problems practice phần 4 docx

Since both 4 and 6 are even, the units’ digit must be even and the number is divisible by 2 and so A is the answer... The only answer where the fraction is equal to the number on the rig

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25 X and Y are both double digit numbers If in each, the tens digit is greater by 3 than the tens digit, what is the greatest possible difference between X and Y?

The best answer is C

Let’s pick the largest and the smallest numbers: 96 and 30

The difference between them is therefore (96 – 30 = 66) and so C is the right answer

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a  and so A is the right answer

4 If every digit of a whole number is either a 4 or a 6, the number must

The best answer is A

An even number is a number where the units’ digit is even

Since both 4 and 6 are even, the units’ digit must be even and the number

is divisible by 2 and so A is the answer

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The best answer is D

Two parallel lines have the same slope

The only two lines with the same slope among all the lines are b and d and therefore D is the right answer

7 In which of the following answers, the fraction is equal to the number? (A) 4/5, 0.45

(B) 5/6, 0.825

(C) 1/5, 0.1

(D) 7/8, 0.875

(E) 8/9, 0.9

The only answer where the fraction is equal to the number on the right is

The best answer is E

Let X be the cost of the t-shirt

The jeans cost X + 15.75 The jeans and the t-shirt both cost

2X + 15.75 = 64.25  2X = 64.25 – 15.75 = 48.5  X= $24.5

9 Daniel bought a camera with 3 films for $104 If the camera costs 10 times more then a film, how much will 2 cameras and 2 films cost? (A) $56

(B) $112

(C) $158

(D) $176

(E) $182

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Let X be the price of a film and so the price of a camera is 10X

10X + 3X = 96  13X = 104  X=$8 and a camera costs $80

2 cameras with 2 films will cost 2(80) + 2(8) = $176

10 When an integer X is divided by a prime number Y, the result is always a fraction Which of the following proves that the statement above

X=21, which is an integer and Y=7, which is a prime number

X/Y = 21/7 = 3, which is not a fraction and therefore it proves that the statement is not definitely true

11

In the figure above, there are three shaded squares blocked inside a larger square What is the ratio between the shaded area to the area of the square ABCD?

The area of the shaded region is the sum of the areas of three squares:

2 x 2 + 1 x 1 + 2 x 2 = 9

The area of the rectangle ABCD is (5) x (5) = 25

The required ratio is therefore 9:25

12 In which of the following sets of numbers is the sum of the numbers greater than their product?

2 2

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Go over the sets of numbers and look for one set in which the sum is greater than the product In the third set, the sum is greater than 5 yet the product is

0.5 x 2 x 3 x 1/3 = 1 and therefore this is the right answer

13 In a local pool there are 5000 liters Water is being poured into the pool at a rate of X liters per second and also, because of a leak, water is leaking out at a rate of Y liters per minute After Z seconds, how many liters are in the pool?

In the beginning there are 5,000 liters in the pool

Every second, X liters come in and Y/60 liters come out

And therefore after Z seconds there are 5,000 + ZX – ZY/60 liters in the pool, or Z(X – Y/60) + 5,000 and so E is the right answer

14 55 percent of 80 are the same as 25 percent of which of the following numbers?

55 percent of 80 are 50% of 80 + 5% of 80, which is 40 + 4 = 44

In order to find the number, divide 44 by 0.25 to get 44 x 4 = 176 and therefore A is the right answer

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In the triangle above, which of the following must be true?

The third angle of the triangle is therefore 180 – 34 – 43 = 103o

A side in front of a larger angle is larger and therefore AB > AC > CB The only answer that is true is C

16 Let @X be defined as (2X – 8)/(3X) If @A = -2, which of the

following values can replace A?

Try out the answers to figure this one out

Try A=1: @1 = (2 – 8)/3 = -2 and so C is the right answer

17 A new action was defined for every X:

[X] = the largest possible integer, that is small or equal to X

The expression is also equal to 6 x 0 + 2 x 7 = 14 and so A is the answer

18 Let ~(A, B)~ be defined as the remainder of the partition A/B

What is the value of ~(163, 7)~ ?

(A) 1

(B) 2

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(C) 3

(D) 4

(E) 5

The best answer is B

163 divided by 7 is 23 with a remainder of 2

The answer to the question is the remainder and so B is the right answer

19 If 2X = 3Y, 2Y = 3Z, 2Z = 3R and 2R = 3S, what is the value of X/S? (A) 16/81

X/S = (81/16) and therefore E is the right answer

20 Which of the following is equal to the sum of two consecutive odd integers?

Let X be one of the integers and X + 2 the other

We know that X + X + 2 = 2X + 2 is equal to one of the answers and so subtract 2 from the answer, divide by two and see if you get an odd integer

All the answers except for C and D are disqualified since they’re odd Check C: 382 – 2 = 380  380/2 = 190, which is even

Check D: 396 – 2 = 394  394/2 = 197, which is odd and this is the answer

21 If X is even and Y is odd, which of the following must be an integer? (A) XY/2

(B) (X + Y)/2

(C) XY/3

(D) (X + Y)/3

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(E) (X + Y)/XY

In these types of questions, the easiest way is to pick up numbers

Let X=2 and =1

Answer A is an integer since XY/2 = 2/2 = 1 and so this is the right

answer

22 E is an even number To which of the following E3 doesn’t

necessarily divide to with out a remainder?

