Grid questions practice
This file contains 120 Grid questions with full answers and explanations.
Good luck on your test.
1. A determinant of [a, b, c, d] is defined as (a x d – b x c).
What is the value of the determinant of [2.5, 2, 1, 5]?
According to the rules of the determinant in the question, the result of the determinant of [2.5, 2, 1, 5] = (2.5 x 5 – 2 x 1) = 10.5.
2. The mathematical function #(X, Y, Z) is defined as (X2 – Y2)/ Z2.
#(7, 5, 6) = ?
Use the pattern in the question. #(7, 5, 6) = (49 – 25)/36 = 24/36 = 2/3.
3. 2.25 grams of sugar can be found in a can of Juicy-juice.
How many grams of sugar can be found in a dozen cans?
If one can consists of 2.25 grams, a dozen (12) cans consists of
12 x 2.25 = 27 grams of sugar.
4. The local race track is 6 miles long. How long is the track in kilometers assuming that 1 mile = 1.6 kilometers?
The transition is 1 mile = 1.6 kilometers.
6 miles = 6 x 1.6 kilometers, which is 9.6 kilometers.
5. A cake recipe requires 
tablespoons of chocolate powder. How many teaspoons of chocolate powder should you put in the cake assuming that 1 teaspoon is 1/3 tablespoon?
According to the question, one tablespoon is three teaspoons.
We require 
teaspoons.
6. If 0.2X = 15, what is the value of 
?
Solve the equation: 0.2X = 15 è X = 15/0.2 = 15 x 10/2 = 75.
X/10 = 75/10 = 7.5.
7. If 0.66X = 4 – 0.34X, what is the value of 
?
Solve the equation: 0.66X = 4 – 0.34X è X = 4.
= 3 x 2 = 6.
8. If X + 2Y = 24 and Y – 3X = 10, what is the value of X?
From the second equation, Y = 3X + 10, replace this with Y in the first equation: X + 2(3X + 10) = 24 è X + 6X + 20 = 24 è 7X = 4 è X=4/7.
9. If X + Y = 15 and X – Y = -5, what is the value of X/Y?
Add the equations to get: 2X = 10 è X = 5.
Y = 15 – X = 10.
X/Y = 5/10 = ½.
10.
The triangle in the figure above is not drawn to scale.
If the measurement of angle 4 is 115.5o, what is the measurement of angles 1 and 2 (in degrees) ?
Notice that angles 3 and 4 are vertical angles and thus equal.
The sum of the angles in the triangle is 180o and therefore we can write the following connection: angle 1 + angle 2 + 115.5o = 180oè the sum of angle 1 and 2 is equal to (180 – 115.5 = 64.5o).
11. ABC is an isosceles triangle, AB = BC. If the measurement of angle ABC is between 102 and 105, what is the value of the measurement of angle BCA minus CAB?
Draw a sketch of the triangle.
Since the triangle is an isosceles, angles BCA = CAB and therefore the answer to the question is always zero no matter what the third angle is.
12. If the sum of two numbers is 6 and their difference is 2, what is the square of their product?
Let X and Y be the two numbers.
X + Y = 6 and Y – X = 2 are the two equations.
è Y = 4 and X = 2.
Their product is 8 and the square of their product is 82 = 64.
1. If A = 13 is a solution to the equation A2 + 2A – B = 0, what is the value of B?
Solve the equation with while replacing A with 13:
169 + 26 – B = 0 è B = 169 + 26 = 195.
2. If 
, what is the value of 
?
Take , and multiply it by 10X è 14X2 – 5X2 = 100
9X2 = 100 è X = 
and so the right answer is 10/3 or 3.33.
3.
AI: The
On the basis of the information in the graph above, if the
(When gridding, disregard the $ sign)
The
(100% – 35% – 50% = 15%) of the total expanses.
The total expanses are $825, and so 1% of the total expanses are:
825/15 = $55. If 1% is $55, then 35% are 35 x 55 = $1925 and that is the answer.
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4.
Note: Figure not drawn to scale
If line AB and DC are parallel, AB + DC = 26 and the area of triangle ACD is equal to the area of the triangle ABC, what is the length of AB?
Triangles ACD and ABC have a common height and so we can compare the areas of the two triangles: (DC x H)/2 = (AB x H)/2 è DC = AB
è 2AB = 26 and so AB = 13.
5. What is one possible value of A, which A < 5 < 1/A ?
It is obvious that A is a fraction since 1/A > A.
Try A = 1/5: 1/A = 5, which is equal to 5 and not greater and so we should take a smaller fraction, anything between 1/5 and 0.
Any answer between 0 and 1/5 is acceptable, for example 1/8.
6. For all numbers A and B, let AKB be defined as 
.
If 5KX = 2/5, what is the value of X?
