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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 =
(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?
(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.
1. If 5X + 7 + Y = 28 and 2X + Y = 0, what is the value of X?
(A) -14
(B) -7
(C) 0
(D) 7
(E) 9
The best answer is D.
From the second equation, Y = -2X. Replace -2X with Y in the first equation to get the following: 5X – 2X = 21 è 3X = 21 è X = 7 and so D is the right answer.
2. If A + 2B = 50 and B – 3A = 10, what is the value of (A + B)?
(A) -4/27
(B) -3
(C) 0
(D) 7/120
(E) 190/7
The best answer is E.
From the second equation: B = 10 + 3A.
A + 2B = 50 è A + 2(10 + 3A) = 50 è A + 20 + 6A = 50 è 7A = 30 è
A = 30/7. Plug A into one of the equations to get: B = 160/7.
A + B = 30/7 + 160/7 = 190/7 and so E is the answer.
3. If 
and 
, what is the value of (Y – 2X)?
(A) 4/3
(B) 28/3
(C) -15
(D) 
(E) 
The best answer is D.
The first equation can be simplified to 3X + 3Y = 24Y or X + Y = 8Y è X = 7Y.
The second equation can be simplified to 2Y + X = 12.
Replace X with 7Y in the second equation: 2Y + 7Y = 12 è Y = 12/9 = 4/3.
(Y – 2X) = (Y – 14Y = -13Y = -13 x 4/3 = -52/3 = -17 and a third.
4. If it takes Jackson 24 minutes to eat two cheeseburgers by him self, how many minutes does it take him to eat half a cheeseburger?
(A) 4 minutes
(B) 6 minutes
(C) 12 minutes
(D) 14 minutes
(E) 48 minutes
The best answer is B.
In two cheeseburgers there are 4 half’s. It takes 24 minutes to eat 4 half’s and so it will take (24/4 = 6) minutes to eat one half.
5. It takes Mandy 3 hours to bake 3 cakes while Sandy can do it in only 2.
How long (in minutes) will it take Sandy and Mandy, working together, to bake 3 cakes?
(A) 24
(B) 45
(C) 60
(D) 72
(E) 120
The best answer is D.
Each hour, Mandy bakes 1/3 of the job (baking 3 cakes) while Sandy finishes 1/2 of the job. Together, their combined power is 1/3 + 1/2 = 5/6.
1/(5/6) = 6/5 is the time in hours it will take them to bake 3 cakes.
In minute, the time is 6 x 60 / 5 = 6 x 12 = 72 minutes.
6. If it takes 4 workers 5 hours to build 2 platforms, how many hours would it take 3 workers to build 3 platforms?
(A) 6
(B) 8
(C) 10
(D) 12
(E) 14
The best answer is C.
è Hours = 5 x 4 / 2 = 10 hours.
7. If it takes 3 people 14 minutes to drink 2 liters of water, how many minutes will it take 2 people to drink 2.5 liters of water?
(A) 24 minutes and 10 seconds
(B) 25 minutes
(C) 26 minutes and 15 seconds
(D) 26 minutes and 25 seconds
(E) 28 minutes and 25 seconds
The best answer is C.
è minutes = (2.5 x 14 x 3)/(2 x 2) = 105/4 = 26.25 = 26 minutes and 15 seconds.
8.
Two lines intersect in the figure above. What is the value of Z + X – Y?
(A) 180
(B) 225
(C) 285
(D) 310
(E) 345
The best answer is C.
X and Z are vertical angles and therefore equal.
X and 25 are adjacent angles and so X = 180 – 25 = 155o.
Y and 25 are vertical angles and therefore equal.
Z + X – Y = 155 + 155 – 25 = 285o and so C is the answer.
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9.
Two lines intersect in the figure above. What is the value of 2Z – 2Y?
(A) 78o
(B) 112o
(C) 156o
(D) 192o
(E) 214o
The best answer is D.
X and Z are vertical angles and therefore equal.
X and 42 are adjacent angles and so X = 180 – 42 = 138o.
Y and 42 are vertical angles and therefore equal.
2Z – 2Y = 138 x 2 – 42 x 2 = 276 – 84 = 192o and so D is the answer.
10. If 3X + 17 = 373, what is the value of 3X – 17?
(A) 356
(B) 339
(C) 313
(D) 298
(E) 273
The best answer is B.
Since 3X + 17 = 373, 3X = 373 – 17 = 356.
