2015-08-21
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Chapter 1:
1. If 3X + 7 = 4, what is the value of (X – 2)3?
(A) -81
(B) -27
(C) 0
(D) 12
(E) 27
The best answer is B.
Solve: 3X + 7 = 4 è 3X = -3 è X = -1.
(X – 2)3 = (-3)3 = -27 and so answer B is correct.
2. A, B, C and D are all different digits between 0 and 9.
If AB + DC = 7B (AB, DC and 7B are double digit numbers), what is the value of C?
(A) 0
(B) 1
(C) 2
(D) 3
(E) 5
The best answer is A.
Start adding from the right: B + C = B and so C = 0 since it can't be 10, 20 or anything more than 9.
3. If 37 = 92T, what is the value of T?
(A) 0.5
(B) 1.5
(C) 1.75
(D) 2.25
(E) 2.5
The best answer is C.
37 = 92T è 37 = (3)2 x 2T è 7 = 4T è T = 7/4 = 1.75.
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4.
In the figure above, if the line segment AB is perpendicular to the x-axis and has a length of 8, what is the value of s/r?
(A) 0.25
(B) 0.5
(C) 0
(D) 1
(E) 2
The best answer is B.
If the line has a length of 8 and it’s parallel to the x-axis the coordinates of point B will be (2, 1) and so S=1 and r=2.
The value of s/r = ½.
5. Everyday, Kevin divides $180 equally across different charity organizations. On Sunday Kevin divided his money to 9 different organizations, while on Monday, he divided his money to only 7.
How much more money (in dollars) did each organization receive on Monday than on Sunday?
(A) 3.5
(B) 
(C) 
(D) 7.5
(E) 9
The best answer is B.
Calculate the amount that each organization received each day:
On Sunday: 180/9 = $20.
On Monday: 180/7 = $ 
, which is larger by from Sunday and so B is the right answer.
6. The income of a doctor is X times larger than that of a teacher. If we’ll cut 20% of the doctor’s income and we’ll increase the teacher’s income by 20% their income will be even. X =
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(A) 1.5
(B) 2
(C) 3
(D) 2.5
(E) 3.5
The best answer is A.
Let the doctors income be D and the teachers income be T.
The doctors' income after a decrease is equal to the teachers' income after an increase: 
And therefore X = 1.5.
7. Dorothy has 200 marbles. If 10% of the marbles are large and the rest are small, how many small marbles should Dorothy remove so the portion of the large marbles will be 20% of the total?
(A) 40
(B) 80
(C) 100
(D) 140
(E) 150
The best answer is C.
10% of the marbles are big, thus 20 marbles. The rest (200 – 20 = 180) are small. We want 20 marbles to be 20% of the total and so the total should be 100. In order for the total to be 100, 100 small marbles should be removed and so the right answer is C.
8. A giant wheel with a radius of 
, breaks loose from its axis and starts rolling with out slipping on a flat surface. What is the distance that the wheel will cross after three complete revolutions?
(A) 260
(B) 540
(C) 780
(D) 1040
(E) 2520
The best answer is C.
The circumference of the wheel is 
.
After three whole revolutions, the wheel will cross a distance of 260 x 3
= 780 and so C is the right answer.
9. For which of the following values of X will 5X + 8 be smaller than 14?
(A) 4
(B) 3
(C) 2
(D) 1.5
(E) 1
The best answer is E.
We want 5X + 8 < 14 è 5X < 6 è X < 6/5 or X < 1.2
The only number that answers these criteria’s is answer E.
10. Which of the following numbers is smaller than 1/7 and larger than 1/8?
(A) 0.2
(B) 0.08
(C) 0.13
(D) 0.75
(E) 0.12
The best answer is C.
One number between 1/8 and 1/7 is 1/(7.5) = 
.
2/15 is like 20/150, which is approximately 0.133 and so C is the right answer.
11. If 
and both X and Y are integers, which of the following is true?
(A) X = -Y
(B) X = -1
(C) X is positive
(D) X is negative
(E) X is a fraction
The best answer is D.
The first answer that pops into mind is that X is a fraction, but the question says that X is an integer. The only way that X makes Y smaller is if it is negative and so D is the right answer.
12. If 
, what is the value of X?
(A) 5
(B) 15
(C) 30
(D) 35
(E) 45
The best answer is C.
can be written as: 
. Divide both sides by 
to get: 
and so X = 30.
13. If X is positive even number and Y is a positive odd number, which of the following expressions is not even?
(A) (XY)Y
(B) Y2
(C) X5
(D) Y2X3
(E) 4Y
The best answer is B.
Pick up random numbers for X and Y: X=2 and Y=1.
The only odd answer is B, 12 = 1 and so this is the right answer.
14. A, B and C are integers. If the expression A + B + C is even and the expression A – C is odd, which of the following can be said on B?