According to the pattern, 5KX can be written as 
= 2/5 è
Cross multiply to get: 50 = 6X + 6 è 6X = 44 è X = 22/3.
7. Meg skated a total of 124 miles in 5 days.
Each day she traveled twice the distance she traveled the day before.
How many miles did Meg travel on her last day?
Let X be the distance Meg traveled on her first day.
If she traveled X on her first day, she traveled 2X on her second day, 4X on the third, 8X on the fourth and 16X on the fifth.
Sum the distances to 124 miles to find X:
X + 2X + 4X + 8X + 16X = 124 è 31X = 124 è X = 4 miles.
Meg traveled 16 x 4 = 64 miles on her last day.
8.
On the axis system above there are two lines.
If the length of AB is one and a quarter longer than BC, what is the value of the Y coordinate in point C?
The length of AB is (8 – 2 = 6), which is 1.25 longer than BC and therefore the length of BC is 6/1.25 = 
= 4.8.
The Y coordinate is therefore 3 + 4.8 = 7.8 or (3 + 24/5 = 39/5).
9. Joe the greengrocer has many tomatoes in his shop.
At 10:00 in the morning, Sandra came and bought 1/8 of the tomatoes.
At noon, Ricky came and bought 1/3 of the rest.
At 16:30, Joe ate half the tomatoes that were left in his shop, leaving only 14 tomatoes. What is the original number of tomatoes at Joe's shop?
At 10:00, Sandra bought 1/8, leaving 7/8 of the tomatoes.
At noon, Ricky bought 1/3 of 7/8, leaving 2/3 of 7/8, which is 14/24 = 7/12.
At 16:30, Joe ate half of 7/12, leaving 7/24 of the tomatoes.
14 tomatoes are 7/24 of the original number of tomatoes and therefore
there were 14 x 24 / 7 = 48 tomatoes in the beginning.
10. Lilac has two times more braids than Tiffany. If Tiffany would make 8 more braids, she would still have 8 less braids than Lilac.
How many braids does Tiffany have?
Let L be the amount of braids that Lilac has and T, the amount Tiffany has.
According to the question: L = 2T and T + 8 = L – 8.
T + 8 = 2T – 8 è T = 16.
Therefore Tiffany has 16 braids.
11. Paul forgot his girlfriend's 7-digits phone number.
Paul remembers the first 5 digits and that the last two digits were different from one another. If each digit can be between 0 to 9, how many arrangements are possible for Paul's girlfriend number?
Each of the last two digits can be between 0 and 9, thus 10 possibilities.
One of the digits has 10 arrangements and the other has 9 (since they must be different) and so there are 90 possible arrangements.
12. P(x) is defined as the greatest possible prime number that is less than x.
What is the value of P(3)/P(100)?
The greatest possible prime number that is smaller than 3 is 2.
The greatest possible prime number that is smaller than 100 is 97.
The answer to this question is 2/97.
1. If the product of two numbers is 9 and their difference is 0, what is their sum?
Let X and Y be the numbers.
We can write the following equations: XY = 9 and Y – X = 0.
Y = X è X2 = 9 è X = Y = 3.
The sum of the numbers is 3 + 3 = 6.
2. If 645 = 22X, what is the value of X?
Rewrite the expression: 645 = 22Xè (26)5 = 22Xè 230 = 22X
Compare the powers: 2X = 30 è X = 15.
3. If 1253 = 5Y, what is the value of 
?
Rewrite the expression: 1253 = 5Yè (53)3 = 5Yè 59 = 5Y
Compare the powers: Y = 9.
The value of 3Y/2 = 27/2 = 13.5.
4. If the perimeter of a rectangle is four times the length of the rectangle, then the width of the rectangle is what percent of the length?
Let W be the width and L the length of the rectangle.
The perimeter of the rectangle is 2W + 2L.
We can write the following connection: 2W + 2L = 4L è W = L (square).
Therefore the width is 100% of the length and so the answer is 100.
5. In a certain rectangle, the length is three times the width and the perimeter is equal to 64. What is the value of the length of the rectangle?
We can write the following connections: L = 3W and 2L + 2W = 64.
Replace L with 3W and write: 2(3W) + 2W = 64 è 8W = 64 è W=8.
L = 3 x 8 = 24.
6. There are 50 blue balls and 120 red balls in a jar containing 170 balls only. If only blue balls are to be added to the jar so that the probability of randomly picking a blue ball from the jar becomes 1/2, how many blue balls must be added to the jar?
Let X be the number of blue balls that must be added.
We want the portion of the blue balls to be half of the entire amount of balls in the jar and therefore 50 + X (the new number of blue balls) divided be 170 + X (the entire number of balls) should be ½.
. And so if 70 balls are added there'll be 120 blue balls and 240 balls in general.
7. A bag contains 15 red marbles, 12 red marbles and 18 blue ones.
What is the probability of drawing two red balls one after the other?
