РЕГИОНЫ РОССИЙСКОЙ ФЕДЕРАЦИИ 1 страница
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.
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.
A: B: a AI: The side of the square that is inside the R radius circle is The shaded region is equal to the white one.
The best answer is C. The area of the square is The area of the shaded region is Since the areas are equal we can write the equation: And therefore a =
4. A: X B: Y 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:
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%.
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 B: AI: X7 = 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. We cannot determine which is greater. If X = ½, Q is bigger but if X = 8 for example, W is bigger.
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. The area of the white circle is The area of the shaded region is X2 - 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.
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. A: B: 2 – 2X AI: X > X2
The best answer is A. 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:
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:
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.
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.
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.
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 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 AI: X and Y are positive.
The best answer is D. There are only two numbers that are equal to their own square root- zero and one. Since X and Y are positive, X/Y is equal to 1. X and Y can be both 5 or even 1, the answer cannot be determined.
9. A: Y/X B: 1 AI: X and Y are positive.
The best answer is C. There are only two numbers that are equal to their own square root- zero and one. Since X and Y are positive, X/Y is equal to 1. Y/X is also 1 and therefore the answer is C.
10. A: T B: 9 AI:
The best answer is D. Simplify the expression: T can be 12 or even -347 and therefore the answer cannot be determined.
11. A: 1 B: The best answer is A. Open the parenthesis:
12. A: X – Y B: 0 AI: (X + 3)2 = (Y – 2)2
The best answer is B. You can remove the power from both sides of the equation to get: X + 3 = Y – 2 à X – Y = -5 and so column B is greater.
13. A: The difference between Dorothy and Amber’s grade point average B: 18 AI: The average grade point average of Dorothy and Amber is greater by 10 points from Amber’s grade point average.
The best answer is A. Since no numbers are given, we’ll make them up. Let’s say that the grade point average (G.P.A) of Amber is 5. The average G.P.A of Amber and Dorothy is therefore 15 and the sum of their grades is (15 x 2 = 30). 30 – 5 = 25, which is Dorothy’s G.P.A and the difference in their grades is (25 – 5 =20), which is greater then column B.
14. A: B: AI: K > 1
The best answer is C. Simplify each of the columns: Column A = K x K1.5 = K2.5. Column B = K x K0.5 = K2.5. Both columns are equal to one another.
15. A: A B: B AI: 6B + 24A = 84 A and B are positive.
The best answer is D. There are many options for A and B that will result in 84 and therefore the answer cannot be determined distinctively.
16. A: X B: 60 degrees AI: X, Y and Z are three angles in a triangle.
The best answer is D. Take two sets of numbers: X=20, Y=20 and Z=140 or X=60, Y=60 and Z=60. We can see that in both cases the result is different and so the answer is D.
17. A: Y B: 60 degrees AI: X, Y and Z are three angles in a triangle. Z > Y > X.
The best answer is B. The only case that Y could be 60 degrees is when the triangle is an equilateral. In every other case Y < 60 and so column B is greater.
18. A: The amount of alcohol in a liter of cocktail, which consists of 50% Tequila and 30% Vodka B: The amount of alcohol in a liter of cocktail, which consists of 30% Tequila and 50% Vodka AI: The concentration of alcohol in Tequila is half of the concentration in Vodka. A cocktail consists of Tequila, Vodka and water only.
The best answer is B. In both columns there is 1 liter of cocktail, which consists of (100% - 30% - 50% = 20%) water. Since the amount of alcohol in Vodka is twice than that in Tequila, the cocktail in column B consists of a greater percent of Vodka and therefore it has more alcohol
19. A: The side of the triangle B: The base of the triangle AI: Triangle ABC is an isosceles, which one of its angles is 40 degrees and it’s perimeter is 20 inches.
The best answer is D. We have two options to create such a triangle: 40, 40 and 100 or 70, 70 and 40. In the first case the side is greater than the base and in the second case it’s the opposite and so the answer cannot be determined.
20. A: 320 B: The number of different combinations that Ruth can dress assuming that she wears a shirt, pants and shoes or boots AI: Ruth has 10 pairs of pants, 8 shirts, 3 pairs of shoes and one pair of boots
The best answer is C. The number of combination consists of 10 pants, 8 shirts, 3 shoes + 1 boots. The calculation is simple: 10 x 8 x 4 = 320 combinations.
21. A: The weight of Courtney (In Kg) B: 56 AI: The average weight of Tiffany and Jennifer is 48 Kg. The average weight of Tiffany, Jennifer and Courtney is 50 Kg.
The best answer is B. The easiest way is to try the answer. Say that Courtney weighs 56 Kg (column B), the average weight of Courtney, Tiffany and Jennifer would be (56 + 48)/2 = 52, this conflicts the additional data and therefore Courtney must weigh less then 56 and thus column B is greater.
22. A: The number of candies that Jacob took at the most B: 23 AI: Three friends have 60 cadies together. Michael took more than a third of the candies while Paul took only 17.