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(A) odd
(B) even
(C) negative
(D) positive
(E) None of the above
The best answer is A.
Since A – C is odd, A + C is also odd (the sign doesn’t matter).
By adding B to A + C, the expression becomes even and therefore B must be also odd and so A is the right answer.
15. How many square feet of marble are needed to cover a rectangular floor that is 60 inches by 132 inches? (1 foot = 12 inches)
(A) 32
(B) 44
(C) 55
(D) 68
(E) 75
The best answer is C.
The area that is needed to be covered is 60 on 132 inches, in feet its
60/12 on 132/12, which is 5 feet on 11 feet and so the area is 55 and the right answer is C.
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16.
If M is the diameter of the circle and ACB is a right triangle at C, what is the sum of X + Y + Z in degrees?
(A) 90
(B) 120
(C) 180
(D) 270
(E) 360
The best answer is C.
Since M is the diameter of the circle, Z as the inscribed angle, is 90o.
The sum of the angles in the ABC is 180o and so X + Y + 90 = 180 è
X + Y = 90 and so X + Y + Z = 180o and so C is the answer.
17. On a blueprint, 3 cm represent 125 meters. If a road is 437.5 meters, how many cm will represent it on the blueprint?
(A) 5
(B) 7.5
(C) 9
(D) 10.5
(E) 12
The best answer is D.
437.5 / 125 = 3.5. And therefore we should stick to the given proportion and just multiply 3.5 by 3 cm to get 10.5 cm on the blueprint.
18. If 53X+2 = 125 and X2Y – 1 = X6, what is the value of Y/X?
(A) 8
(B) 9.5
(C) 10
(D) 10.5
(E) 11
The best answer is D.
53X+2= 125 è 53X+2 = 53 and so 3X + 2 = 3 è X = 1/3.
X2Y – 1 = X6 è 2Y – 1 = 6 è Y = 7/2.
Y/X = (7/2)/(1/3) = 10.5.
19. In Pillsbury Island the weather during a 30 day month is distributed as followed: 12 days of rain, 7 days of sunny weather and the rest of the days have thunderstorms. When a randomly chosen tourist comes to the island, what is the probability that there would be thunderstorms?
(A) 6/15
(B) ¼
(C) ¾
(D) 19/30
(E) 11/30
The best answer is E.
There are (30 – 12 – 7 = 11) days of thunderstorms in a month and so if a day is chosen at random, there is a probability of 11/30 that there’ll be thunderstorms.
20. If the volume of a cube is 27, what is the shortest distance from the centre of the cube to one of its corners?
(A) 
(B) 2 
(C) 
(D) 
(E) 
The best answer is D.
Draw a cube and also draw a line from its center to one of the corners.
Use the Pythagoras principle to find the diagonal of the base of the cube.
The volume is 27 and so each side is 3.
The diagonal of the base is 
.
The diagonal of the cube is again calculated using the Pythagoras principle; this time with the height of the cube and the diagonal of the base:
The diagonal of the cube is 
.
Half the diagonal is the shortest distance between the center and the corner and so the answer is 
= .
21. When X is divided by 6, the remainder is 3. Which of the following represents the number that when is divided by 5 will yield a remainder of 3?
(A) X + 1
(B) X + 2
(C) X + 3
(D) X + 4
(E) X + 5
The best answer is D.
Pick a number for X and try the answers: X = 9 and so when divided by 6, the remainder is 3.
Only (X + 4), or in our case 13, when divided by 5, the remainder is 3 and therefore the right answer is D.
22. How many positive double-digit numbers are there where both the tens and the units' digits are prime numbers?
(A) 15
(B) 25
(C) 35
(D) 45
(E) 55
The best answer is B.
The possible prime digits are: 1, 2, 3, 5 and 7.
The numbers that can be put together are: 11, 22, 33, 55, 77, 12, 13, 15, 17, 21, 23, 25, 27, 31, 32, 35, 37, 51, 52, 53, 57, 71, 72, 73 and 75.
Altogether we have 25 numbers and so the right answer is B.
23.
If A2, A and A-3 lie on a number line in the order show in the figure above, which of the following can be a value of A?
(A) -4
(B) -0.25
(C) 0
(D) 0.25
(E) 2
The best answer is D.
The only way that A2 is smaller than A and that A-3 is larger than A is if A is a positive fraction.
Take answer D for example: A = 0.25.
A2 = 1/16 and A-3 = 64 and so this answer is good.
24. A snake crawled for three minutes. In the first minute the snake traveled 65 feet, which was 30% farther than it traveled in the second minute. In the third minute the snake traveled at a speed of 5 feet per second. What is the total distance that the snake traveled in feet?
(A) 385
(B) 415
(C) 460
(D) 500
(E) 525
The best answer is B.
In the first minute it traveled 65 feet.