The probability of drawing a red marble is the number of red marbles divided by the entire number of marbles in the bag.
The probability of drawing the first red marble is (15)/(45) = 1/3.
The probability of drawing the second red marble is (14)/(44) = 7/22.
The joint probability is the multiplication of the probabilities, and therefore the answer is 
.
8. What is the probability of getting a number larger than 3 tossing a fair dice?
While throwing a dice there are 6 results: 1, 2, 3, 4, 5 and 6.
Only three results are over 3: 4, 5 and 6 and therefore the probability is 3 out of 6 or ½.
9. The average (arithmetic mean) of 6 positive integers is 110. The value of two of the integers is 24 and 28 and the other integers are greater than 30.
If all the numbers are different from one another, what is the greatest possible value for any of the 6 integers?
We know the value of 2 integers. If we want one of the integers to be as large as possible, take all the others as small as possible. In other words, take the two integers that are given (24 and 28), take three more integers greater than 30: 31, 32 and 33 and the fourth one would be the greatest.
Write the average formula: 
è
24+28+31+32+33+X = 660 è X = 512, which is the largest possible value since we took the rest as small as possible.
10.
If the sum of 4 consecutive numbers is 220, what is the average
(arithmetic mean) of the first and the last among those numbers?
Let x, x+1, x+2 and x+3 be the four numbers.
We can write the equation: x + x + 1 + x + 2 + x + 3 = 220.
è 4x + 6 = 220 è x = 52.
The average arithmetic mean of the first and the last numbers is
(52 + 55)/2 = 53.5.
11. What is the time elapsed from 12:12 to 23:43, in minutes?
Start from 12:12, add 11 hours to reach 23:12.
Add 31 more minutes to reach 23:43.
Altogether, its 11 hours and 31 minutes.
In minutes its: 11 x 60 + 31 = 691 minutes.
12. What is the angle between the large and the small hand of the clock at 12:30, in degrees?
At 12:30, the angle is not 180o since the hour hand (the small hand) rotated a bit clockwise. Every hour the small hand of the clock moves 30o and so in 30 minutes, it moved 15o.
The angle between the hands of the clock is (180 – 15 = 165) degrees.
It might go easier if you draw a sketch of a clock.
1. If X > 6 and X3X2.5XY = X8, what is the value of X?
These questions are only solved by comparing the powers of both sides, in our case of X.
X3X2.5XY = X5.5 + Y = X8è 5.5 + Y = 8 è Y = 2.5.
2. If 2X+2 = 4X-1, what is the value of X?
These questions are only solved by comparing the powers of both sides.
2X+2 = 4X-1è 2X+2 = 22(X-1)è X+2=2(X-1) è X+2=2X – 2 è X = 4.
3. If X = (0.5)2 and Y = X2, what is the value of (X + 4Y)?
X = (0.5)2 = 0.25.
Y = X2 = (0.25)2 = 0.0625.
X + 4Y = 0.25 + 4(0.0625) = 0.25 + 0.25 = 0.5.
4. If A=1/X and B=X/Y and if X=1/4 and Y=1/5, what is the value of (A+B)?
A = 1/X = 1/(1/4) = 4.
B = X/Y = (1/4) / (1/5) = 5/4.
A + B = 4 + 5/4 = 21/4.
5. Nikki and Mike bought a new house for $120,000.
Their families paid 42% of the price and the rest was divided equally and annually across six years. How many thousand of dollars did Nikki and Mike pay each year?
Their families paid 42% of $120,000 and so all they had to pay themselves is
(100% – 42% = 58%) of $120,000.
0.58 x 120,000 = $69,600.
Each year they would pay a sixth of that amount, thus (69,600/6 = 11,600) and so the answer is 11.6 thousands of dollars.
6. A new computer costs a thousand dollars including tax. If Travis paid for three quarters of his new computer every month for a year, how much did he spent each month assuming that the payments were equal?
Travis paid for only 75% of his computer, thus $750.
He paid that price in 12 equal payments, each ($750 / 12 = $62.5) and so the final answer is 62.5.
7.
The following figure is of a parallelogram.
What is the value of X + Y + Z ?
Look at the upper left triangle, the sum of the angles there should equal 180o and so X + Y + 115 = 180 è X + Y = 65o.
Since the shape is a parallelogram, Z = 115o and so X + Y + Z = 180.
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8.
An isosceles triangle was attached to a rectangle.
If X = 3.5 and Y = 1.5, what is the perimeter of the figure above?
The perimeter of the figure above is made from two sides of the triangle and three more sides of the rectangle, thus X + X + X + Y + Y = 3X + 2Y, which is also equal to 3(3.5) + 2(1.5) = 13.5.
9. If 
and 
, what is one possible value for Y2?