The best answer is B. If Michael took more than a third, the situation where Jacob took the most is if Michael took the least amount of candies, thus 21 candies. It turns out that Jacob can take (60 – 21 – 17 = 22) candies, which is smaller than column B.
23. A: The perimeter of the triangle divided by three B: The average (arithmetic mean) of the sides of the triangle AI: The triangle is a right triangle The best answer is C. Take a known group of sides in a right triangle, for example: 3, 4 and 5. Column A = (3 + 4 + 5)/3 = 4. Column B = 4 and therefore the columns are equal.
24. A: The average amount of stamps that Josh and Roy have before giving Sharon any stamps B: The amount of stamps that Sharon has after receiving them from Josh and Roy AI: Josh and Roy have stamps while Sharon has none. They decided to give Sharon half of the amount that Josh has and half of the amount that Roy has.
The best answer is C. Let X be the number of Stamps that Josh had prior to giving Sharon stamps and Y the number of stamps that Roy had prior to giving Sharon stamps. Column A = (X + Y)/2. Column B = X/2 + Y/2, which is equal to the amount in column A.
25. A: A x C x E B: B x D x F AI: A < B < C < D < E < F
The best answer is D. Take two cases: 1) A=1, B=2, C=3, D=4, E=5 and F=6, in this case column A equals 15 while column B equals 48. 2) A=-2, B=-1, C=0, D=1, E=2 and F=3, in this case column A equals 0 while column B equals -3. The answer is dependent on the selection of the parameters and therefore it is not distinct.
26. A: X B: 0 AI: The best answer is D. The additional data can be written as 9X2 = 1 è X2 = 1/9 è X = 1/3 or -1/3 and so 0 is right in the middle so the answer cannot be determined.
27. A: M + N B: MN AI:
The best answer is A. Since it’s hard to find numbers to replace M and N, use your logic. In column A, we take M and increase its value by adding a positive number. In column B, we take M and reduce its value by dividing it using a fraction and therefore column A is greater.
28. A: The probability of pulling a black ball B: 1/5 AI: A jar contains balls in different colors. The number of black balls is five times smaller than the number of the other colored balls.
The best answer is B. Don’t be mistaken, the answer is not B. Let’s say there’re 5 black balls and 25 balls in other colors. The probability of pulling a black ball is 1 out of 30, which is 1/6. 1/5 > 1/6 and so the answer is B.
29. A: The diameter of the circle B: AI: The circumference of a circle is
The best answer is C. The circumference of the circle is
30. A: X B: Y AI:
The best answer is D. Plug in numbers that validated the inequality. For example: (X, Y) = (0, 0) and (2, 3). In the first case X=Y and in the second case X<Y and so the answer cannot be determined.
31. A: B: AI: 0 < X < Y < 1
The best answer is B. Plug in some numbers: X=1/9 and Y=1/4. Column A = 1/3 and column B = ½ and so the answer is B.
32. A: X – Y B: 13 AI: 1 < X < 8 and -4 < Y < 4
The best answer is B. Find the range of numbers that X – Y can get. The minimum is when X is the largest and Y is the smallest: X=8, Y=-4 and so the maximum is 12. The minimum is found in the same way, it’s -3. We can see that (X – Y) cannot be 13 and therefore column B is greater.
33. A: Y B: 10 AI: 40% of Y is equal to 60% of (Y + 20)
The best answer is B. Translate the percentages to an abstract form: This expression can be written as:
34. A: U + V B: U – V AI: 2/3 < U < 1 0 < V < ½
The best answer is A. You can subtract U from both sides, A = V and B = -V. V is positive and therefore column A is greater than column B.
35. A: X5 B: X2 + X3 AI: X > 0
The best answer is D. Take two cases: X = 1 and X = 2. In the first case: column A = 1 and column B = 2. In the second case: column A = 32 and column B = 12. We got two different answers for X’s that are valid and so the answer cannot be determined.
36. A: The probability that the product of the numbers that will be received from three tosses will be even B: The probability that the product of the numbers that will be received from three tosses will be odd AI: A coin is marked with 1 on one side and with 2 on the other side
The best answer is A. There are 23 = 8 combinations. Column B: The only case in which the result will be odd is if all the tosses turn out 1, there is a probability of (1/2)3 = 1/8 that this would happen. Column A: The probability here completes 1/8 to 1 and therefore it’s 7/8. We can see that column A is greater.
37. A: 6 B: 2W AI:
The best answer is B. Simplify the expression: W is a little over 3 and therefore 2W are a little over 6 and so column B is greater.
38. A: The number of cookies that Ethan can bake while Heather bakes 14 cookies B: 20 AI: While Heather bakes 8 cookies, Ethan bakes 12 cookies.
The best answer is A. The ratio between Ethan and Heather is 3:2. And so when Heather bakes 14 cookies, Ethan can bake (14 x 3/2 = 21) cookies and so column A is greater.
39. A: X B: AI: X, Y < 0
The best answer is C. Simplify the expression in column B to: XY/Y = X and so both columns are equal.
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