A: The two cars form a triangle where the distance is the hypotenuse. After three minutes car X traveled 3 Km while car Y traveled 4 Km, the hypotenuse will be 5 Km, which is the value of column A. B: Each car is traveling in a different direction and therefore the distance between the cars is the sum of both distances. Car X traveled 2 Km and Car Z traveled 4, the total is 6 Km, which is greater than column A. Lines and angles 3. l AA B C mm A: B B: A + C AI: A, B and C are three angles as drawn. M and l are parallel lines. The best answer is C. The easiest way to see that the columns are equal is to draw another parallel line that crosses the angle B. The upper angle that the line forms is equal to A and the lower angle is equal to C, both complete the angle B. 4545 ThiNganHang.com
4. r x z t A: x + z. B: 135 degrees. AI: In the drawing above, x, z and 45 are angles. r and t are parallel lines. The best answer is C. 45 is a vertical angle to the triangle and therefore the sum x + z + 45 = 180. We can also write the equation: 45 = 180 – x – z è x + z = 180 – 35 = 135. The columns are equal. 5. y k z p x A: z + y B: 360 – (x + y) AI: k and p are two parallel lines. x, z and y are angles. The best answer is C. The line that crosses the two parallel lines is a transversal line and so x + y =180 and x is equal to z (vertical angles). A = z + y = x + y = 180. B = 360 – (x + y) = 360 – (180) = 180. And therefore the answer is C. Triangles 6. A: 7 B: The length of the third side AI: The length of two sides of a triangle is 16 and 9 The best answer is B. Any side of a triangle must be greater than the difference of the other two sides and therefore the third side is greater than the difference (16 – 9 = 7). 7. A: 2 B: The ratio of the diagonal to a side of a rectangle The best answer is D. Only in a square the ratio is 1:1: 2 , since this is a rectangle and not a square the ratio cannot be distinctly determined. ThiNganHang.com
8. z y x A: x + y B: z AI: x, y and z are three angles in a triangle The best answer is D. If x = 10, y = 10 and z = 160, A < B. If x = 60, y = 70 and z = 50, A > B and therefore the answer is D. 9. B C A A: BC B: AB AI: the angle BAC = 44 degrees and the angle CBA = 85 degrees. The best answer is B. The side in front of the larger angle is also greater. IF BAC = 44 and CBA = 85, ACB = 180 – 44 – 85 = 51 degrees and therefore the side which is in front of ACB is greater than that of BAC and the answer is B. 10. A B E D C A: The area of EBC B: The area of AEB AI: ABCD is a rectangle The best answer is C. Calculate the area of each of the triangles. Define X and Y as the length and the width of the rectangle. X XY A = ×Y ×0.5 = . 2 4 X XY B = ×Y ×0.5 = and therefore the answer is C. 2 4 Notice that each of the individual triangles has the same area. ThiNganHang.com
1. A: The perimeter of an octagon, each of whose sides is Z B: The perimeter of a 30-60-90 triangle whose larger leg is 6Z The best answer is B. A = 8Z since the shape is an octagon with eight legs. If the larger leg is 6Z, the smaller one is 3Z and already the perimeter is bigger than 8Z and thus the answer is B. 2. A: The area of a right isosceles triangle whose shorter leg is X B: The area of a square whose perimeter is 4X The best answer is B. A = X2/2 since the triangle is an isosceles and the area can be calculated using the two shorter sides. B = X2 since each side of the square is (4X / 4 = X) and the area is the multiplication of the sides. 3. A: 40 B: The perimeter of a rectangle whose area is120 The best answer is B. The perimeter of a rectangle can be as large as we want, but the smallest perimeter would be if the rectangle would be a square. In that case, each side would be 120 , which is more than 10 and therefore the perimeter is over 40. 4. A: The volume of Mike’s cylinder B: The volume of Tiffany’s cylinder AI: Mike and Tiffany each roll a piece of 25 X 30 cardboard to form a cylinder. Mike attaches the two 25 centimeters sides together and Tiffany attaches the two 30 centimeters sides together. The best answer is A. The volume of a cylinder is π × R2 × H . If Mike attaches the two 25 centimeters sides together than the radius of the cylinder is bigger since 30 = 2×π × R and therefore the radius of Mike’s cylinder is bigger. Since the radius is being powered by two the one with the larger radius wins and therefore A is the answer. 5. ThiNganHang.com
