For more material and information, please visit Tai Lieu Du Hoc at www.tailieuduhoc.org Grid questions practice This file contains 120 Grid questions with full answers and explanations. Good luck on your test. For more material and information, please visit Tai Lieu Du Hoc at www.tailieuduhoc.org 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 (X 2 – Y 2 )/ Z 2 . #(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 4 3 5 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 4 69 3 4 23 3 4 3 5 =× ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ =× ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ teaspoons. 6. If 0.2X = 15, what is the value of 10 X ? Solve the equation: 0.2X = 15 Î 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 X3 ? Solve the equation: 0.66X = 4 – 0.34X Î X = 4. For more material and information, please visit Tai Lieu Du Hoc at www.tailieuduhoc.org X3 = 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 Î X + 6X + 20 = 24 Î 7X = 4 Î 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 Î 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 115.5 o , 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 180 o and therefore we can write the following connection: angle 1 + angle 2 + 115.5 o = 180 o Î the sum of angle 1 and 2 is equal to (180 – 115.5 = 64.5 o ). 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. Î Y = 4 and X = 2. For more material and information, please visit Tai Lieu Du Hoc at www.tailieuduhoc.org Their product is 8 and the square of their product is 8 2 = 64. 1. If A = 13 is a solution to the equation A 2 + 2A - B = 0, what is the value of B? Solve the equation with while replacing A with 13: 169 + 26 – B = 0 Î B = 169 + 26 = 195. 2. If X XX 10 25 7 =− , what is the value of X ? Take X XX 10 25 7 =− , and multiply it by 10X Î 14X 2 – 5X 2 = 100 9X 2 = 100 Î X = 3 10 ± and so the right answer is 10/3 or 3.33. 3. AI: The Rutherford’s yearly expenses Electricity 35% 50% Education On the basis of the information in the graph above, if the Rutherford family spends $825 per year on education, how much do they spend on electricity? (When gridding, disregard the $ sign) The Rutherford family spends $825 on education, which is (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. For more material and information, please visit Tai Lieu Du Hoc at www.tailieuduhoc.org A B D C 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 Î DC = AB Î 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 A.B be defined as 33 2 +B A . If 5.X = 2/5, what is the value of X? According to the pattern, 5.X can be written as 33 52 +⋅ ⋅ X = 2/5 Î Cross multiply to get: 50 = 6X + 6 Î 6X = 44 Î 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 Î 31X = 124 Î X = 4 miles. Meg traveled 16 x 4 = 64 miles on her last day. For more material and information, please visit Tai Lieu Du Hoc at www.tailieuduhoc.org 8. )3,2(A )3,8(B ),8( YC 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 = 5 24 4 5 6 = = 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 Î T = 16. Therefore Tiffany has 16 braids. 11. Paul forgot his girlfriend’s 7-digits phone number. For more material and information, please visit Tai Lieu Du Hoc at www.tailieuduhoc.org 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 Î X 2 = 9 Î X = Y = 3. The sum of the numbers is 3 + 3 = 6. 2. If 64 5 = 2 2X , what is the value of X? Rewrite the expression: 64 5 = 2 2X Î (2 6 ) 5 = 2 2X Î 2 30 = 2 2X Compare the powers: 2X = 30 Î X = 15. 3. If 125 3 = 5 Y , what is the value of 2 3 Y ? Rewrite the expression: 125 3 = 5 Y Î (5 3 ) 3 = 5 Y Î 5 9 = 5 Y 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. For more material and information, please visit Tai Lieu Du Hoc at www.tailieuduhoc.org We can write the following connection: 2W + 2L = 4L Î 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 Î 8W = 64 Î 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. 701702100 2 1 170 50 =⇒+=+⇒= + + XXX X X . 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 66 7 22 7 3 1 =× . 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. For more material and information, please visit Tai Lieu Du Hoc at www.tailieuduhoc.org 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: 110 6 3332312824 = +++++ X Î 24+28+31+32+33+X = 660 Î 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. Î 4x + 6 = 220 Î 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 180 o since the hour hand (the small hand) rotated a bit clockwise. Every hour the small hand of the clock moves 30 o and so in 30 minutes, it moved 15 o . 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. For more material and information, please visit Tai Lieu Du Hoc at www.tailieuduhoc.org 1. If X > 6 and X 3 X 2.5 X Y = X 8 , what is the value of X? These questions are only solved by comparing the powers of both sides, in our case of X. X 3 X 2.5 X Y = X 5.5 + Y = X 8 Î 5.5 + Y = 8 Î Y = 2.5. 2. If 2 X+2 = 4 X-1 , what is the value of X? These questions are only solved by comparing the powers of both sides. 2 X+2 = 4 X-1 Î 2 X+2 = 2 2(X-1) Î X+2=2(X-1) Î X+2=2X – 2 Î X = 4. 3. If X = (0.5) 2 and Y = X 2 , what is the value of (X + 4Y)? X = (0.5) 2 = 0.25. Y = X 2 = (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. [...]... is the highest probability A B D 7 C 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 30.25 = 5.5 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. .. $1,200 The answer should be in thousands of dollars and therefore the right 4 answer is 1.2 2 2 4 5 4 2 2 4 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... 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 13Q=52 Q=4 and so 4 x 7 = 28 is the number of hard questions in the exam 9 Every month Paul works 60 hours... will be the largest and B the smallest (2A + B) would be equal to (2 x 32 - 1) = 63 5 If 5 ⋅ 6 = 30 Y , what is the value of Y? Compare the powers from both sides of the equation 5 ⋅ 6 = 30 Y 30 = 30Y 30 0.5 = 30 Y Y = 0.5 •B 6 A• 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 2 , what is the area of the square? The radius of the circle... 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 5X = 220 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... 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... 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 6 3 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... 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: 2 and therefore in this specific square, 2 144 ⎛ 12 ⎞ = 72 each side is (12/ 2 ) and the area is ⎜ ⎟ = 2 ⎝ 2⎠ 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. .. 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 y 3 (x, y) x (5,0) 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... integers is 120, what is the smallest possible value of the median among the three numbers? All the numbers are integers greater than 0 For more material and information, please visit Tai Lieu Du Hoc at www.tailieuduhoc.org 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 357 the median is 2 2 . 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. 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