3X – 17 = 356 – 17 = 339 and so B is the right answer.
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11.
In the figure above, a line is to be drawn through point A so that it never crosses the Y-axis. Through which of the following points must the line pass?
(A) (-3, -5)
(B) (-3, 5)
(C) (6, -5)
(D) (3, 5)
(E) Answers A and C.
The best answer is D.
We are told that the line crosses point A yet it never crosses the Y-axis, which means that its X coordinate is constant. Look among the answers for a point with the same X-coordinate, the only possible answer is D.
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12.
In the figure above, a line is to be drawn through points O and M.
Through which of the following points must the line pass?
(A) (1, 0)
(B) (0, 1)
(C) (5, 5)
(D) (10, 5)
(E) (5, 10)
The best answer is C.
We are told that the line crosses point O and M and therefore it is in 45o relative to the X-axis and the Y-axis. Look through the answers for a coordinate that its X value is equal to its Y value. Only answer C fits this description and so this is the right answer.
13.
In the figure above, a line is to be drawn through point B so that it doest cross point O.
Through which of the following points can’t the line pass?
(A) (1, 0)
(B) (0, 1)
(C) (6, 6)
(D) (10, 5)
(E) (5, 10)
The best answer is C.
We are told that the line crosses point B and not O and therefore it is not in a 45o relative to the X-axis and the Y-axis. Look through the answers for a coordinate that its X value is equal to its Y value. Only answer C fits this description and so this is the right answer.
14. The ratio between 17.5 and 7 is equal to the ratio between 5 and what number?
(A) 9.5
(B) 7
(C) 6.5
(D) 3
(E) 2
The best answer is E.
We can write the following relations: 
, where X is the number we’re looking for. X = 
and so E is the best answer.
15. The ratio between 20 and 34 is equal to the ratio between 10 and what number?
(A) 15
(B) 17
(C) 21
(D) 24
(E) 28
The best answer is B.
We can write the following relations: 
, where N is the number we’re looking for. N = 
and so E is the best answer.
16. When X is deducted by 9, the result is 2 times X.
X =?
(A) -9
(B) -6
(C) -3
(D) 0
(E) 9
The best answer is A.
Write an equation from the question above: X – 9 = 2X è X = -9 and so A is the right answer.
17. What number increased by 15 equals 4 times that number?
(A) 3
(B) 5
(C) -3
(D) -5
(E) 0
The best answer is B.
Let the number we are looking for be X.
Write the following equation: X + 15 = 4X è 3X = 15 è X = 5 and so B is the answer.
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18.
What is the area of the rectangle ABCO in the figure above?
(A) 4
(B) 8
(C) 12
(D) 16
(E) 18
The best answer is D.
According to the coordinates of the rectangle, we know that one of the sides is equal to 4 and so is the other. The area of the rectangle is therefore 4 x 4 = 16 and so D is the right answer.
19.
What is the area of the triangle OAB in the figure above?
(A) 22
(B) 24
(C) 26
(D) 28
(E) 30
The best answer is D.
The area of a triangle is determined by the multiplication of the base and the height divided by two. Using the coordinates, we can see that the base is 7 and the height is 8 and therefore the area of the triangle is 7 x 8 / 2 = 7 x 4 = 28.
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20.
O is the origin of the circle in the figure above. If the circle crosses point A, what is the area of the circle?
(A) 
(B) 
(C) 
(D) 
(E) 
The best answer is
We need to find the radius of the circle. Draw an imaginary right triangle so that the hypotenuse is the line AO. We can see that the right triangle has the following sides: 3 and 4. Use the Pythagoras principle and find the line AO: 
.
Since the radius is 5, the area of the circle is 
.
21. The distance from Christy’s home to school is 4.5 Km and the distance from Christy’s school to work is 5.5 Km. Which of the following could not be the distance, in Km, from Christy’s home to work?
(A) 1
(B) 3
(C) 4.5
(D) 9
(E) 11
The best answer is E.
When we are given a distance, it is presumable that it’s a straight line.
From home to school its 4.5 Km and from school to work its 5.5 Km.
The distance between school and work can be between (5.5 – 4.5 = 1 Km) and the sum of the two (5.5 + 4.5 = 10 Km). The only answer that is not in that range is E, 11 Km.
22. Car A drives at an average speed of 15 mph. It takes car A, at average speed, 30 minutes to drive from point A to point B and 2 hours to drive from point B to point C.