A: The volume of a cube whose edges are 7.5 B: The volume of a box whose edges are 6, 7 and 10. The best answer is A. A = The volume is (7.5)3 = (56.25)2, which is a little over 420. B = The volume is 6 x 7 x 10 = 42 x 10 = 420 and therefore A is greater. 6. A: X B: Y AI: X is the surface area of a cube in inches squared and Y is the volume of the box in inches cubed. The best answer is D. Plug in numbers. Take the side of the cube 1, then X = 6 and V = 1. Take the side of the cube 8, then X = 384 and V = 512 and therefore the answer is not distinct. 7. A: The volume of a cube whose edge is X B: The volume of a sphere whose radius is X The best answer is B. A = X3. 4 3 B = π × X and therefore the answer is B. 3 To those of you, who don’t know the volume of a sphere, try to visualize the two shapes; the cube can easily fit inside the sphere since the diameter of the sphere is 2X and so its volume must be bigger. 8. A: S is the slope of one of the edges of a triangle B: 1 The best answer is D. You don’t know the position of the triangle and therefore the slope of one of his edges can be anything, even 0 or 1. When the answer is not distinct, the answer is D. 9. B(x,y) o A(-4, 4) o A: 0 B: X – Y The best answer is A. ThiNganHang.com
We can see that the line as a 45 degrees angle which means that all the points to the left of it have a greater X-coordinate than a Y-coordinate in absolute value. Take a representative point for X and Y, for example (-8, 6). In this case X – Y = -14, which is smaller than 0 and therefore the answer is A. 10. A B A: The slope of line B B: The slope of line A The best answer is A. We can see that line A is steeper than line B but their slopes are negative. Let’s say that line A has a slope of –2 and line B has a slope of –1/2 and therefore A is the answer. 11. A B A: Y B: Q AI: The coordinates of A are (X, Y) and the coordinates of B are (Q, W). The slope of the line is -¾. The best answer is A. Since A is on the vertical axis, X = 0 and since B is on the horizontal axis, W = 0. 0 −Y Use the slope formula: = −0.75 ⇒ Y = 0.75Q and therefore Q is greater and the Q −0 answer is B. Probability questions 12. A: The probability of getting a head after five consecutive tails B: The probability of getting a tail after four consecutive heads AI: Seven fair coins are flipped The best answer is C. The coins have an attribute of a lack of recollection, in other words, each time you flip the coin the result is independent on the previous ones. Since the coin is a fair coin, the probability of column A and column B are the same. The coin doesn’t “know” that it landed on the same side 5 times. 13. ThiNganHang.com
A: The probability that an even number is drawn out B: The probability that an odd number is drawn out AI: All the numbers from 1 to 103 inclusive are each written on a card and placed in a box. One card is drawn out. The best answer is B. The first number is odd and the last number is odd and therefore there is one more odd number than an even number. Theoretically, the probability of drawing an odd number is greater and therefore the answer is B. 14. A: The probability of randomly pulling a red ball B: The probability of randomly pulling a blue ball AI: In a vase there are 4 red balls, 2 green balls and 5 blue balls. The best answer is B. The probability of pulling a red ball is 4 out of (4 + 2 + 5 = 11), 4/11. The probability of pulling a blue ball is 5 out of 11, which is greater. 