Tài liệu The Grid question practice ppt

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Grid questions practice

This file contains 120 Grid questions with full answers and
explanations.


Good luck on your test.

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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.
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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.
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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.




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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.
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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.
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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.
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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.

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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.



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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.