In the second minute it traveled X feet, X + 0.3X = 65 à X= 65/1.3 = 50 feet. In the third minute it traveled (5 x 60 = 300) feet.
The total distance is (65 + 50 + 300 = 415) feet.
25. Two birds are flying towards one another. The first bird is traveling at a speed of 40 mph, which is 25% faster than the second bird. If the distance between the birds is 20 miles, how long will it takes the birds to meet
(In minutes)?
(A) 35
(B) 30
(C) 25
(D) 20
(E) 15
The best answer is D.
The speed of the second bird is X, (X + 0.25X = 40) è
X = 40/1.25 = 32 mph. In order to find the time it will take the birds to meet, divide the total distance by the sum of the bird's speeds:
24 / (40+32) = 24/72 = 1/3 hour, which is 20 minutes and so the best answer is D.
Chapter 2:
1. The total weight of John and his brother is 180 kilograms. If John’s weight is 15 Kilograms more than half of his brothers weight, what is the differences between their weights?
(A) 25
(B) 30
(C) 40
(D) 50
(E) 70
The best answer is C.
Let J be the weight of John and B the weight of his brother.
We can write the following two equations:
J + B = 180 and J = 15 + B/2.
è 15 + B/2 + B = 180 è 3B/2 = 165 è B = 
Kg and so J = 70 Kg. The differences between their weights is (110 – 70 = 40 Kg).
2. Bill and Ted have $25,000 together. If Bill has 5 thousand dollars more than 3 times the money Ted has, how much money does Ted have?
(A) $5,000
(B) $8,500
(C) $10,000
(D) $12,500
(E) $20,000
The best answer is A.
Let B the money Bill has and T the money Ted has.
We can write the following equations:
B + T = 25,000 and B = 5,000 + 3T è 5,000 + 3T + T = 25,000 è
4T = 20,000 è T = $5,000 and so B = $20,000.
Ted has $5,000.
3. If 
, what is the value of 
?
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(A) 
(B) 
(C) 
(D) 
(E) 
The best answer is E.
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4. What is the value of 
?
(A) X2/Y2
(B) X/Y
(C) X2Y2
(D) Y2/X-2
(E) X-2Y2
The best answer is E.
= 
and so E is the answer.
5.
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In the figure above, A, B and C are three angles as drawn. M and l are parallel lines.
What is the value of the expression (A + C – B)?
(A) 0o
(B) 5o
(C) 45o
(D) 90o
(E)The answer cannot be determined from the given data
The best answer is A.
The easiest way to see that A + C = B 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.
6.
Five equilateral triangles were attached to a regular hexagon.
If the side of the hexagon is 5, what is the length of the outer perimeter of the shaped that was formed?
(A) 25
(B) 35
(C) 50
(D) 60
(E) 75
The best answer is D.
Since the triangles are equilaterals, each of their side is equal to the hexagon’s side. The total perimeter of the shaped that was formed is made up of 12 sides, thus 12 x 5 is the length. Therefore the answer is D, 60.
7. If X and Y are integers and 
, what must be true?
I. X is negative
II. Y is positive
III. X is a fraction
(A) 1 only
(B) 2 only
(C) 3 only
(D) 1 and 2
(E) 2 and 3
The best answer is E.
Since Y is greater than an absolute value, it must be positive and so 2 is true.
Inside the absolute value, X makes Y smaller and therefore X is a fraction.
1 and 2 are correct and therefore the answer is E.
8. If X and Y are integers and 
, what must be true?
I. X is negative
II. Y is positive
III. X/Y is negative
(A) 1 only
(B) 2 only
(C) 3 only
(D) 2 and 3
(E) 1, 2 and 3
The best answer is E.
Since Y is greater than an absolute value, it must be positive and so 2 is true.
If an absolute value of a number is greater than the number itself, the number is negative and so 3 is also true.
Since X/Y is negative and Y is positive, X must be negative and so all three claims must be true.
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9.
The triangle in the figure above is an isosceles (AB = BC).
If the area of the triangle is 100, what is the length of side BC?
(A) 
(B) 12.5
(C) 
(D) 16
(E) 
The best answer is C.
The area is simply the base multiplied by the height and divided by two.
.
BC can be calculated using the Pythagoras principle: BC2 = 202 + 55 è
BC = 
and so C is the correct answer.
10.
In the figure above, three identical triangles with a radius of R are blocked inside a larger circle. What is the area of the larger circle using R?
(A) 24R2
(B) 
(C) 
(D) 
(E) 
The best answer is E.
The radius of the larger circle is made up of 6 small radiuses.
The area of a circle is 
, which in our case is equal to 
.
11. If (A – 3)(A + 7) – (A + 2)(A + 10) = 12, what is the value of A?
(A) -5.5
(B) 
(C) 
(D) -10.5
(E) 
The best answer is B.
Open the parenthesis and find the value of A.