15. A: The probability of randomly choosing 2 consecutive roses B: ¼ AI: A vase contains 6 roses, 6 Lilies and 2 violets The best answer is B. The probability of choosing two consecutive roses is the multiplication of the probabilities: (Choosing a rose)(Choosing a rose out of what’s left). A = (6/14)(5/13) = 30/182 = 15/91, which is smaller than ¼ and therefore the answer is B. 16. A: The probability of randomly pulling a short stick and then a long stick B: The probability of randomly pulling a short stick twice AI: In a box there are 10 short sticks and 10 long sticks The best answer is A. The probability of choosing two consecutive short sticks is the multiplication of the probabilities: (Choosing a short stick)(Choosing a short stick out of what’s left). B = (10/20)(9/19) = 9/38. The probability of column A is a little different, after we pulled a short stick, there are 10 long sticks left and so the entire probability is (10/20)(10/19) = 10/38 and therefore column A is greater. 17. A: The probability of getting 11 consecutive heads B: The probability of not getting 11 consecutive heads AI: A fair coin is tossed 11 consecutive times The best answer is C. ThiNganHang.com
Since the coin is fair, the probability of getting a head or a tail in each toss is always ½ and therefore the probability of getting 11 heads is equal to the probability of getting 11 consecutive tails. 18. A: The number of ways to arrange 5 people in 5 seats B: The number of ways to arrange 5 people in 4 seats The best answer is C. Use the counting principle: If there are 5 seats the first seat has 5 possibilities, the second has 4, the third 3 and so on. The combination is the multiple of 5-4-3-2-1. Using 4 seats, the number of combinations is 5-4-3-2, which is equal to the first combination. 19. A: The number of ways you can divide 6 different candies to 4 different people B: The number of ways you can divide 7 different candies to 3 different people The best answer is A. Use the counting principle in this one. Take A for example; there are 4 people, the first can get 6 candies, the second 5 and so on. The number of combinations is the multiplication of 6 x 5 x 4 x 3 = 360 ways. B = 7 x 6 x 5 = 210 ways only and therefore A is greater. 20. A: The probability that 3 people chosen at random were born in the same month B: The probability that 2 people chosen at random were born in the same day of the week The best answer is B. A = (1)(1/12)(1/12) = 1/144. B = (1)(1/7) and therefore the answer is B. 21. A: The probability that 4 people chosen at random are males B: The probability that 3 people chosen at random were born in the same day of the week The best answer is A. A = (1/2)(1/2)(1/2)(1/2) = (1/2)4 = 1/32. B = (1)(1/7)(1/7) = 1/49, which is smaller than column A. 22. A: The probability that a number chosen at random from the primes between 200 and 399 is odd. B: 0.89 The best answer is A. Every prime number (except 2) is odd and therefore the probability in column A is equal to 1, which is greater than B. ThiNganHang.com
Some Algebra 23. A: The product of all the positive divisors of 23 B: The sum of all the positive divisors of 23 The best answer is B. 23 has two divisors, 1 and 23. A = 1 x 23 = 23. B = 1 + 23 = 24. 24. A: The sum of all the positive divisors of 20 B: The product of all the positive divisors of 20 The best answer is B. 20 has the following positive divisors: 1, 2, 4, 5, 10 and 20. Their sum is much smaller than their product and therefore B is the answer. 25. A: 101(X + Y) B: 101X + Y The best answer is D. Subtract 101X from both columns to get 101Y against Y, if Y=0 the columns are equal, otherwise they’re not and therefore the answer cannot be determined. 26. A: The average (arithmetic mean) of all the positive multiples of 3 less than 16 B: The average (arithmetic mean) of all the positive multiples of 4 less than 21 The best answer is B. A = the average of 3, 6, 9, 12, 15 which is the middle number. A = 9. B = the average of 4, 8, 12, 16, 20 which is the middle number. B = 12. 27. A: The number of 5-cent stamps that he used B: The number of 7-cent stamps that he used AI: Ruth put exactly 50 cents worth of postage on an envelope using only 5-cent stamps and 7-cent stamps. The best answer is D. Let X be the number of 5-cent stamps that he used and Y the number of 7-cent stamps that he used. We can write the following equation: 5X + 7Y = 50. There are two combinations: X=3, Y=5 and X=10,Y=0 and therefore the answer cannot be determined. ThiNganHang.com