Every even number in the power of 3 is divisible by 1, itself, itself in the power of 3 and 8 and therefore E is the only answer left

Try any number in order to validate this principle

23 If the first angle in a triangle is equal to two thirds of the second angle and the second angle is equal to one fifth of the third angle, what is the measurement of the median angle in the triangle?

Let the median angle (the one we are looking for) be X

We know that the first angle is 2X/3 and that the third angle is 5X

The sum of the angles in the triangle is equal to 180o and therefore

X + 2X/3 + 5X = 180  20X/3 = 180  X = 27o and so the right answer

is B

24 It takes Gloria 7.5 hours to knit 6 sweaters If Cynthia can knit three times faster, how many minutes would it take her to knit 3 sweaters? (A) 15

(B) 25

(C) 50

(D) 75

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(E) 100

Cynthia can knit three times faster and therefore it takes her 2.5 hours to knit 6 sweaters and half that time to knit 3 sweaters

Half of 2.5 hours is one hour and a quarter, which is 75 minutes

25 Three carpenters, working at a constant rate, can build a cabin in 7 hours

How many hours would it take 7 carpenters, working at a constant rate, to built the same cabin?

Instead of three carpenters, now we have 7, which is 2.5 times the

manpower If there are 2 and a third times the workers it would cut the time by 2 and a third also and therefore it would take 7/

3

1

2 = 3 hours to built the same cabin and B is the right answer

First add the numbers in the parenthesis and only then relate to the power (7 + 11)2 = 182 = 324 and so E is the right answer

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Answer C can also be written as  

) 19 17 )(

19 17 ( 19 17

3 The average (arithmetic mean) of 4 numbers is 100

If three of the numbers are 20, 60 and 80, what is the fourth number? (A) 120

(B) 180

(C) 220

(D) 240

(E) 360

Let the fourth number be X

4

60 80 20

X = 400 – 20 – 80 – 60 = 240 and so D is the right answer

4 If the average (arithmetic mean) of 7 numbers is 343, what is sum? (A) 49

(B) 187

(C) 343

(D) 1520

(E) 2401

The sum of the numbers is actually the average multiplied by the amount

of numbers, thus 343 x 7 = 2401 and so the right answer is E

5 If 6X + 9Y = 23, what is the value of (15X + 22.5Y)?

Take 6X + 9Y = 23 and multiply it by 2.5:

(6 x 2.5)X + (9 x 2.5)Y = 23 x 2.5  15X + 22.5Y = 57.5 and so the right answer is A

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6 The movie “Back to the past” is two hours and 15 minutes long How many times can the theater run the entire movie in 12 hours assuming that each movie starts right after the previous one?

(A) Once

(B) Twice

(C) Three times

(D) Four times

(E) Five times

Each movie is 2.25 hours

We were asked how many whole times is 2.25 fit inside 12

12/2.5 = 4 and a remainder and therefore D is the right answer

7 It takes Adam one hour and 40 minutes to put together one puzzle How many complete puzzles can Adam put together in 14 hours? (A) 9

(B) 8

(C) 7

(D) 6

(E) 5

One hour and 40 minutes is 1

3

2 hours

42 5

3 14

3 5 14

3

2 1

And so Adam will complete only 8 complete puzzles

8 If (-1 < A < 0) and (0 < B < 1), which of the following must be true? (A) A + B = 0

(B) AB > 0

(C) AB < 0

(D) B – A < 0

(E) A – B > 0

Pick up a number for A and for B

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For example: A= -1/2 and B= ½

The only true statement is that AB < 0 and so answer C is the right

(E) All the answers are correct

All of the answers are good, since the only wrong answer would be: X is negative There is no such answer among the answers and so all of them are correct

10 Z is a prime number larger than 10 If Y = 3Z + 2, which of the

following must be true?