A2 + 7A – 3A – 21 –(A2 + 2A + 10A + 20) = -8A -41 = 12 è 8A = -53
A = -53/8 = and so B is the correct answer.
12. If 2X < Y and 2X2 > XY, which of the following must be true?
(A) X > 1
(B) X < 0
(C) X > 0
(D) Y < 0
(E) Y > 1
The best answer is B.
We know that 2X < Y, after multiplying both sides by X the sign of the inequality changed and therefore X must be negative and B is the answer.
13. If 
, which of the following must be true?
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(A) X > 12
(B) X > 1
(C) X < -12
(D) X < -4
(E) -6 < X < 6
The best answer is D.
Simplify the expression given: è 20X + 32 < 12X – 3 è
8X < -35 è X < -35/8, which is approximately -4.4.
Since X is smaller than -4.4, it must also be smaller than -4.
Therefore the answer is D.
14. If 0 < Y < 0.25 and YR = 20, which of the following must be true?
(A) R > 80
(B) 0.2 < R < 5
(C) 0.25 < R < 20
(D) 0 < R < 1
(E) -2 < R < 8
The best answer is A.
Pick a value for Y, for example Y = 0.2.
Input this value into the second equation: 0.2R = 20 è R = 100 and therefore the only answer that includes this range is A.
15. The first number in a sequence is -8 and every term after the first is 6 more than the term preceding it. What is the value of the 102’st term?
(A) 598
(B) 606
(C) 600
(D) 592
(E) 608
The best answer is A.
If the sequence started in 0, the 102 number in the sequence would simply be
101 x 6 = 606. Since the first number wasn’t zero, it was -8 the whole series of numbers is smaller by that number and therefore the answer is
(606 – 8 = 598).
16. In a certain class, the average height is 5 feet and 2 inches. Which of the following must be true?
I. The tallest member in the classroom must be at least taller by an inch then the shortest member in the class.
II. If there are 20 members in the class, the sum of the class members’ heights is approximately 340 inches.
III. If there is a 5 feet tall member in the class, there also must be a 5 feet and 4 inches member.
(A) None
(B) 1 only
(C) 2 only
(D) 3 only
(E) 2 and 3
The best answer is C.
All the members can be the same height and so 1 doesn’t have to be true.
The sum is the average multiplied by the number of people and in our case
The sum is 5 feet and 2 inches (which is 17 inches) x 20 = 340 inches and so this statement is good.
3 doesn’t have to be true since the distribution around the average doesn’t divide equally by pairs of people.
We can see that only 2 is true and so C is the answer.
17. The price of a car was X pounds in February. If the price reduces by 20% each month, what will the price be in May?
(A) 0.4X
(B) 0.6X
(C) 0.64X
(D) 0.512X
(E) 0.4096X
The best answer is D.
This question can be solved easier by choosing numbers.
Let say that the price in February was 100 (X = 100 pounds).
In March, the price reduced by 20% to (0.8 x 100 = 80).
In April, the price reduced again by 20% to (0.8 x 80 = 64).
In May, the price reduced again by 20% to (0.8 x 64 = 51.2).
51.2 is 0.512X and so the answer is D.
18. The value of a certain stock was Q dollars on Sunday.
If on Monday the price increased by 10% and on Tuesday it decreased by 10%, what is the value of the stock at the end of Tuesday?
(A) Q
(B) 0.99Q
(C) 1.1Q
(D) 1.01Q
(E) 1.21Q
The best answer is B.
Pick a number for Q, for example $100.
On Monday, the value increased by 10% to 110.
On Tuesday, the value decreased by 10% to (0.9 x 110 = $99).
And therefore the value is not Q, its 0.99Q and so B is the right answer.
19. If X is an integer and X5 = 243, what is the value of X?
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5
The best answer is C.
Use the answers. Take 2 for example, 25 = 32 and so this is not the answer.
It can’t be 4 or 5 since the units digit is 3 and therefore the answer must be 3 and really 35 = 81 x 3 = 243.
20. 5 ice-creams are sold for $2.30 and 3 snack bars are sold for $1.65.
What will be Tim’s change if he purchases 15 ice-creams and 6 snack bars using a hundred dollar bill?
(A) $92.80
(B) $91.20
(C) $89.80
(D) $55.60
(E) $50.40
The best answer is C.
15 ice-creams cost 3 x 2.30 = $6.90.
6 snack bars cost 2 x 1.65 = $3.30.
The total price is 6.90 + 3.30 = $10.20.
Tim paid with a $100 bill and therefore his change was (100 – 10.20 = $89.80).
21. The price of soda can is 42 cents. If you buy 73 cans using a $50 bill, how much change should you expect?
(A) $30.66
(B) $28.54
(C) $22.12
(D) $19.34
(E) $16.84
The best answer is D.