1. A: The amount Tim paid for the car B: The amount Din paid for the car AI: Tim and Din bought the same new car. Tim got a 10% discount from the dealership and added 7% tax. Din added the 7% tax and only then got a 10% discount. The best answer is C. Lets pick up numbers. Lets say the car costs $1,000. Tim got a 10% discount and so the price is $900 and added 7% tax to $963. Din added the 7% tax to $1,070 and then got a 10% discount to $963 and therefore the answer is C. 2. A: 5,000 B: Y – X AI: X is the sum of integers from 101 to 150 and Y is the sum of integers from 201 to 250 The best answer is C. X = (101 + 102 + 103 + 104 + … + 150). Y = (201 + 202 + 203 + 204 + … + 250). Y – X = (201 – 101) + (202 – 102) + … + (250 – 150) = 100 + 100 + … + 100 = 100 x 50 = 5,000. And therefore the answer is C. 3. R A: 2 B: a AI: The side of the square that is inside the R radius circle is a π . The shaded region is equal to the white one. The best answer is C. The area of the square is (a π )2 = πa2 . The area of the shaded region is πR2 −πa2 . Since the areas are equal we can write the equation: πa2 =πR2 −πa2 ⇒ 2a2 = R2 . R And therefore a = . 2 4. A: X B: Y ThiNganHang.com
AI: X and Y are negative integers. XY = 9 The best answer is D. Since the following combinations fit the terms in the question: X=1, Y=9 or X=3, Y=3, the answer is not distinct and so we cannot determine which is greater. 5. A: 88% of X B: 7 X 8 The best answer is A. 1/8 is (100/8 = 12.5%) and therefore 7/8 is 87.5% of X, which is less than 88%. 6. 4 4 A: The area of the quadrilateral B: 16 The best answer is D. Since we don’t know that the quadrilateral is a rectangle or a square (we know nothing of its angles), the area cannot be found. As far as we’re concerned the quadrilateral can also be a parallelogram and therefore more data is needed in order to calculate the area. 7. A: X 217 B: 33 7 21 AI: X = 33 The best answer is X. The fraction given is too complicated for any calculations. Take ½ for example instead. X7 = ½ è X = (1/2)-7 = 27 = 128. On the other hand, (1/2)7 is 1/128 and therefore the answer is A. 8. A: Q B: W AI: The circumference of a square is Q and the area is W The best answer is D. Define X as the side of the square. Q = 4X and W = X2. ThiNganHang.com
We cannot determine which is greater. If X = ½ , Q is bigger but if X = 8 for example, W is bigger. 9. A: The area of the shaded region B: The area of the white region AI: The white circle is inscribed inside the square The best answer is B. Define X as the side of the circle. X 2 πX 2 The area of the white circle is π × = . 2 4 πX 2 X 2 The area of the shaded region is X2 - ≈ . 4 4 And therefore the answer is B. 10. A: The circumference of the shaded region B: The circumference of the white region AI: The white circle is inscribed inside the square The best answer is A. There is no need to calculate anything. The circumference of the shaded region includes the circumference of the circle and therefore column A is greater. 11. A: The number of prime numbers from 2 to 10 inclusive B: 3 The best answer is A. The prime numbers in the specified range are 2, 3, 5 and 7. 12. A: AB B: 1 AI: A + B = ¾ The best answer is B. There are few possibilities: A or B is negative and the other is positive, one of them is zero or they’re both positive fractions. In either case, AB will be smaller than 1 and so column B is greater. ThiNganHang.com
13. A: A + B B: 35 AI: AB = 35. The best answer is D. From the additional information we can conclude that there are two possible combinations: A=35, B=1 or A=7, B=5. Each case resolves in a different answer and so the answer is D. 14. A: The average (arithmetic mean) of A, B and C B: The average (arithmetic mean) of A, B, C and A AI: C > B > A The best answer is A. Pick up easy numbers for this one. A=1, B=2 and C=3. A = (1+2+3)/3 = 2. B = (1+2+3+1)/4 = 1.75 and therefore column B is smaller. 15. A: -(X + Y) – Z B: -(X – Y) – Z AI: X < Y < 0 < Z The best answer is A. Compare the two columns: -X –Y –Z vs. –X +Y –Z è 0 vs. 2Y and since Y < 0 column B is smaller and the answer is A. 16. A: -(B – A – C) B: A + B + C AI: C > 1, B < 1 and A > 0. The best answer is D. Subtract A + C from both columns to get: Column A = -B and Column B = B. Now since B can be positive or negative the answer cannot be determined because if B is negative column A is greater and otherwise B is greater. 17. 2 A: 1− X B: 2 – 2X AI: X > X2 The best answer is A. ThiNganHang.com