Which of the following could not be the distance, in miles, between point A and C?
(A) 22.5
(B) 25
(C) 32
(D) 35
(E) 40
The best answer is E.
Distance = Time x speed.
The distance between A and B is 15 x 0.5 = 7.5 miles.
The distance between B and C is 15 x 2 = 30 miles.
The distance between A and C is between (30 – 7.5 = 22.5 miles) and (30 + 7.5 = 37.5 miles), look for an answer that is not in that range.
E is the only answer that is out of the specified range.
23. Three quarters of a meter of fence are used for creating a fence box.
Which of the following lengths of fences, in meters, could be used to build a fence box with the least amount of remaining fence?
(A) 3/2
(B) 1/2
(C) 13/12
(D) 5/4
(E) 19/20
The best answer is E.
We need only ¾ of a meter in order to build the fence box. Look for the number which is closest (yet greater) than 3/4.
All the answers but E are greater than 1 meter or smaller than ¾ and therefore E is the best answer. 19/20 would leave only 1/5 meter of residue.
24. 1.5 meters squared of plywood are needed in order to build a desk. Which of the following plywood areas, in meters squared, could be used to build two desks with the least amount of remainders?
(A) 13/4
(B) 10/3
(C) 16/5
(D) 19/6
(E) 7/2
The best answer is D.
We need 3 meters squared of plywood. Look among the answers for the number that is closest to 3 meters. The best answer is D; 19/6 = 3 and 1/6.
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25.
If X is equal to 60o, what is the value of the angle Y?
(A) 20o
(B) 30o
(C) 40o
(D) 50o
(E) 60o
The best answer is B.
Angle subtended by an arc at the center of a circle is double the angle subtended by the same arc on the circumference, therefore X = 2Y.
Since X = 60, Y = 30o and so B is the best answer.
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.
1. If A = 0.2B, B = 6C and C = 0.6D, what is the value of A when D = 70?
(A) 40.8
(B) 45.5
(C) 48.2
(D) 50.4
(E) 62.2
The best answer is D.
A = 0.2(6(0.6(70))) = 0.2 x 6 x 0.6 x 70 = 0.72 x 70 = 50.4 and so D is the right answer.
2. The center of a certain circle with a radius of 5 is located at O(0,0).
Which of the following points is located on the circumference of the circle?
(A) (2, 3)
(B) (-5, 5)
(C) (5, 5)
(D) (3, 4)
(E) (0, 10)
The best answer is D.
Only coordinate (3, 4) is located on the circumference of the circle.
Draw an axis system with the relevant point.
Complete the point to a triangle with the following coordinates: (3,0) and (0,0) so that the hypotenuse is equal to 
and so this point is located on a circle with a radius of 5.
3.
The rectangle in the figure above is divided into equal squares.
What percent of the rectangle is shaded?
(A) 
(B) 22%
(C) 
(D) 32.5%
(E) 
The best answer is A.
There are 12 x 3 = 36 identical squares in the figure above.
We can count that 6 of them are shaded and therefore 1/6 of the rectangle is shaded. 1/6 in percent terms is 
and so A is the right answer.
4. If X and Y are positive integers, which of the following expressions is equivalent to 
?
(A) 52Y
(B) 5(XY)/X
(C) 5
(D) 52XY-X
(E) X5Y
The best answer is D.
can be simplified to 
, which is equal to 52XY – X and therefore D is the right answer.
5.
The ratio between the radiuses of the small circle to that of the large one is 1:2.
If the radius of the small circle is 
, what is the area of the white region?
(A) 2
(B) 3
(C) 6
(D) 8
(E) 12
The best answer is A.
The radius of the large circle is therefore 2 x = 
.
The area of the small circle is = 
and in the same way the radius of the large circle 8.
The area of the white region is equal to 8 – 4 – 2 = 2 and so A is the right answer.
6. If X + 2X + 3X + 4X + 5 = Y + 2Y + 3Y + 4Y + 6,
what is the value of (X – Y)?
(A) 0.01
(B) 0.1
(C) 1
(D) 10
(E) 100
The best answer is B.
Simplify the expression in the question:
10X + 5 = 10Y + 6 è 10X – 10Y = 1 è 10(X – Y) = 1 è
X – Y = 0.1.
7.
Approximately what percent of the square is dotted?
