Студопедия — РЕГИОНЫ РОССИЙСКОЙ ФЕДЕРАЦИИ 7 страница
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РЕГИОНЫ РОССИЙСКОЙ ФЕДЕРАЦИИ 7 страница






 

39.

A: The minimum number socks that are needed to be taken out of a drawer in order to get two socks of the same color

B: 8

AI: In a sock’s drawer, there are 7 pairs of socks in different colors.

The best answer is C.

Take the worst case scenario, after taking 7 socks out of the drawer, the 8’Th sock must be in the same color as one of the other socks and therefore the columns are equal to one another.

 

40.

A: 5 degrees

B: The measurement of the smaller angle between the clock’s hands at 14:10

 

The best answer is C.

Draw the clock with the relevant time.

We can see that the large hand is on 2 and the smaller hand is a little clockwise, the question is how much. Ten minutes have passed from the time the small hand was exactly on the 2 bar and therefore it moved by

degrees, which is 5 degrees.

The columns are equal.

 

41.

A: The number poles needed to support the cable

B: 67

AI: poles support a 20 Kilometers long electric cable.

The distance between the poles cannot surpass 300 meters and a pole is

Required in the both edges of the cable for support.

 

The best answer is A.

20 Km are 20,000 meters, which are divided into 66 - 300 meters long parts and another segment of 200 meters. Let’s not forget that we also need a pole in the beginning and in the end, which sums up to (66+1+1 = 68) poles and therefore column A is greater.

 

42.

A:

B: The time it will take to build a fence that is X meters long with a pole every Y meters

AI: Sticking a pole in the ground takes s seconds.

Installing a meter of fence takes t seconds.

 

The best answer is C.

The number of poles is .

The time it would take to stick all of these poles is .

It takes seconds to install the fence.

Sum those two to get exactly the same expression written in column A.

 

43.

A: The number of people in the bachelor party including Brad

B: X(2Y + 1)

AI: Brad invited X people to his bachelor party.

With each of the guests arrived Y more friends, and everyone of them

Also brought his date with him.

 

The best answer is A.

The number of people invited: X.

The number of friends that joined (including dates): 2YX.

The total number of people in the party, including Brad =

1 + X + 2XY è X(2Y + 1) + 1, which is greater by 1 than column B and therefore the answer is A.

 

Geometry

 

44.

 

A:

B:

 

The best answer is C.

The two right triangles are similar triangles and therefore the angles and equal to one another.

 

45.

A: (AB + BC)2

B: AC2

AI: A, B and C are three corners of a right triangle where AC is the hypotenuse.

 

The best answer is A.

Compare the columns:

(AB + BC)2? AC2

AB2 + 2AB x BC + BC2? AC2

According to the Pythagoras theory, AC2 = AB2 + BC2. Subtract there items:

2AB x BC? 0

AB and BC are positive numbers and therefore column A is greater than column B.

 

     
     
     

46.

 

 

A: Shaded region

B: White region

 

The best answer is B.

Let’s say that the measurements of the white cube are X on X.

Column A = (2X)(3X)(0.5) + (X)(2X)(0.5) = 4X2.

Column B = (3X)(3X) = 9X2 – 4X2 = 5X2.

Since X must be positive, column B is greater.

 

   
     
     

47.

 

 

A: The area of the triangle divided by the area of the white region

B: ½

AI: Each of the squared cubes has an area of 1 meter squared.

 

The best answer is A.

The area of the triangle is (3)(3)(0.5) = 4.5 meters squared.

The area of the while region is 9 minus the area of the triangle,

Thus 4.5 meters squared. Column A is greater than column B.

 

48.

A: (Z2)/(X2 + Y2)

B: 1

AI: X, Y and Z are three sides of a right triangle

Z > Y

 

The best answer is D.

From the given information we cannot determine if Z or X are the hypotenuse and therefore the right answer is D.

 

49.

A: (Z2)/(X2 + Y2)

B: 1

AI: X, Y and Z are three sides of a right triangle

X > Y > Z

 

The best answer is B.

Let’s set X=5, Y=4 and Z=3 as the sides of the triangle.

Column A = (9)/(25 + 16) = 9/41 < 1 and therefore column B is greater.

 

50.

A: The largest possible size of a base angle in an isosceles triangle

B: 89 degrees

 

The best answer is A.