Since X > X2, X is a positive number and so divide each wing by X to get X < 1. Pick a number for X, for example X = ½ and plug it in. A = 4 and B = 1 and therefore column A is greater. 18. A: W – Z B: X – Y AI: X < W, Z < Y The best answer is A. Move variables from one column to the other: A = Y – Z, which is positive and B = X – W, which is negative and therefore column A is greater. 19. A: 0 B: Z AI: X > Y, YZ > XZ The best answer is A. If X > Y then Z must have changed the sign in the inequality YZ > XZ and so Z must be negative. 20. A: X4 B: X5 AI: X × Y < −X The best answer is A. In the extra information, since a positive number is multiplying X and the result is smaller than -X itself, X must be negative. Since X is negative, X4 results in a positive number and therefore column A is greater. 21. A: (5X)3 B: 5X56 AI: 2 < X < 3 The best answer is B. Pick up a number for X, for example X = 2.5. A = 52.5 x 3 = 57.5. B = 52.5+6 = 58.5, which is greater than column A. 22. A: X(Y – X) B: XY AI: X ≠ 0 The best answer is B. Subtract XY from both sides to get: A=-X2, B=0 and since X cannot be zero, A is a negative number, which is smaller than B. ThiNganHang.com
A B 1. D C A: The measurement of the angle DAC B: The measurement of the angle DBC The best answer is B. The angle DBC is an outer angle to the triangle DBA and therefore it’s equal to the sum of the two inner angles, which are not adjacent to it. If DBC = DAC + BDA then DBC > DAC. 2. 32o 8 68o A: The perimeter of the triangle B: 24 The best answer is A. The third angle in the triangle is (180 – 32 – 68 = 80) degrees. The side 8 is in front of the angle 32, which is the smallest one in the triangle and therefore the other sides must be greater than 8 and so their sum must be greater than 24 (8 x 3). 3. P X Y Q R A: X/2 – 90 B: Y AI: The triangle is an isosceles X and Y are outer and internal angles respectively The best answer is C. We need to find a connection between X and Y. Since the triangle is an isosceles its angles are Y, Y and 180 – 2Y. Focus on the upper corner of the triangle, the sum of the angles around it should be 360 degrees and therefore we can write: X + 180 – 2Y = 360 è X – 2Y = 180 è Y = X/2 – 90 and so the columns are even. 4. A: The number of sunflowers in the shop ThiNganHang.com
B: The total number of daffodils and roses in the shop AI: In a flower shop there are three types of flowers: roses, daffodils and sunflowers. The ratio between the number of roses to the number of flowers that are not roses is 1:3. The ratio between the number of daffodils to the number flowers that are not daffodils is 1:2. The best answer is B. From the ratio 1:3 we can conclude that the roses are a quarter of the flowers in the shop. From the ratio 1:2, the daffodils are a third of the flowers in the shop. 1/3 + ¼ = 7/12, which is more than half and therefore column B is greater. 5. A: The height of John B: The height of Kristy AI: The average height in the class didn’t change after John and Walter left and Kristy and Julia came. The best answer is D. From the additional data we can only conclude that the total height of John and Walter is equal to that of Kristy and Julia but we don’t know how it divides among the kids and so the answer is D. 6. A: X5 – X3 B: X4 – X2 AI: X > 1 The best answer is A. Plug in a number for X. For example X = 2. Column A = 25 – 23 = 32 – 8 = 24. Column B = 24 – 22 = 16 – 4 = 12. 7. A: The price of a movie ticket B: The price of a slice of pizza AI: Travis received an allowance. With a fifth of the money he bought a ticket to a movie, and with a quarter of what’s left he bought a slice of pizza. The best answer is C. The easiest way is to plug in numbers. Let’s say that he got 100 dollars. A fifth ($20) was spent on the ticket. A quarter of $80 was spent on the pizza, which is $20 so the columns are equal. 8. A: X B: 1 X X AI: = Y Y ThiNganHang.com
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