РЕГИОНЫ РОССИЙСКОЙ ФЕДЕРАЦИИ 4 страница
1. A: B:
The best answer is Simplify each of the expressions. Column A can be written as: Column B can be written as:
2. A: B:
The best answer is A. Column A is simple, the result of the expression is 2.002. In column B there are two numbers to be divided: 10 and 0.001. The first results in 2 and the second results in 0.0002 and so column B is equal to 2.0002, which is smaller than column A.
3. A: 2.5 x 3.5 x 8.8 x 4.5 B: 5 x 3.5 x 4.4 x 9
The best answer is B.
Simplify column B into: 2.5 x 2 x 3.5 x 4.4 x 4.5
Divide by common items. Column A = 2 and column B = 4 and therefore the answer is B. 4. A: The distance from nest M to nest O B: 10 miles AI: A flock of birds flies north from nest M to nest N at an average speed of 8 miles per hour and then west from nest N to nest O at an average speed of 3 miles per 30 minutes.
The best answer is C. Draw a sketch of the path, you can see that the birds form a right triangle with three corners: M, N and O. The distance from M to O can be calculated using the Pythagoras principle as followed: MO =
5. A: B:
The best answer is C. Remember the expression: (X + Y)(X – Y) = X2 – Y2. Now use this formula in both cases to get: A = B =
6. A: B: 0 AI: n is a positive integer greater than zero
The best answer is D. When n=0, column A equals to zero but in all the other cases it doesn’t and therefore the answer cannot be determined.
7. A: B: 0 AI: X is a negative integer The best answer is B. X can be -1, -2, -3 and smaller. Take X = -1, in this case column A = 3 – 3.5 = -0.5. All other X’s give even a smaller result and therefore column B is greater.
A: The area of the shaded region B: The are of one and a half circles AI: Two circles with the same radius are blocked inside a rectangle.
The best answer is B. The area of the circle is The area of the rectangle is (4R)(2R) = 8R2. The area of the shaded region is approximately 8R2 – 2 x 3.14R2 = 1.7R2 Column A = 1.7R2 and column B = approximately 4.7R2 and therefore the answer is B.
A: The perimeter of the rectangle B: The circumference of the two circles AI: Two circles with the same radius are blocked inside a rectangle.
The best answer is B. The circumference of the two circles is The perimeter of the rectangle is 2(4R) + 2(2R) = 12R and therefore column B is greater.
10. A: 4 + 42 + 43 + 44 B: 45
The best answer is B. Don’t try to solve these columns, instead compare. 45 = 4 x 44 = 44 + 44 + 44 + 44, which is definitely greater than column A. Lets do some questions according to subjects… Algebra
11. A: Y2 B: AI: -1 < Y < 0
The best answer is B. Take a representative number: Y = -1/2. Column A = ¼ and column B = ½ and therefore the answer is B.
12. A: (A + B)2 B: (A – B)2 AI: A and B are different from zero.
The best answer is D. Simplify both columns and subtract common items. A = A2 + 2AB + B2. B = A2 - 2AB + B2. Compare between 2AB to -2AB. There is no room to compare since we don’t know if A and/or B are positive or negative.
13. A: 3X B: 2Y AI: X > Y
The best answer is A. If X > Y than 3X is definitely greater than 2Y and so the answer is A.
14. A: 3Y B: 2X AI: X > Y
The best answer is D. Take two sets of numbers: (X, Y) = (2, 1) and (3, 2). For the first pair, A = 3Y = 3 and B = 2X = 4 è Column B is greater. For the second pair, A = 3Y = 6 and B = 2X = 6 è Both columns are equal and therefore the answer cannot be determined.
15. A: B: AI: 0 > X > Y.
The best answer is B. Column A: The numerator of the fraction is positive and the denominator is negative and therefore the whole expression is negative. Column B: The numerator is negative and so is the denominator and so the whole expression is positive and greater than column A.
16. A: B: AI: 0 > X > Y.
The best answer is B. Both variables are negative, where Y is “more” negative. Since we subtract from Y a larger number, the result will definitely be smaller than that in column A but don’t forget the absolute value, which makes it greater.
17. A: B: AI: 0 > Y > X.
The best answer is D. Take two sets of numbers: (X, Y) = (-2, -1) and (-15, -10). For the first pair: A = 9 and B = 9 and for the second pair: A = 22 and B = 18 and therefore the answer cannot be determined.
18. A: 273 B: 327
The best answer is B. Column A can be written as 33x3 = 39 and thus column B is greater.
19. A: B(B – A) B: A(A – B) AI: 0 > A > B
The best answer is A. Column A: The variable out side the parenthesis is negative and the expression in side the parenthesis is negative and therefore the entire expression is positive. Column B:: The variable out side the parenthesis is negative and the expression in side the parenthesis is positive and therefore the entire expression is negative.
20. A: 16X2 B: 4X AI: 0 < X < ¼
The best answer is B. Input a number for X, for example: X = 1/8. Column A = Column B = ½.