Simplify the expression: è 
è
1 = (XY)Y2è Y2 = 1/(XY) = 1/100 è Y2 = 0.01.
10. If 
, what is one possible value for A?
Compare the powers of 5 in each side.
è 535A < 54-A è 53+A < 54-Aè 3+A < 4-A è 2A < 1 è
A < 0.5 and so one possible value would be anything smaller than 0.5, for example 0.25 or 1/6.
11. The volume of a cylinder is 3 cubic feet.
The radius was increased by three times and the height was increased by 2 times, what is the new volume of the cylinder in cubic feet?
When the radius is increased by 3, the area of the base of the cylinder increases by 9 times. The new volume of the cylinder is 9 x 2 = 18 times greater than before and so the new volume is 3 x 18 = 54 cubic feet.
12.
What is the volume of water needed in order to fill a third of the box in the figure above?
Calculate the volume of the box: V = 12 x 4 x 7 = 336.
A third of that volume is equal to 336/2 = 112.
1. A, B and C are digits between 0 to 9.
CA and AB are double digit numbers and ABA is a three digit
number.
What is the value of ABA?
In the tens digit, we can see that A + B = A and thus B=0.
The sum of two double digit numbers is a three digit number and so its hundreds digit must be one, thus A=1.
The number is therefore 101.
2. A and B are digits between 0 to 9.
A2 = 4B (4B is a double digit number)
The only number squared with a tens digit of 4 is 7 (72 = 49).
And so A=7 and B=9.
A x B = 7 x 9 = 63.
3. X and Y are two digits between 0 and 9. When 36 is multiplied by another double digit number, the result is 3XY.
What is one possible value for Y?
Start with an easy number, 36 times 10 = 360 add 36 to get 396.
And therefore one answer is 0 and the other can be 6.
4. If A and B are positive integers, A < 34 – B and A > 17, what is the greatest possible value of (2A – B)?
Since A + B < 34, take A as 32 and B as 1 and this way A will be the largest and B the smallest. (2A + B) would be equal to (2 x 32 – 1) = 63.
5. If 
, what is the value of Y?
Compare the powers from both sides of the equation.
è 
è 
è Y = 0.5.
6.
In the figure above, B is the center of the circle and A is the center of the square. If the radius of the circle is 
, what is the area of the square?
The radius of the circle is actually half the diagonal of the square and so the diagonal is equal to 
.
The ratio between the sides of a square to its diagonal is 1: and so each of the sides of the square are equal to 2.
The area of the square is simply 2 x 2 = 4.
7. A chocolate box contains only white, sweet and bitter chocolate in the following ratio: 2:3:4 respectively. The sweet chocolate is either with or with out walnuts, and 4 times as many sweet chocolate are with walnuts than with out. If a chocolate is chosen at random, what is the probability that it would be a sweet chocolate with walnuts?
In this question it is smart to plug in numbers.
Say that there are 90 chocolates in the box.
According to the ratio, there are (3/9) x (90) = 30 sweet chocolates.
Since there are four times as many chocolates with walnuts as there are with out there are (4/5 x 30 = 24) walnut chocolate and (1/4 x 30 = 6) with out.
The probability of pulling a sweet chocolate with walnuts is 24/90 = 4/15 or 0.266 or 0.267.
8. The area of a certain rectangle (which is not a square) is 25 inches squared, what is one possible length of its smaller side?
If the rectangle was a square each side would be 5 and so 5 x 5 = 25.
Since the rectangle is not a square, the larger side is bigger than 5 and the smaller side is smaller than 5.
The acceptable answers are 4, 3, 2 and 1.
9. What is the area of a square if the sum of the diagonals is 24
We know (using the Pythagoras principle) that the ratio between the sides of a square to its diagonal is 1:1: and therefore in this specific square, each side is (12/) and the area is 
.
10. The ratio of 8.25 to 66 is the same as the ratio of X to Y. What is the value of (X/Y)2 ?
Since we want to know what is the value of X/Y replace X with 8.25 and Y with 66 è X/Y = 8.25/66 = 1/8.
(X/Y)2 = 1/64 or 0.016.
11. If 2 < X < 7 and -2 < Y < 8, what is the greatest possible value of
(Y – X)?
We want the greatest possible value of (Y – X) and therefore we will take the largest Y and the smallest of X è Y = 7 and X = 3.
The value would be (7 – 3) = 4.
12. If 6 < A < 20 and -12 < B < -5, what is the greatest possible value of AB in absolute value?
Since B is negative and A is positive, AB will also be negative.
If we take the most negative B and the largest A, the result would be the greatest in absolute value. And so A = 20 and B = -12 è AB = -240 and so the answer is 240.
1. If (X4 + 2X2 + X)(X2 + X) = aX6+bX5+cX4+dX3+eX2, what is the value of
(c – a)?