One soda can costs 42 cents and so 73 cans will cost (73 x 42 = 3066 cents), which is $30.66. If we pay using a $50 bill, the change will be
(50 – 30.66 = $19.34) and so D is the answer.
22. D is the midpoint of line segment AB and C is the midpoint of line segment DB. If the length of AB is 42, what is the length of AC?
(A) 45
(B) 42.5
(C) 38.25
(D) 34
(E) 31.5
The best answer is E.
Draw the line and mark A, B, C and D.
AC is made up of two segments: AD and DC.
D is the midpoint of AB and therefore it is equal to 21.
DC + CB = 21 and DC = CB è DC = 10.5
AC = 21 + 10.5 = 31.5.
23. If A + C = 2B, what is the value of 
? (A>B>C)
(A) 0.
(B) 1.
(C) –A.
(D) C + A.
(E) -1.
The best answer is E.
Simplify the expression: 
Therefore the answer is E.
24. Steve has 7 more than 4 times the number of posters that Sanders has.
If 2X represents the number of Steve’s posters and if 2Y represents the number of Sanders posters, which of the following is a correct expression relation X and Y?
(A) X = 2Y + 7
(B) X = 4Y + 7
(C) X = 4Y + 3.5
(D) X = 2Y + 3.5
(E) X = 4(Y + 7)
The best answer is C.
According to the question, Steve has 7 more than 4 times the number of posters. In other words, 2X = 4(2Y) + 7 è X = 4Y + 3.5 and so C is the correct answer.
25. In the repeating decimal
The digits 13579 are repeated, 3.135791357913579…
Which digit is in the 1,999’Th place to the right of the decimal point?
(A) 1
(B) 3
(C) 5
(D) 7
(E) 9
The best answer is D.
Every 5’Th digit is always 9 and therefore the 2,000’Th digit will be 9.
The 1,999’Th digit is one to the left of 9, which is 7 and so the answer is D.
Chapter 3:
1.
In the figure above, A, B, C, D and E are centers of 5 circles.
If each of the circles has an area of 15, what is the area of the rectangle?
(A) 40
(B) 
(C) 
(D) 
(E) 
The best answer is D.
Each of the circles has an area of 15, which is equal to 
, and therefore the radius of each of the circles is 
.
The area of the rectangle is the multiplication of its sides.
One side is equal to 2 radiuses of the circle and the other side is made from 6 radiuses. The area of the rectangle is (2R) x (6R) = 12R2, which is equal to
and so D is the answer.
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2.
In the figure above, A, B, C, D and E are 5 centers of the two circles. If each circle has an area of 2, what is the perimeter of the square?
(A) 256
(B) 
(C) 
(D) 
(E) 
The best answer is E.
Each of the circles has an area of 2, which is equal to , and therefore the radius of each of the circles is 
.
The perimeter of the square is 4 times the length of one side.
One side is equal to 4 radiuses of the circle. The perimeter of the square is (16) x (R), which is equal to 
and so E is the answer.
3. There are 5 roads from city A to city B and 6 roads from city B to city C.
If Michael walks from city A to city C and back, passing through city B each time, and doesn’t travel any road twice, how many different routes for the walking trip are possible?
(A) 900
(B) 750
(C) 720
(D) 600
(E) 540
The best answer is D.
Let’s go over each road at a time.
From A to B: there are 5 options.
From B to C: there are 6 options.
From C back to B: there are (6 – 1) options, thus 5.
From B back to A: there are (5 – 1) options, thus 4.
The number of possibilities is the multiplication of these numbers:
5 x 6 x 5 x 4 = 600 possibilities and therefore the answer is D.
4. Linda has 12 shirts, 5 pants and 3 belts. If Linda wears one of each of the items, in how many different ways can she dress?
(A) 160
(B) 175
(C) 180
(D) 190
(E) 200
The best answer is C.
The number of possibilities is actually the multiplication of the number of options in each item.
The number of possibilities is (12 shirts) x (5 pants) x (3 belts) = 180 combinations and so C is the answer.
5. If a jet airplane can travel 450 feet per second, how many yards can he fly in a quarter of a minute?
(A) 1550
(B) 1750
(C) 1900
(D) 2150
(E) 2250
The best answer is E.
A quarter of a minute is 15 seconds.
In one second it flies 450 feet, which is 150 yards (1 yard = 3 feet).
(150 yards per second) x (15 seconds) = 2250 yards.
6. If a turtle travels 25 meters in half a minute, how much time would it take him to travel 20 meters, in seconds?
(A) 21
(B) 22
(C) 23
(D) 24
(E) 25
The best answer is D.
The turtle can travel 25 meters in 30 seconds, which makes his rate 25/30 = 5/6 meters per second. It would take him (20) / (5/6) = 24 seconds to travel 20 meters.
7. James just bought a ball from a local sports shop that sells only basketballs, soccer balls and bowling balls. Which of the following must be true?