The base angles in an isosceles triangle are less than 90 degrees and therefore they can even be 89.8, which is greater than 89 degrees and therefore column A is greater.

 

51.

 

 

 

A: The area of the shaded region

 

 


B: The area of the shaded region

 

AI: The shaded region is blocked inside the square

 

The best answer is C.

In both shapes there is a complete circle that is blocked inside a square.

The area of the circle must be the same and therefore both columns are equal.


52.

 

A: The area of the triangle ABC

B: The area of the triangle ADC

AI: AC is the median of BD

 

The best answer is C.

The two triangles have the same height and the same base since AC is the median of the side BD and therefore they have the same area.

 

53.

 

 

A: The area of the triangle ABC

B: The area of the triangle ADC

AI: AC is the bisector of the angle DAB

 

The best answer is D.

The two triangles have the same height but we know nothing of their bases.

The fact that AC is the bisector of the angle DAB tells us nothing about the base and therefore the answer cannot be obtained.

 

54.

A: The area of a pentagon that is blocked in a circle

B: The area of a hexagon that is blocked in a circle

AI: The blocking circle is identical in both columns

 

The best answer is B.

The more sides the polygon has, the closer it is to a shape of a circle and thus its area is larger. A hexagon has 6 sides while a pentagon has only 5 and therefore the area of the hexagon will be greater.


55.

 

A: AB + DC

B: BC + AD

AI: ABCD is blocked inside a circle

 

The best answer is C.

There is a rule that implies that opposite sides of a quadrilateral that is blocked inside a circle are equal to the other opposite sides.

Therefore, AB + DC = AD + BC.

56.

A: The sum of the angles in a pentagon

B: 4 angles in a regular octagon

 

The best answer is C.

The pentagon has 5 angles, each 108 degrees, they sum up to 540 degrees in total. The octagon has 8 angles of 135 each. 4 angles of an octagon are 54 degrees and therefore the two columns are equal.

 

57.

A: The sum of the angles an hexagon and a square

B: The sum of the angles in three squares

 

The best answer is C.

A hexagon has 6 angles of 120 degrees each, which sum up to 720.

Column A = 720 + 360 = 1080 degrees.

Column B = 360 x 3 = 1080 degrees.

The answer is C.

 

58.

A: Four angles of a hexagon

B: Three angles of an octagon

 

The best answer is A.

A hexagon has 6 angles of 120 degrees each, 4 of them are exactly 480 degrees. An octagon has 8 angles of 135 each, 3 of them are 405 degrees and therefore column A is greater than B.


59.

 

A: Z

B: W

AI: X < Y

 

The best answer is A.

Opposite angles of a quadrilateral that is blocked in a circle are equal to 180 degrees è Z + X = 180 and W + Y = 180.

Since X < Y, Z must be greater than W.


60.

 

 

A: The white region

B: The shaded region

AI: A shaded rectangle is placed inside a white rectangle.

The perimeter of the outer rectangle is twice the perimeter of the

Inner rectangle.

 

The best answer is D.

Let’s name X and Y as the sides of large rectangle.

Let’s name Q and P as the sides of the small rectangle.

We can write the following equation:

2X + 2Y = 4Q + 4P è This equation is insufficient in order to determine the answer and therefore D is the right answer.

 

61.

 

 

A: The dotted area

B: The area of the rectangle ABCD

AI: AC = R/2, where R is the radius of the circle

 

The best answer is D.

Since the question didn’t imply that AB is the diameter of the circle we cannot calculate the area of the rectangle or the dotted area.

More information is required in order to answer the question.

 

62.

 

 

A: The area of the rectangle ABCD

B: The area of the shaded region

AI: CD = R/2, where R is the radius of the circle

AC is the diameter of the circle

 

The best answer is B.

The area of the rectangle is (2R)(R/2) = R2.

The area of the half circle is , which is greater than R2 and therefore column B is greater.

 

63.

A: The radius of the car’s wheel

B: 50 centimeters

AI: A car travels at 3000 R.P.M (rounds per minute).

The car’s speed is 4000 meters per minute.

 

The best answer is B.

The perimeter of the wheel is .

If we divide the distance that the car traveled in one minute by the perimeter of the wheel, we’ll get the number of rounds that the wheel did during that period of time: è meters.

Column B is greater than column A.

 

Note: Is it too much of an engineering question?

 

64.

