Tài liệu The Grid question practice ppt

If each digit can be between 0 to 9, how many arrangements are possible for Paul’s girlfriend number?. Px is defined as the greatest possible prime number that is less than x.. The great

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

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

7 If 0.66X = 4 – 0.34X, what is the value of 3 X ?

Solve the equation: 0.66X = 4 – 0.34X Î X = 4

3 = 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 115.5o, 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 = 180o Î 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

Î 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:

50%

Educatio n

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

A B

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

3 3

2 +

B

A

If 5.X = 2/5, what is the value of X?

According to the pattern, 5.X can be written as

3 3

5 2 +

⋅

⋅

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

8

) 3 , 2 (

) , 8

( Y

C

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

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 X2 = 9 Î 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 = 22X Î (26)5 = 22X Î 230 = 22X

Compare the powers: 2X = 30 Î X = 15

3 If 1253 = 5Y, what is the value of

2

3Y

?

Rewrite the expression: 1253 = 5Y Î (53)3 = 5Y Î 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 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

70 170

2 100 2

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

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

6

33 32 31 28 24

= + + + +

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 180o since the hour hand (the small hand)

rotated a bit clockwise Every hour the small hand of the clock moves 30oand 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 = X8 Î 5.5 + Y = 8 Î 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

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

o X

o Y

o

Z

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

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 X ⋅Y = 100 and 2 4 4 2 5 7

Y X Z Y X

Z ⋅ ⋅ = ⋅ ⋅ , what is one possible value for Y2?

Simplify the expression: 2 4 4 2 5 7

Y X Z Y X

125 , what is one possible value for A?

Compare the powers of 5 in each side

< 54-A Î 3+A < 4-A Î 2A < 1 Î

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

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

CA and AB are double digit numbers and ABA is a three digit +AB

number ABA

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)

?

=

⋅ B

A

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 5⋅ 6 =30Y, what is the value of Y?

Compare the powers from both sides of the equation

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?