(A) The ball is a basketball
(B) The ball is a bowling ball
(C) The ball is not a leather basketball.
(D) The ball is not a professional bowling ball
(E) The ball is not a tennis ball
The best answer is E.
Since all the balls are either basketballs, soccer balls or bowling balls, the only answer that must be true is E since there are no tennis balls in the shop.
8. If P is an even prime number, what is the next greater prime number in terms of P?
(A) P + 3
(B) P – 1
(C) P + 1
(D) 2P – 1
(E) 2P + 3
The best answer is C.
There is only one even prime number, 2.
The next greater prime number is 3, which is 2 + 1 and in terms of P it is
P + 1 and so the answer is C.
9. A certain pipe is divided evenly into 5 parts, each 4 feet and 2 inches.
What is the length of the original pipe, in feet? (1 foot = 12 inches).
(A) 
(B) 
(C) 
(D) 
(E) 
The best answer is B.
If one part is 4 feet and 2 inches, which is 
feet, the original length of the pipe should be 5 x = and so B is the answer.
10. Of the following numbers, which is the least?
(A) ¾ + 1/5
(B) ½ - 1/3
(C) 1/3 + 1/12
(D) 1/6 – 1/5
(E) 1 – ¾
The best answer is D.
All the answers apart from D are positive and we know that a negative number is smaller than a positive one and so the right answer must be D.
11. [1, 3, 6, 8, 9]
How many different pairs of unequal numbers can be chosen from the set above so that their multiplication is greater than 12? (Do not consider 1,3 and 3,1 as different pairs.)
(A) 6
(B) 8
(C) 12
(D) 18
(E) 24
The best answer is A.
The following pairs answer the criteria: (3,6), (3,8), (3,9), (6,6), (6,8) and (6,9). The answer, therefore, is A.
12.
If the shape in the figure above is to be rotated by 270o around an axis that is found in the middle of the rectangle, which of the following represents the resulting circle?
(A)
 | ||||
 | ||||
 |
(B)
(C)
(D)
(E)
The best answer is D.
Look at the shape as if it was a clock. The original figure is set, so the arrow points at the hour 5 approximately.
If you turn the arrow 270o, the arrow should point to the hour 1.
There are two probable choices: D and E.
Only D is good since the circles in E are not in the same orientations as they were in the original figure.
13. A printing machine can only work from 8:00 A.M. to 13:00 P.M. and from 16:00 P.M. to 20:00 P.M every day due to the noise. If it takes 4 minutes for a printing machine to print one newspaper, how many days would it take 2 machines to print 810 newspapers?
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5
The best answer is C.
One printing machine can print (60 / 4 = 15) newspapers in one hour.
In one working day there are 5 + 4 = 9 hours.
One printing machine can print 15 x 9 = 135 newspapers everyday.
Two machines can print 270 newspapers in one day and so it would take
(810 / 270 = 3) days in order to complete the task.
14.
In the figure above, the line segment joining the points (4,5) and (9,5) is the radius of a certain circle. Which of the following represents the circumference of the circle?
(A) 
(B) 
(C) 
(D) 
(E) 
The best answer is C.
The length of the line segment is 5 according to the coordinates.
The circumference of a circle is calculated using 
, which in our case is equal to .
15. Four brothers are to share a heritage of $850,000 in the following ratio 1:2:3:4. What is the amount of the greatest share?
(A) $510,000
(B) $340,000
(C) $255,000
(D) $170,000
(E) $85,000
The best answer is B.
Find the share unit of heritage, thus ($850,000) / (1 + 2 + 3 + 4) = $85,000.
The greatest share was 4 units, or 4 x $85,000 = $340,000 and so answer B is correct.
16. If X is a prime number, which of the following is a factor of 4X?
(A) X2
(B) 2X
(C) 3X
(D) 11X
(E) 12X
The best answer is E.
Since X if a prime number, only if you multiply it by a number that is dividable by 4, the result would be a factor of 4X and so the only possible answer is E.
17.
In the figure above, what is the perimeter of the triangle?
(A) 
(B) 
+ 22
(C) 
(D) 22 +
(E) 32
The best answer is D.
First find, using the Pythagoras principle, the length of the missing side.
Name the missing side M and write: M2 = 142 – 82 and so 
.
The perimeter of the triangle is therefore 8 + 14 + = 22 + and so D is the right answer.
18. What is the least possible prime number for which 30% of that number is greater than 4.5?
(A) 11
(B) 13
(C) 17
(D) 23
(E) 27
The best answer is C.
Take 30% of each of the answers and see which one will result in a number that is greater than 4.5.
30% of 13 is 3.9 < 4.5
30% of 17 is 5.1 > 4.5 and so C is the right answer.
19.
Which of the following is equal to the area of the shaded region above?
(A) 2XY
(B) X2Y2
(C) 4XY/9
(D) 8XY/9
(E) XY/9
The best answer is D.