A: The area of a square whose sides are

B: Twice the area of an equilateral whose sides are 4

 

The best answer is B.

Column A: The area of the triangle is x = 12.

Column B: The area of the equilateral triangle is = , which is approximately 7 and therefore twice the area is approximately 14, which is greater than column A.

 

 

65.

A: The measurement of the angle Z

B: 90

AI: X, Y and Z are three angles in a triangle.

Y = X + Z.

 

The best answer is B.

Since Y = X + Z, Z cannot be 90 degrees because then the sum of the three angles is over 180 and that’s not possible.

 

66.

 

 


A: The measurement of the angle X

B: 20 degrees

AI: Y = 3X

 

The best answer is C.

The sum of the angles in a triangle is 180 degrees.

180 = Y + 90 + X + 10 è 80 = 4X è X = 20 and therefore the columns are equal to one another and the answer is C.

 

67.

 

 

A: 58 degrees

B:

AI:

The best answer is D.

The size of the inner left angle in the triangle is equal to 58 degrees.

Since we have no further information on the triangle, we have no correlation between this angle and .

The right answer is D.

 

68.

A: 31 inches

B: The perimeter of the triangle

AI: In the triangle ABC, AB = 6 inches and BC = 9 inches

 

The best answer is A.

The third side (AC) has a range that it can be in:

3 < AC < 15 (Between the other sides difference to their sum).

Therefore the perimeter also has a range:

18 < Range < 30 and so 31 is out of the rage and thus greater than the perimeter of the triangle.


69.

 

 

A: 14 inches

B: BC

AI: AB = 12 inches

 

The best answer is A.

The angle 80 is a supplementary angle to the angle inside the triangle, which is equal to (180 – 80 = 100 degrees). Since the sum of the angles in the triangle is equal to 180 degrees, the angle in front of the side BC is smaller than 100 and so BC < AC < 12 inches.

Column A is greater than column B.

 

 

70.

 

A:

B: 90 degrees

 

The best answer is A.

If = 90, the side in front of it would be 5 and not 6.

Since the side of the triangle in front of is bigger than 5 then is bigger than 90 degrees and column A is greater than column B.

 

71.

 

A: The area of the trapezoid

B:

AI: A trapezoid is made up of three equilateral triangles with a side of 2

 

The best answer is B.

The area of the trapezoid is (Base1 + Base2)(Height)/2.

The area is equal to (2 + 4)()/2 = .

Column B is greater than column A.


72.

 

 

A: 4 inches

B: BC

AI: The area of ACD is 6 inches squared.

The area of ABCD is 16 inches squared.

 

The best answer is B.

Let’s name the height of the trapezoid, which is also the height of the triangle ADC. The area of ADC = 6 = 3H/2 è H = 4 inches.

The area of the trapezoid is equal to (AD + BC)(H)/2 = (3 + BC)(4)/2 = 16 è BC = 8 – 3 = 5 inches and therefore column B is greater than column A.


73.

 

 

A: 2AC

B: The perimeter of the rhombus

 

The best answer is B.

Look at ABC: AB + BC > AC

Look at ADC: AD + DC > AC

Add these two: AB + BC + AD + DC = perimeter of the rhombus > 2AC.


74.

 

A: Twice the perimeter of the trapezoid AEFD

B: The perimeter of the rectangle ABCD

AI: DF = EB = X

 

The best answer is A.

The perimeter of the rectangle is 2(BC + AB) = 2(BC + AE + X).

The perimeter of the trapezoid twice = 2(BC + AE + X + FE).

We can see that the perimeter of the trapezoid includes all the sides that the perimeter of the rectangle includes plus one more (FE) and therefore it is bigger.

 

 

75.

 

 

A: The area of the shaded region

B:

AI: The radius of the circle is 1

 

The best answer is C.

The angle of the shaded region is 360 - .

The area is therefore: .


76.

 

A: FG

B:

AI: The side of the regular hexagon is S

 

The best answer is B.

Look at the triangle AGF; the angle GAF is (120/2) since the whole triangle is symmetric on the hexagon, thus AG is the median.

If GAF is 60 degrees and AGF is 90 degrees è AFG is 30 degrees.

Since AFG is a 30 degree angle in a right triangle, AG = S/2.

Use the Pythagoras principle: S2 = (S/2)2 + FG2 è FG = .

We can see that column B is greater than column A.

 

77.