21. A: B: AI: A > 0.
The best answer is B. Column A can be simplified to: Column B can be simplified to: The power of A in column B is greater than that in column A and therefore the answer is B.
22. A: B: AI:
The best answer is D. There isn’t enough data on the unknowns and so we’ll input numbers. Take X=2, Y=1: Column A= 1/3 while column B= 3/2, the answer is B. Take X=2, Y=-3: Column A=1 while column B=5/6, the answer is A. We received two different answers and so the answer is D.
23. A: B: AI: Q < -5
The best answer is B. Input a number for Q, for example Q = -8. A = 4 and B = -9 and therefore column B is bigger.
24. A: X2 B: (X – Y)2 AI: X > 0 > Y
The best answer is B. Column B can be written as X2 – 2XY + Y2. Subtract X2 from both columns to get 0 in column A and Y2 – 2XY in B. Add 2XY to both columns: Now we compare 2XY Vs. Y2. The first is negative and the second is positive and therefore Column B is greatrer.
25. A: A – B B: A + B AI: A < -10 and B < -1.
The best answer is A. A – B? A + B subtract A from both sides. -B? B Now, B is negative and therefore column A will be positive, which is greater.
26. A: B: AI: R > 1.
The best answer is B. Plug in a number for R, for example R = 2. Column A = 2 + 1/3. Column B = 2 + 1 + ½, which is greater than column A.
27. A: W2 – S2 B: S – W AI: S > W > 3.5
The best answer is A. W2 – S2? S – W (W – S)(W + S)? -(W – S) divide both sides by (W – S) (W + S)? -1 Both W and S are positive and therefore column A is greater.
Problems
28. A: A B: 70 percent of B AI: A and B are positive numbers.
The best answer is A. Replace A with 100 (for example) è Y = 130. Column A = 100. Column B = 70% of 130, which is 91 and therefore column A is greater.
29. A: 75% of X B: Y AI: X and Y are positive numbers.
The best answer is B. Replace Y with 100 è X = 125. Column B = 100. Column A = 75% of 125, which is less than 100 and therefore the answer is B.
30. A: 28% B: The percent of guys Mary went out with that didn’t dye their hair AI: 30% of the boys that Mary went out with are bold. 60% of the boys with hair that Mary went out with, dye their hair.
The best answer is C. Input 100 as the number of boys that Mary went out with. 30% are bold, thus 30 boys. The number of not bold boys is (100 – 30 = 70) boys. Out of these 70, 60% dye their hair, thus 42 boys. The rest (28) is the percent of boys that don’t dye their hair and therefore the answer is C.
31. A: The number of different sets that can be made where the color of the hat is identical to the color of the shirt B: The number of different sets that can be made where the color of the hat is different from the color of the shirt AI: In a certain closet, three green hats, a black shirt, a green shirt and a blue shirt.
The best answer is B. Column A: Three hats multiplied by one green shirt is three combinations only. Column B: 3 sets from the black shirt with all the hats and 3 more from the blue shirt with all the hats. 6 sets in total and therefore this column is greater.
32. A: The number of possibilities of choosing a pair of a boy and a girl. B: The number of possibilities of choosing a pair of two boys. AI: There are 7 boys and 6 girls.
The best answer is A. Column A: 7 x 6 = 42 is the number of possibilities. Column B: For the first boy there are 7 options and for the second one there are 6, and again there are 42 possibilities. Don’t forget to divide by 2 because there is no meaning to the order of the possibilities.
33. A: The number of possibilities of choosing a couple of cards B: The number of possibilities of choosing three cards AI: There are 5 cards in the deck.
The best answer is C. The cards can be divided into two groups of 2 and 3. When you pick up two cards there are 3 left, which are actually also chosen. In the same manner, when you choose 3 cards, there are 2 left and therefore the number of possibilities is the same.
34. A: ½ B: The probability that George will choose a shirt and pants in different colors AI: George has red shirt and a blue shirt. George has a red pair of pants and a brown one as well.
The best answer is B. Lets enumerate all the options in column B: (Red, Red), (Red, Brown), (Blue, Red), (Blue, Brown). Three out of four combinations fit the criteria’s of column B and therefore the answer is B.
35. A: The probability that the sum of the numbers that will be received from two tosses will be even B: The probability that the sum of the numbers that will be received from two tosses will be odd AI: A coin is marked with 0 on one side and with 1 on the other
The best answer is C. Write all the combinations: (1,1) à even (0,0) à even (0,1) à odd (1,0) à odd The probability for an even sum and an odd sum is 1/2.
36. A: The sum of all the natural numbers from 1 to 40 inclusive B: 810
The best answer is A. Use the arithmetic progression formula. The sum is =
37. A: 27 B: The number of germs at 17:30 AI: A germ multiplies it self every round hour. 3 germs were placed in a small plate at 14:30.
The best answer is A. At 14:30 there were 3 germs. At 15:00 there were 6 germs. At 16:00 there were 12 germs. At 17 there were 24 germs, which is smaller than 27.