The area of the shaded region is equal to the area of the great rectangle minus the area of the white section.
The area of the entire rectangle is XY and the area of the white section is (X/3)(Y/3) = XY/9.
The area of the shaded region is therefore, XY – XY/9 = 8XY/9 and so D is the answer.
20. If 3R = 6 + P/3, then which of the following can be equal to R assuming that 0 < P < 9?
(A) 9/11
(B) 45/13
(C) 7/6
(D) 20/7
(E) 1/9
The best answer is D.
From the question è R = 2 + P/9, and since P is a positive number smaller than 9, P/9 is a fraction and so R is between 2 and 3.
Look among the answers for a number that is between 2 and 3, the only possible answer is of course D; 20/7 = 
.
21. Subtract 7 from Y, divide the result by 6, multiply everything by 2 and add 11 to this quotient.
Which of the following is the possible result obtained by performing the operations described above?
(A) 
(B) 
(C) 
(D) 
(E) 
The best answer is C.
Do the following one step at a time:
Y – 7 è 
è 
è + 11 è 
and so C is the answer.
22. If the ratio between X and Y is 3:7, which of the following could represent X and Y?
(A) X=4, Y=36
(B) X=4, Y= 
(C) X=7, Y=3
(D) X=1.5, Y=3
(E) X=0, Y=14
The best answer is B.
The ratio should be 3:7 and so if we divide Y by X, the result should be
7/3 = 2.5. Take answer B: /4 = 
and so this is the right answer.
23. A barrel with a radius of R feet is rolling down a slope.
If the barrel rolled a distance of S yards, how many complete revolutions did it make? (1 yard = 3 feet)
(A) 
(B) 
(C) 
(D) 
(E) 
The best answer is A.
The distance that the barrel rolled is 2S feet.
When the barrel travels , it completes one revolution.
And therefore the barrel completed revolutions.
24. Two sisters went shopping, the bigger sister spent Y dollars and the smaller sister spent X dollars. If the difference between the two expenses is equal to their average, what is the value of the expression (X/Y)2?
(A) 2
(B) 3
(C) 6
(D) 9
(E) The answer cannot be determined from the information given
The best answer is D.
If the difference is equal to the average, then we could write the equation:
X – Y = (X +Y)/2.
è X – 3Y = 0 àX/Y = 3 è (X/Y)2 = 9 and so the answer is D.
25. If X is a positive even number, Y is a positive odd number (X > Y) and
Z = X – Y, which of the following expressions must be even?
(A) (XZ)2
(B) (XY)Z
(C) YZ
(D) ZY
(E) Z4
The best answer is B.
In these types of questions it is easy to pick up numbers.
For example: X=2, Y=1 and Z = 2 – 1 = 1.
Check out all the answers, B is equal to 21 = 2 and so this is the right answer.
Chapter 4:
1. If X + X/3 + X/4 = Y + Y/3 + Y/4 and Y = 60, what is the value of X?
(A) 14.5
(B) 16
(C) 20
(D) 45
(E) 60
The best answer is E.
Replace Y with 60 and solve:
X + X/3 + X/4 = 60 + 60/3 + 60/4 = 60+20+15 = 95 è multiply both sides by 12 to get: 12X + 4X + 3X = 95 x 12 = 1140 è 19X = 1140 è X=60.
2. The price of 15 books is $45. If the price of 8 tapes and 4 books is $44, what is the price of one tape?
(A) 3
(B) 4
(C) 5
(D) 6
(E) 7
The best answer is B.
15 books cost $45 and therefore one book costs $3.
8(tapes) + 4(3) = $44 è 8(tapes) = $32 è The price of a tape is $4.
3. Tom is three times older than Shelly. If in three years Tom would be twice as old as Shelly, how old is Shelly today?
(A) 3
(B) 4
(C) 6
(D) 9
(E) 12
The best answer is A.
Let T be the age of tom today and S the age of Shelly.
We can write the following equations: T = 3S and (T + 3) = 2(S + 3) è
3S + 3 = 2S + 6 è S = 3 and so the correct answer is A.
4. How many double-digit numbers are there where the product of their digits is 8?
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5
The best answer is D.
Since there aren’t so many numbers, write them all: 18, 24, 42 and 81 and so the right answer is D.
5. How many odd double-digit numbers, which the sum of their digits is 6, exist?
(A) 1
(B) 3
(C) 5
(D) 7
(E) 9
The best answer is B.
Since there aren’t many numbers, write them all down:
15, 33 and 51 and so B is the right answer.
6.
| Inches | Centimeters |
| 17.78 | |
| X | 53.34 |
Using the following conversion table, what is the value of X?
(A) 14
(B) 17
(C) 21
(D) 28
(E) 30
The best answer is C.
Since 17.78 x 3 = 53.34, X = 7 x 3 using the same ratio.