 

 

A: The area of the parallelogram

B: 45

AI: A Parallelogram is blocked inside a rectangle.

The area of the rectangle is 60 feet squared.


The best answer is B.

Draw the following guide lines:

 

We can see that the rectangle is made up of 6 identical triangles and therefore 6 of those are equal to 10 è each = 10.

The parallelogram is made up of 4 triangles and so its area is

4 x 10 = 40 feet squared.

Column B is greater.

 

 

1.

A: P

B: ½

AI:

 

The best answer is C.

Simplify the expression: è èP = 1/2.

The columns are equal to one another.

 

2.

A: X

B: 3

AI:

 

The best answer is B.

Simplify the expression: .

2/3 = 2X è X = 1/3 and therefore column B is greater.

 

3.

A:

B: 1

 

The best answer is C.

There is no other way, simplify the expression in column A:

and therefore the columns are equal to one another and the answer is C.

 

4.

A: -5

B: Y

AI:

 

The best answer is D.

Y can be -5 but it can also be -2 for example and therefore there is more than one answer and the right answer is D.

 

5.

A:

B: 22

 

The best answer is A.

Simplify the expression in column A: and therefore column A is greater than column B.

 

6.

A:

B:

 

The best answer is C.

Column A = .

Column B = .

The columns are equal.

 

7.

A:

B:

 

The best answer is A.

Simplify the expression in column A:

.

We can see that column A is greater than column B.

 

8.

A: 3X – Y – Z

B: Y – 4

AI: X, Y and Z are three consecutive integers (X < Y < Z)

 

The best answer is C.

Replace X with Y – 1 and Z with Y + 1:

3X – Y – Z = 3(Y – 1) – Y – (Y + 1) = 3Y – 3 – Y – Y – 1 = Y – 4.

We can see that both columns eventually are equal to Y – 4 and therefore the answer is C.

 

9.

A: 872 - 862

B: 172

 

The best answer is A.

We can write column A as (87 – 86)(87 + 86) = 1(87 + 86) = 173.

Column A is greater than column B.

 

10.

A: The number of cola cans that Leroy can buy after the increase in the price

B: 10

AI: Leroy buys 12 cans of cola everyday. The price per can suddenly increased by 50%

 

The best answer is B.

Let’s say that each can cost $2 prior to the increase in the price.

After the price increased, each can cost (1.5 x 2 = $3).

Leroy spends 12 x $2 = 24 and now that would be enough for only 8 cans and therefore column B is greater than column A.

 

11.

A: The number of double digit numbers that the sum of their digit is smaller than 4

B: 5

 

The best answer is A.

The best way is to think of the numbers, there are 6 numbers that answer the criteria’s: 10, 11, 12, 20, 21 and 30.

Column A is greater than column B.

 

12.

A: How much percent did Cindy add to her weight

B: 25%

AI: Cindy gained 15 Kg and now she weighs 65 Kg

 

The best answer is A.

Cindy’s initial weight was 50 Kg. The question is how much is 15 out of 50 in percent terms: and therefore column A is greater than column B.

 

13.

A: X

B: 15

AI: X percent of X are X/8

 

The best answer is B.

Rewrite the extra data in percent terms: . Multiply both sides by 100 to get: 12.5 X = X2 è X = 12.5 and therefore column B is greater than column A.

 

14.

A: The total decrease in the car’s value

B: 60%

AI: The value of a car decreased by 20% and then it decreased by 40% more

 

The best answer is B.

Let’s say the car cost $100.

At first its price decreased to $80, then it decreased by 40%:

(0.6 x $80 = $48). The total decrease is (100 – 48 = 52%).

The car decreased by 52% and not by 60% and so column B is greater.

 

15.

A: The total increase in the stock’s value

B: 40%

AI: The value of a certain stock increased by 20% on January and by 20% more on February

 

The best answer is A.

Let’s say that the value of the stock was $100.

In January it increased to $120.

In February it increased to (1.2 x 120 = $144).

All together, the stock’s value increased by 44%, which is greater than 40%.

 

16.

A: By how much percent are rollerblades more expansive than ice-skates

B: 80%

AI: Rollerblades cost 40% more than skateboards and skateboards cost 40% more than ice-skates

 

The best answer is A.

Let’s say that ice-skates cost $100 since there are no numbers in the question. Skateboards cost 40% more, thus $140.







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