Open the parenthesis to get: X6+X5+2X4+3X3+X2, and therefore
a=1 and c=2 and so (c – a) = 1.
2. If (X2 + X)(X2 + X) = aX4+2X3+cX2+dX, what is the value of
a + b + c + d ?
Open the parenthesis to get: X4+2X3+X2 and so a=1, b=2, c=1 and d=0.
a + b + c + d = 1 + 2 + 1 = 4.
3.
In the figure above, (x,y) is the coordinate of a point found in the middle of the side of the square. 2y-x =
Since the figure is of a square y=5.
The coordinate is of a point in the middle of the side and therefore x=2.5.
2y-x = 10 – 2.5 = 7.5.
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4.
In the figure above, (x,y) is the intersection point of the two lines.
What is the value of y/x?
In order to find the intersection point compare the functions of the line:
3x + 4 = -x +8 è 4x = 4 è x=1 and y = 7.
y/x = 7/1 = 7.
5. A coin is marked with the number 8 on one side and the number 9 on the other. What is the probability of receiving an odd number on the second toss?
The probability every single toss is ½ for 8 and ½ for 9.
Therefore the probability of receiving an odd number on the second toss is 0.5 or ½.
6. A jar contains 4 red balls, 3 green balls and 7 blue balls.
What is the probability of not drawing out a white ball?
Since all the balls are not white, there is not chance of pulling a white ball and the probability is therefore 1.
This question checks if you know that 1 is the highest probability.
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7.
If the area of the square is 30.25, what is the perimeter of the rectangle ABCD?
The area of the square is its side squared and therefore the side of the square is 
. The diameter of the circle is equal to the side of the circle since they are both blocked under the same rectangle.
One side of the rectangle is 5.5 and the larger side is 5.5 x 2 = 11.
The perimeter is 2(5.5 + 11) = 2 x 16.5 = 33.
8. There are 52 questions in a certain exam. If the ratio between the easy questions to the hard questions is 6:7, how many hard questions are there?
If the number of easy questions is 6Q and the number of hard questions is 7Q, we can write: 6Q + 7Q = 52 è 13Q=52 è Q=4 and so 4 x 7 = 28 is the number of hard questions in the exam.
9. Every month Paul works 60 hours for 30 pounds per hour. Due to cutbacks his wage was decreased to 20 pounds per hour. How many additional hours would Paul have to work in order to make the same amount of money each month?
Every month Paul made 60 x 30 = 1,800 pounds.
If he still wants to make the same amount of money he should work
1,800 / 20 = 90 hours.
90 are 30 hours more than 60 and therefore the answer to the question is 30.
10. The denominator of a certain fraction is bigger by 5 then the numerator.
If 3 is added to the numerator and to the denominator, the denominator would be two times bigger than the numerator. What is the original fraction?
Solve this one from the end.
Take a fraction where the numerator is half the denominator, for example 1/2. This fraction is not good; they both need to be over 4 and so this time take 5/10. If you subtract 3 from the denominator and the numerator, the fraction would be 2/7, which does fulfill the requirements.
Therefore the original fraction is 2/7.
11. Every hour Ana sneezes 5 sneezes more than Reese, and each one of them sneezes an equal number of times every hour. If during a whole day both girls sneezed 360 times, how many times did Reese sneeze each hour?
Reese sneezed X times per hour and so Ana sneezed X+5 sneezes.
In one hour, they sneezed X + X + 5= 2X + 5.
In a whole day (24 hours), they sneezed 24(2X + 5) = 48X + 120.
48X + 120 = 360 è 48X = 240 è X=5 and this is the answer.
12. When Tim was 8 he was two times older than Rick during that time.
If today Tim is 14 years of age, how old is Rick today?
During the time that Tim was 8, Rick was half his age, thus 4.
If today Tim is 14 years old (6 years later), Rick is also 6 years older, thus 10 and so the right answer is 10.
1. If the average of three different positive integers is 120, what is the smallest possible value of the median among the three numbers?
All the numbers are integers greater than 0.
The smallest possible value of the numbers is 1 and the smallest possible value of the median is therefore 2, and that is the answer.
The three numbers is 1, 2 and 357è the median is 2.
2 If X2 = 2XY – Y2 + 14.5, what is the value of (X-Y)2?
The expression X2 = 2XY – Y2 + 14.5 can be written as
X2 – 2XY + Y2 = 14.5 è (X-Y)2 = 14.5 and so the answer is simply 14.5.
3. If 17XY = 34X + 51X, what is the value of (3.5)Y?
Simplify the expression: 17XY = 34X + 51X è 17XY = 85X è
Divide by 17X è Y = 5 and therefore 3.5Y = 3.5 x 5 = 17.5.
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4.
Note: Figure not drawn to scale
In the figure above, what is the value of A in degrees?