C is the correct answer.
7. The product of two odd integers is between 304 and 337 exclusive.
Which of the following cannot be one of the integers?
(A) 5
(B) 13
(C) 35
(D) 39
(E) 43
The best answer is E.
If the number is too big, for example 43, multiplying it with another odd integer wouldn’t be in the specified range.
43 x 7 = 301, which is under the range.
43 x 9 = 387, which is over the range and therefore 43 cannot be one of the integers.
8. If the average of 6 consecutive numbers is X, what is the average of the two median numbers?
(A) X/2
(B) X/3
(C) X – 1
(D) X
(E) X + 1
The best answer is D.
Since the numbers are consecutive, the average of the two middle numbers is the average of all the numbers. Take 1, 2, 3, 4, 5 and 6 for example:
The average is 3.5, now take 3 and 4; their average is also 3.5.
In our case the average stays X.
9. If X and Y are different positive integers and 3X + Y = 14, what is the product of all the possible values of X?
(A) 6
(B) 8
(C) 14
(D) 20
(E) 24
The best answer is E.
Start from the first number:
X=1 è not possible.
X=2 è Y=8.
X=3 è Y=5.
X=4 è Y=2.
X=5 è not possible.
The product is therefore 2 x 3 x 4 = 24.
10.
In the figure above, one side of the square is overlapping with the diameter of the circle. If the radius of the circle is 2, which of the following is the best approximation for the area of the shaded region?
(A) 5
(B) 10
(C) 15
(D) 20
(E) 25
The best answer is B.
The area of the shaded region is derived from the area of the square minus half the area of the circle.
The area of the square is 4 x 4 = 16.
The area of the circle is 
and so half the area would be 6.3.
The area of the shaded region would be approximately 16 – 6.3 = 9.7 and so B is the closest answer.
11. If 
, what is the value of Q?
(A) 98
(B) 104
(C) 108
(D) 112
(E) 118
The best answer is C.
Cross multiply the expression to get 11Q = 12 x 99. Divide both sides by 11 to get: Q = 12 x 9 = 108 and so C is the right answer.
12.
What is the perimeter of the triangle in the figure above?
(A) + 6
(B) 3(3 + )
(C) 6
(D) 9
(E) The data specified is insufficient
The best answer is B.
This is a special right triangle, its sides are in the ratio of 1:2: .
Since the small side is 3, the second side is 6 and the third side is 3 .
The perimeter of the triangle is therefore 3+6+3 = 3(3 + ).
13.
According to the chart above, how many more regular cameras than digital cameras were produced by company P during 1998?
(A) 2,000
(B) 2,500
(C) 3,200
(D) 4,500
(E) 5,200
The best answer is D.
According to the graph 12.5 thousand regular cameras were made and 8 thousand digital cameras, the difference is 4.5 thousand and so D is the right answer.
14. A certain lottery requires guessing 75% of the numbers to win.
If the lottery contains 60 numbers, how many numbers should one guess in order to win?
(A) 30
(B) 35
(C) 40
(D) 45
(E) 50
The best answer D.
In order to win the lottery, one should guess 75% of 60, which is 60 x 3 / 4 = 45 numbers and so the right answer is D.
15.
If in the cylinder above, the radius is equal to 
and the height is 3, what is the volume of the cylinder?
(A) 4
(B) 6
(C) 8
(D) 12
(E) 16
The best answer is D.
The volume of a cylinder is received by 
, which in our case is
and so D is the right answer.
16. If X and Y are coordinates, which of the following pairs (X,Y) answer these conditions: X > Y + 2 and X < 7?
(A) (4, 5)
(B) (7, 3)
(C) (6, 4)
(D) (5, 1)
(E) (3, 7)
The best answer is D.
Look among the answers for two things: X smaller than 7 and X bigger than Y. Take answer D: 5 > 1 + 2 and 5 < 7 and therefore this is the right answer.
17. If X = 2X/Y, then 2Y =
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5
The best answer is D.
Solve: X = 2X/Y, divide by X to get: 1 = 2/Y è Y=2 and so 2Y = 4.
18. If X + 2Y = 6 and Y = 3 – X/2, what is the value of (X + Y)?
(A) 4
(B) 6
(C) 8.5
(D) 10
(E) The answer cannot be determined
The best answer is E.
The second equation can be written as 2Y = 6 – X, which is equal to the first equation and so we have two variables with only one equation and so more sufficient data is required.
19. If 20% of the liquids in the barrel are X liters and 45% of the liquids are Y liters, which of the following must be true?
(A) X = 15
(B) X = Y – 20
(C) 4Y = 9X
(D) X + 30 < Y
(E) None of the answers above are true
The best answer is C.
Lets say that there are 100 liters of liquid in the barrel.
X = 20%, which is 20 liters and Y = 45 liters.
4 x 45 = 180 li