Any prime number larger than 10 is also odd

3Z is an odd number and when you add 2 to an odd number, the result is also odd and so the right answer is D

11 If A is prime number bigger than 3, which of the following is even? (A) A + 4

(B) A2 + 2

(C) (A – 1)2

(D) 2A + 1

(E) A – 2

Take A = 5 for example

Only answer C is even, (5 – 1)2 = 42 = 16 and therefore the right answer is

C

12 3X, 4X and 5X are three angles in a triangle, what is the value of X? (A) 8

(B) 12

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(C) 15

(D) 18

(E) 22

The sum of all the angles in the triangle is 180o

The cost of 3 markers is N cents

420/3 = 140 and so it will cost 140N cents to buy 420 markers

140N = 1.4N dollars and so C is the right answer

14 If

7

13 7

Multiply the expression by 7Y: 49 + 6Y = 13Y  7Y = 49  Y=7 The value of 3Y is therefore 21

15 What percent of 150 are 12?

Use the answers

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Try answer B: 12

10

120 10

15 8 150 100

16 If A(4, 3), B(X, 3) and C(5, X) are three vertices of a triangle, what is its area?

Draw an axis system with the relevant points

The area of the triangle is AB multiplied by the height of the triangle The base of the triangle is X – 4 in length and the height of the triangle

Is X – 3 The area of the triangle is therefore:

6 5 3 5 0 2

12 7 2

X X X

X

and so E is the answer

17 A, B and C are three consecutive numbers (C > B > A)

What is the value of the expression (C – A)2(B – A)2(C – B)2 ?

If the numbers are consecutive, B = A+1 and C = A + 2

(C – A)2(B – A)2(C – A)2 = 22 x 12 x 12 = 4 and so the right answer is D

18 A blue marble is worth $2 and a red marble is worth $3

A jar contains 5 blue marbles and 4 red marbles only If only one marbles

is pulled at once, what is the probability of pulling marbles that are worth

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The only possible way of pulling marbles that are worth $5 is by pulling a red and then a blue marble or the opposite

The probability of pulling a red and then a blue marble is:

9

2 90

20 9

5 10

20 9

4 10

(E) All the answers are correct

Open the parenthesis:

21 Let’s define  as a new action

If (5) = 13, which of the following can be equal to (x)?

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13 can be also 5 x 2 + 1 and 5 x 3 – 2 and therefore

(x) can be equal to 2x + 1 or 3x – 2

22 Let’s define  as a new action

If (5) = 23, which of the following can be equal to (z)?

ABCDEF is a regular Hexagon, as shown in the figure above

If O is the centre of the hexagon, what is the value of the angle ODC? (A) 60

(B) 80

(C) 100

(D) 120

(E) 150

There are 6 main angles in the hexagon (like angle DOC) and so each one

of them, including DOC, is equal to 360/6 = 60o

OD = OC since ABCDEF is a regular hexagon

DOC is an isosceles with an angle of 60o and therefore the rest of the angles are also 60o and so A is the right answer

24 10,000 liters of water contain 3 grams of sodium

How many grams of sodium can be found in 105 liters of water?

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(D) 300

(E) 3,000

10,000 are 104 and so 105 are 10 10

10

10 14

25 If the distance between A and C is 13 and the distance between A and

B is 5, which of the following is true?

(A) BC = 8

(B) BC < 13

(C) BC > 13

(D) 8 < BC < 18

(E) None of the answers is correct

The question didn’t mention that the points are on the same straight line

or that B is between A and C

The largest distance between B and C can be 13 + 5 = 18

The smallest distance between B and C is 13 – 5 = 8 and therefore the best answer is D

1 If 6Y + 3 = 39, what is the value of 3

12

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By simply adding two diagonals to the square, it will divide into 4

identical triangles, each is an isosceles and so the right answer is B

4 A C.N.C machine produces 120 parts in 4 hours At this rate, in how

many minutes can the C.N.C machine produce 15 parts?

The machine produces 120 parts in 4 hours and so it produces 120/4 = 30 parts in one hour

15 parts are half of 30 and therefore it will take the machine 60/2 = 30 minutes to produce this amount of parts

5 A propeller lathe factory produces 4 propellers in 7 hours

At this rate, in how many minutes can the factory produce one part?

A

B

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4 propellers are produced in 7 hours and so one propeller is produced in 7/4 hours, which is 7 x 60 /4 = 105 minutes and so C is the right answer

6 If A = 7, what is the value of A2(B – 3A) in terms of B?

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According to the question:

A = 0.15D, B = 0.22D and C = 0.41D

The value of 2A + B + C is 2(0.15D) + 0.22D + 0.41D = 0.93D and so B

is the right answer

9 If X is 40% of Y and Y is 20% of Z, what is Z in terms of X?

According to the question: X = 0.4Y and Y = 0.2Z

X = 0.4Y = 0.4(0.2Z) = 0.08Z  Z = 1/0.08 x X = 12.5X and so D is the right answer

10 A canister holds four pints of black powder (1 pint = 1/8 gallon)

If a quarter of packet of black powder contains 3 pints, how many

canisters can be filled with 20 packets?

20 packets contain 20 x 3 = 60 pints

Every canister contains 4 pints and therefore with 60 pints (60/4 = 15) canisters can be filled and the right answer is C

11 On a number line, if point A has a coordinate -12 and point B has a coordinate of 24, what is the coordinate of point C, which is located one sixth of the way from B to A?

Point C is located 1/6 of the way from B to A and so the point must be closer to B

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