A, 17o and 97o are all vertical angles to the inner angles of the triangle and therefore equal. The sum of the angles in the triangle is 180o and so we can write the following equation: 180 = 97 + 17 + A è A = 180 – 97 – 17 = 66 degrees.
5. The bank gave Elaine a loan with an interest of 8% on the original amount per month. If Elaine loaned $1,525, how much interest will she pay over a period of three months (in dollars)?
Every month, there is an 8% interest on 1,525, which is 0.08 x 1525 = $122 per month. Elaine will pay 3 x 122 = $366 to the bank.
6. If in a certain jar there are X+1 black marbles and 2X+2 white marbles, what is the probability of randomly pulling a white ball?
The total amount of balls in the jar is X + 1 + 2X + 2 = 3X + 3.
The probability of pulling a white ball is 
or 0.667.
7. There are 30 students in a room. If 3 boys are taken out and now the probability of randomly picking a boy is one third, what is the original number of boys in the room?
After 3 boys are taken out, there'll be 27 students in the room.
The probability of picking a boy now is 1/3 and so there are exactly 9 boys in the room. The original number of boys in the class is therefore 9+3 = 12.
8. If the sum of the first 4 out of 8 consecutive numbers is 40, what is the sum of the rest?
Let the first 4 numbers be: X, X+1,X+2 and X+3.
Their sum is 40 è 4X + 6 = 40 è 4X = 34.
The rest of the numbers are X+4,X+5,X+6 and X+7.
The sum of those numbers is 4X + 4 + 5 + 6 + 7 = 4X + 22 = 56 and this is the right answer.
9. If the points A(5, 5), B(11, 5), C(11, 12) and D(5, 12) are vertices of a rectangle, what is the area of the rectangle?
Draw an axis system with the coordinates, as you can see a rectangle is formed.
One side of the rectangle is (11 – 5 = 6) and the height is (12 – 5 = 7).
The area of the rectangle is 6 x 7 = 42.
10. 100 square feet of a basketball floor parquet costs 5 dollars and 25 cents. How much money will it cost to cover a court with the following dimensions: 60 feet on 100 feet?
The area of 60 feet on 100 feet is 6,000 square feet.
If 100 square feet cost 5.25 dollars, 6,000 will cost
(6,000 / 100 = 60) x 5.25 = $315 and so the answer is 315.
11. The expression 
is how much more than 3Q?
Simplify the expression by joining the two variables: 
.
è 
.
The expression is larger than 3Q by 7/2 or 3.5.
12. 62.5, 50, 40, …
In the sequence above, each term after the 1'st term is 20% less of the term preceding it. What is the value of the 5'Th term of this sequence?
Each term in this sequence is 80% of the previous term.
40 is 80% of 50 and so the fourth term will be 0.8 x 40 = 32 and the fifth term will therefore be 0.8 x 32 = 25.6 and this is the right answer.
1. A, B and C are three points that lie on a straight line. If the distance between A and C is 30 and the ratio between AB and BC is 1:3, what can be the distance between B and C?
Draw a line with the relevant points on it.
If the ratio between AB and BC is 1:3 there are two options, B can be somewhere between A and C or it can be on the outside.
If B is between A and C: AB = 7.5 and BC = 22.5.
If B is on the out side, AB = 15 and BC = 45.
One acceptable answer is 22.5 and the other is 45.
2. What is the difference between the largest and the median among 5 consecutive integers that sum up to 230?
Let the numbers be: X, X+1, X+2, X+3 and X+4.
The sum of these numbers is 5X + 10 = 230 è 5X = 220 è X=44.
The largest of the numbers is 44+4 = 48 and the median is X+2, thus 46.
The difference between the two numbers is 2.
3. If the sum of 6 consecutive odd numbers equals 168, what is the value of the smallest of these numbers?
Let the numbers be: X, X+2, X+4, X+6, X+8 and X+10.
The sum of these numbers is 6X + 30 = 168 è 6X = 138 è
X = 23 and so the smallest number is 23.
4. A certain salesman receives $5 per hour plus 5% bonus on his sales.
If Mark worked for 122 hours on April and had sales of $12,000, what will be his income on April (the answer should be in thousands of dollars with out the dollar sign)?
Mark's income is made of two parts:
1) 122 x $5 = $610.
2) 0.05 x $12,000 = $600.
In total, his income is 610 + 600 = $1,200.
The answer should be in thousands of dollars and therefore the right answer is 1.2.
5.
In the figure above, what is the area of the shaded region?
The area of the shaded region is the area of the large square minus the areas of 4 right triangles.
The area of the large square is (2 + 4)2 = 36.
The area of each of the right triangles is 2 x 4 x 0.5 = 4.
The area of the shaded region is therefore 36 – 4(4) = 20.
6. Jack makes nails out of a steel bar.
If each nail weighs 3% of the total weight of the steel bar, how many whole nail can be made from 2 bars?