(A) 5%
(B) 15%
(C) 25%
(D) 35%
(E) 45%
The best answer is C.
There are two dotted triangles in the figure above out of eight in total.
The question didn’t mention that the figure is not drawn into scale and so we can presume that the percent of the dotted area is 2/8 = ¼, which is 25%.
8. Which of the following fractions is greater than 0.35?
(A) 1/3
(B) 1/6
(C) 2/7
(D) 2/5
(E) 1/8
The best answer is D.
Another way of solving this question is to find the largest fraction, which must be greater than 0.35 anyway.
2/5 is the largest fraction and therefore D is the right answer.
9. Paul is taller than Mike but shorter than Rick. If P, M and R represent each of their height, respectively, which of the following is true?
(A) R < P < M
(B) M < P < R
(C) P < M < R
(D) R < M < P
(E) P < R < M
The best answer is B.
Translate words into inequalities:
Paul is taller than Mike: P > M
But shorter (Paul) than Rick: R > P.
And therefore B is the answer, R > P > M.
10. Travis had X items to sell at an average price of Y dollars per item.
If Travis didn’t sell Z items, which of the following represents the total cents amount he received from the sales?
(A) Y(X – Z)
(B) Z(Y – X)
(C) (100 – Z)XY
(D) 100Y(X – Z)
(E) (Z – X)/(100Y)
The best answer is D.
The amount of items that Travis sold is (X – Z).
If each item costs Y dollars, which is 100 cents, Travis sold 100Y(X – Z) and so D is the right answer.
11.
In the figure above, E is the middle point of side AC of the equilateral triangle ABC. If the perimeter of the triangle is 
, what is the perimeter of the square ADEF?
(A) 6 
(B) 4
(C) 12
(D) 
(E) 16
The best answer is C.
The perimeter of the triangle is and so each of its sides is 
.
Since E is the middle point of side AC, AE = 
, which is the diagonal of the square. The ratio between the diagonal of a square to its sides is :1 and so each of the sides is equal to 3.
The perimeter of the square is therefore 3 x 4 = 12.
12. A clown holds X balls, one on top of the other, on the palm of his hand. The radius of each of the balls is 3.5 inches and the palm of the clown’s hand is 2 feet off the ground.
If the tip of the highest ball is extended up to 45 inches, X =?
(1 foot = 12 inches)
(A) 3
(B) 4
(C) 5
(D) 6
(E) 7
The best answer is A.
The palm of his hand is 2 feet of the ground, which is 24 inches.
The rest of the distance is filled with X balls that extend to an area of (45 – 24 = 21 inches) and so all we need to calculate is how many balls can fit in 21 inches.
The radius of each ball is 3.5 inches and so the diameter is 7 inches. Divide 21 by 7 to get the number of balls that fit in the spacing: 21/7 = 3 balls.
13. The average (arithmetic mean) of 5 numbers is 7. When a sixth number is added, the average of the 6 numbers is 100. What is the sixth number?
(A) 560
(B) 565
(C) 570
(D) 575
(E) 580
The best answer is B.
Let X be the sixth number.
Use the average formula, 
è 35 + X = 600 è X = 565 and so B is the right answer.
14. The remainder of 343 by 16 is divided by 3, resulting in a remainder of which of the following numbers?
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5
The best answer is A.
343 divided by 16 is 21 with a remainder of 7.
When dividing 7 by 3, the result is 2 with a remainder of 1 and so the best answer is A.
15. When 145 is divided by 32, the remainder is X, and when 145 is divided by 22, the remainder is Y. What is the value of Y/X?
(A) 5/7
(B) 12/17
(C) 17/13
(D) 6/11
(E) 13/17
The best answer is E.
When dividing 145 by 32, the result is 4 with a remainder of 17.
When dividing 145 by 22, the result is 6 with a remainder of 13.
The answer is therefore 13/17 and so E is the right answer.
16. The expression (7 x 75) is how many times greater than 49?
(A) 6
(B) 7
(C) 15
(D) 74
(E) 75
The best answer is D.
49 is actually 72 and 7 x 75 is equal to 76.
76 is equal to 74 x 72 and so the right answer is D.
17. In a certain club, only white and black beer is served.
Eight of the club’s members drink white beer and twenty two drink black.
If six of the members that drink black beer also drink white beer, how many members are in the club?
(A) 6
(B) 8
(C) 22
(D) 24
(E) 26
The best answer is C.
8 club members drink white beer.
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