Geometry
A: The height of the triangle to side X
B: The height of the triangle to side Y
AI: The areas of the triangles are equal Y < X
The best answer is B. The area of the triangle in column A is The area of the triangle in column B is Compare the two areas: Now since X > Y, the height of the triangle in column A is smaller than that in column B.
A: BC B: 5
The best answer is A. The sum of the angles in the triangle is 180 and therefore the measurement of the angle BAC is (180 – 40 – 40 = 100 degrees). The side in front of the angle 100 is the largest and therefore BC is larger than 5.
40.
A: 90 degrees B: AI: AB is a straight line
The best answer is C. Since AB is a straight line the sum of the angles upon him equals to 180 and thus we can write the following equation:
41. A: The difference between the lengths of the large side to the short one B: The length of the short side AI: The sum of two long sides in a rectangle is 4 times bigger than the short side in the same rectangle.
The best answer is C. Name the long side X and the short side Y. From the additional data we can write: 2X = 4Y à X = 2Y. Column A = X – Y = Y. Column B = Y, and therefore the two columns are equal.
A: 180 degrees B:
The best answer is B. All the angles in the drawing are vertices angles to the quadrilateral in the centre. The sum of the angles in the quadrilateral is equal to
43.
A: 1.5 meters squared B: The area of the shaded region (half a circle) AI: The side of the white square is 2 meters
The best answer is B. The diameter of the circle is 2 meters and so the radius is 1 meter. The area of the shaded region is
44. A: The area of a triangle with a side of 6 inches and a respective height of B: 9 inches squared
The best answer is A. Find the area of the triangle:
A: The diameter of the circle B: The length of AB
The best answer is A. The largest line in the circle is the diameter and the angle in front of it is 90 degrees. Since the angle in front of the line is smaller than 90 degrees, the line must be smaller than the diameter.
A: The circumference of the inner circle B: Half the circumference of the outer circle AI: A circle is blocked by a square, which is also blocked by an outer circle
The best answer is A. Let R be the radius of the inner circle. The circumference of the inner circle is The side of the square is 2R and so its diagonal is The circumference of the outer circle is
A: (2r)2 B: R2 AI: The radius of the inner circle is r and that of the outer circle is R.
The best answer is A. The ratio between the area of the circle that is blocked by a square to the area of the circle that blocks the square is always 1:2 and therefore the ratio between the radiuses is 1: Column A = (2r)2 = 4r2. Column B =
A: The area of the shaded region B: The area of the white region, divided by two AI: Three identical circles are blocked inside a large circle
The best answer is C. The radius of the big circle (R) is equal to 6 radiuses of the smaller circles (r). Column A = Column B =
49.
A: 270 degrees B:
The best answer is B. The sum of all the outer angles to a polygon is always 360 degrees and therefore in this case
A: The length of the blackened line B: 30 Cm AI: The circumference of the pentagon is 50 Cm
The best answer is A. If one side of the pentagon is (50/5 = 10) Cm then the emphasized line is made up of 2 line segments and one diagonal. Now, we know that any diagonal in a pentagon is larger than the side and therefore the required circumference is more than (10 + 10 + 10) and therefore the answer is A.
1. A: A B: 1/A AI:
The best answer is D. Take two cases: A = 3 and A=1/3. In the first case, column A = 3 and column B = 1/3. In the second case, column A = 1/3 and column B = 3. Since the answer changed, D is the right answer.
2. A: B: AI: X, Y > 0
The best answer is D. Take two cases: X=1, Y=1 and X=2, Y=3. In the first case, column A = 1 and so is column B. In the second case, column A = 2/3 and column B = ¾. Since the answer changed, D is the right answer.
3. A: B: AI: X > -4
The best answer is B. Multiply both columns by 46 and compare: 4X + 7? 5X + 11 Subtract 4X + 7 from both sides. 0? X + 4 Since X > -4, column B is greater than column A.
4. A: B: AI: A and B are positive odd integers.
The best answer is D. If A = B = 1, both columns are equal but in any other case the columns are not even and therefore the answer cannot be determined. D is the right answer in such a case.
5. A: B: AI: X < 0 < Y < Z
The best answer is A. Multiply both columns by YZ (Y and Z are both positive). XY2? XZ2 Divide by X, remember to flip the result in the end since X is negative. Y2? Z2 Z is bigger than Y and so the answer is suppose to be column B. Don’t forget we multiplied both sides by X and therefore the real answer is column A.
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, which is bigger than one and thus greater than column A.
miles and therefore the answer is C.
= 13 – 25 = -12.
52 = -12 and therefore both columns are equal.
8.
and therefore the area of one and a half circles is equal to 1.5 x 
9.
.
.
.
.
, which is greater than column B.
38.
.
.
.
39.
è 
.
42.
and is also equal to 360 degrees and therefore column B is greater.
meters squared and so column B is greater.
inches
, which is more than 9 and therefore column A is greater.
45.
46.
.
, which is also the diameter of the outer circle.
and half of it would be
. We can see that the circumference of the inner circle is greater than this expression and therefore the answer is A.
47.
.
, which is smaller than column A.
48.
.
and so both columns are even.
50.