If each nail is 3% of 1 bar, each bar can make 33 and a third nails and since we are looking for whole numbers, from each bar, 33 nails can be made.
From 2 bars, 66 nails can be made and so this is the answer.
7. A round pizza is equally divided into slices. If each slice is 45o and the total weight of the pizza is 1.5 Kg, what is the weight of one slice of pizza, in Kg?
Since each slice is 45o, there are (360/45 = 8) slices of pizza.
Each slice will weigh 1.5/8 = (3/2)/8 = 3/16 or 0.188.
8. If AC – AD = 3 and BD – BC = 6, what is the value of the expression
(A – B)(C – D)?
Open the parenthesis of the expression:
(A – B)(C – D) = AC – AD – BC + BD = (AC – AD) + (BD – BC) = 3+6=9.
9. If A2 + B2 = 17.5 and AB = 3.2, what is the value of the expression
(A + B)2?
Open the parenthesis of the given expression:
(A + B)2 = A2 + 2AB + B2 = 17.5 + 3.2 = 20.7.
10. For all positive integers x, y and z let R(x, y, z) be defined as 
.
What is the value of R(5, 2, 8) + R(8, 8, 16)?
Use the pattern in the question and plug in the numbers:
R(5, 2, 8) = (5 + 8)/2 = 6.5 and R(8, 8, 16) = (8 + 16)/8 = 3.
The answer is therefore 6.5 + 3 = 9.5 or 19/2.
11. The action X is defined as followed:
If 
, xXy = x – 2y
If 
, xXy = 2x – y
(7X4)X0.25 = ?
(7X3) is equal to 7 – 3 x 2 = 1 since 7 > 4.
(1X0.25) = 1 – 2 x 0.25 = 0.5.
The acceptable answer is 0.5 or ½.
12. The average (arithmetic mean) grades of a certain class is 88. If the average grade of half of the kids is 91.5, what is the average of the rest?
Since half of the kids in the class have an average grade of 3.5 points above the average, the rest must have an average grade of 3.5 points lower than the average, thus 88 – 3.5 = 84.5, and that is the answer.
1. If 3Y + 14 = 5Y + 3, what is the value of Y?
Solve: 3Y + 14 = 5Y + 3 è -2Y = 3 – 14 = -11 è Y = 5.5 or 11/2.
2. There are 230 basketball cards in a stack. One of these cards is to be selected at random. If the probability that an Upper-Decker card will be selected is 2/5, how many Upper-Decker cards are in the stack of cards?
All we need to calculate is how much is 2/5 of 230.
230 x 2/5 = 460/5 = 92 cards and this is the right answer.
3. If A and B are each negative integers and A + B = -4, what is one possible value of 
?
A and B can be (-2, -2) or (-1, -3) or (-3, -1).
can be equal to 0 or 2.
One answer is 0 and the other is 2.
4. In a certain class, there are 3 times more boys than girls.
What is the ratio between the number of girls to the number of boys?
(Grid the ratio as a fraction)
If there are three times more boys than girls, ¼ are girls and ¾ are boys.
è 24 x ¼ = 6 and 24 x ¾ = 18.
The ratio is therefore 6/18 = 1/3.
5. Alan spent half of her allowance on cloths and 1/3 of the rest on fast food.
What fraction of the total is left of Alan's allowance?
Alan spent ½ of the allowance on cloths and 1/3 of what's left (1/2) on fast food. The remainder fraction is (2/3) x (1/2) = 2/6 = 1/3.
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6.
If the volume of the solid figure above is 40.5, the base of the triangle is 6 and the depth of it is 3, what is the height of the dotted triangle?
Let the height of the triangle be H.
The volume of the solid figure above is 
è 6 x H x 3 = 81.
H = (81)/(18) = 4.5 or 9/2.
7. If 5 People came to a certain party and each of them gave all the others a present, how many presents were handed out in the party?
The number of presents given is 5 x 4 = 20.
Pay attention that if person A gives person B a present and person B gives him a present as well, it counts as two presents handed out.
8. How many 4 digits numbers with different digits can be made with
1, 2, 3, 4 and 5 if the two outer digits (the left and the right) are odd?
Start with the outer digits: the left has 3 options (1, 3, 5).
The right has two digits left (after the left took one).
The two inner digits have 3 options and 2 for the last one.
Multiply the possibilities to get all the combinations:
3 x 2 x 3 x 2 = 36.
9. How many phone lines should be stretched between 6 houses that are built in a circle, so that every two houses will be connected?
There are 6 houses and each one is connected to the other 5. Apparently, it seems like the answer is 6 x 5 = 30 but we counted each phone line twice because when house A is attached to house B, house B is also attached to house A and therefore the result should be divided by two.
The right answer is 15.
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10.
What is the perimeter of the figure shown above?
First, we'll have to find S.
The upper part of the figure can be divided into two triangles that each has a side of 6/2 = 3 and (9 – 5 = 4) and so by using the Pythagoras principle,
.
Now that we have S, calculate the perimeter: 6 + 5 + 5 + 5 + 5 = 26.
11. The action X is defined as X X Y = X-Y.
What is the value of 4 X (-1/2) ?
Use the pattern and calculate: 4 X (-1/2) = 40.5 = 2.
12. If X/Y = 3, W/S = 2 and S/Y = 4, what is the value of X/W ?
Cross multiply to get the answer:
3 x ¼ x ½ = 3/8 or 0.375.
1. A beer barrel can contain up to 50 liters of beer when full.
The barrel is 3/5 full. If one liter of beer costs one dollar and 40 cents, how many dollars worth of beer must be purchased to fill the remainder of the barrel? (Disregard the $ sign when gridding your answer)
All we need to do here, is to calculate how many liters are missing in the barrel and multiply that number be 1.4 (the price per liter beer).
2/5 of 50 liters is 20 liters of missing beer, which cost 20 x 1.4 = $28 and so the answer to the question is 28.
2. If 15,000 + 13 x 104 = Q x 104, what is the value of Q?
Solve: 15,000 + 13 x 104 = 15,000 + 130,000 = 145,000.
Compare: 145,000 = X x 10,000 è X = 14.5 or 29/2.
3. If 
,
M =
Simplify the expression: 
. When comparing the powers, we can see that M = 0.5 or ½.
4. 
= ?
Simplify the expression: 
or 2.2.
5. If A + 4B = 10 and 4A + B = 16, what is the value of A + B ?
Solve the equations.
Multiply the first by (-4) and add the second equation:
(-4A – 16B = -40) + (4A + B = 16) = -15B = -24 è B = 24/15 = 8/5.
A = 12 – 4B = 12 – 32/5 = 28/5.
A + B = 8/5 + 28/5 = 36/5 or 7.2.
6. What is the number that satisfies the following four conditions?
1. It is an integer between 111 and 999 inclusive.
2. All of its digits are prime numbers.
3. The number is divisible by 7.
4. All of its digits are equal.
There are several options: 111, 222, 333, 555 and 777.
The only number among those five numbers that is divisible by 7 is 777.
7. For all integers K, let #K# be defined as follows:
#K# = 3K – K3 if K is odd.
#K# = 2K – K2 if K is even.
If #4# – #3# = H, what is the value of H2 ?
4 is even and so #4# = 8 – 16 = -8.
3 is odd and so #3# = 9 – 27 = -18.
And so H = -8 + 18 = 10.
H2 = 102 = 100.
8. For all integers S, let @S be defined as follows:
@S = 2S2 if S is divisible by 2.
@S = 3S2 if S is divisible by 3.
@S = 0 if S is not divisible nor be 2 or 3.
What is the value of @1 + @2 – @3 + @4 ?
Calculate each of the variables:
@1 = 0 since 1 is not divisible by 2 or 3.
@2 = 2 x 22 = 8.
@3 = 3 x 32 = 27.
@4 = 2 x 42 = 32.
The expression @1 + @2 – @3 + @4 = 0 + 8 – 27 + 32 = 13.
9.
A solid cube with dimensions of 8 inches by 3 inches by 2.5 inches is to be painted on all of its faces. If painting one inch squared costs $1.2, how much will it cost to paint the entire cube?
(Disregard the $ sign when gridding your answer)
The total surface area of the cube is 2(8 x 3) + 4(3 x 2.5) =
2(24) + 4(7.5) = 78 inches squared.
The paint should cost 78 x $1.2 = $93.6.
10. A police car, traveling at a speed of 120 mph, is chasing a suspicious car that is traveling at a speed of 100 mph. If the police car caught up with the suspicious car after two hours, what was the initial distance between them in the beginning of the pursuit?
This is a motion problem with an emphasis on relative speed.
The difference in their speeds is 20 mph and so the distance between them is getting shorter by the second.
If it took two hours to reduce the initial distance, the distance was 20 x 2 = 40 miles.
11. If the average (arithmetic mean) of 2Y, 3Y + 4, 2X – 1 and 3X + 2 is 7, what is the value of (X + Y)?
Use the average formula: 
è
5Y + 5X + 5 = 28 è 5(X + Y) = 23 è X + Y = 23/5 or 4.6.
12. If the average (arithmetic mean) of 3A, 5A – 4, 3A + 2, 7A – 3, A and 5A – 5 is 2, what is the value of the second to largest among these numbers?
First find the value of A by using the average formula:
= 2 è 24A – 10 = 12 è
24A = 22 è A = 22/12 = 11/6.
The second largest of these numbers is 3A + 2, which is equal to
3 x 11/6 + 2 = 5.5 + 2 = 7.5 or 15/2.
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