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1
Q

If 2 similar triangles have sides in the ratio of x/y, what is the ratio of their areas?

A

x2/y2

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2
Q

(x2 + 6x + 9) / (x + 3) = 7

x = ?

A

x = 4

We cannot accept - 3 as a result because it would make the denominator of the original equation = 0 and division by 0 is undefined.

Check Manhattan Algebra p. 89

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3
Q

The median of a triangles is 2/3 long from ___ to___, and 1/3 long from ___ to ___.

A

The median of a triangles is 2/3 long from the vertex to centroid, and 1/3 long from cenroid to the midpoint.

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4
Q

Triangles

What is the center of an circumscribed circle?

A

The center of the circumscribed circle is the intersection of altitudes.

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5
Q
A
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6
Q

Solve:

(x(x - 3)(x2 +5)) / (x - 4) = 0

A

x(x - 3)(x2 + 5) = 0

x = 0, or x = 3, or x2 + 5 = 0

x2 + 5 cannot equal 0 for any x, thus x = 0 or 3.

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7
Q

What is the formula for the sum of factors of a certain number?

A

First we have to find out the prime factors of the number. Let those be: x2 * y3 * z1

Thus, the formula is:

((x2+1 -1)(y3+1 -1)(z1+1 -1)) / ((x - 1)(y - 1)(z - 1))

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8
Q

Given that |x - 2| < 5, what is the range of possible values of x?

A

x - 2 < 5

x < 7

-(x - 2) < 5 | multiply both sides by -1

x - 2 > -5

x > -3

-3 < x < 7

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9
Q

xy = 8, xz = 9, yz = 2

What is the aproximate value of xyz?

A

xy * xz * yz = 8 * 9 * 2

x2y2z2 = 144

xyz = 12

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10
Q

Factor:

38 - 28 =

A

38 - 28 =

(34 + 24)(34 - 24) =

(34 + 24)(32 + 22)(32 - 22) =

(81 + 16)(9 + 4)(9 - 4)

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11
Q

y2/36 - 9/x2 = 18

y/6 + 3/x = 4

Solve for x, explain the process.

A

First of all, we have to detect the quadratic equation:

y2/36 - 9/x2 = 18

(y/6 - 3/x)(y/6 + 3/x) = 18

We know that: y/6 + 3/x = 4 (1st equation)

So we can solve the first equation:

(y/6 - 3/x)4 = 18

(y/6 - 3/x) = 18/4 = 9/2 (2nd equation)

Now we have a 2 variables, 2 equations situation and can solve for x.

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12
Q

A hexagon is composed of six identical _____.

A

Equilateral triangles with each side = the length of one side of the hexagon.

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13
Q

Explain direct and inverse proportionality.

A

Direct proportionality = two quantities always change by the same factor. For example X and Y are directly proportional. If X is tripled, Y is tripled too.

Inverse proportionality = two quantities change by reciprocal factors. When X is tripled, Y is multiplied by 1/3 (or divided by 3).

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14
Q

xr/s =

A
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15
Q

(xr)(yr) =

A

(xy)r

For example:

22 * 32 = 62 = 36

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16
Q

We know that 0 < ab < ac, is a negative, if b > c? Explain.

A

ab and ac in order to be greater than zero all elements must be positive or negative.

Knowing that b > c is enough to conclude that a is negative. To make both statements true, all elements individually must be negative. Test cases if you want.

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17
Q

-(2)4 =

A

-16

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18
Q

What are the cycle of powers (the unit digit cycle when we add an infinite exponent x) of:

4

9

A

4: 4, 6
9: 9, 1

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19
Q

0.05 * 10m - k = 5 * 107

What is the value of m - k?

A

0.05 * 10m - k = 5 * 107

5 * 10m - k - 2 = 5 * 107

m - k - 2 = 7

m - k = 9

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20
Q

Solve the fastest way possible the following equation:

(x + 4)2 = 9

A

Because the quadratics in most cases have two possible solutions and the square of 2 of a number leave the posibility that the number can be positive or negative, we can rewrite the equation as follows:

x + 4 = 3

or x + 4 = -3

Thus,

x = -1 or x = -7

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21
Q

Triangles

The midpoint of a right triangle is equidistant from _______.

A

All 3 vertices.

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22
Q

(z + 3)2 = 25

z = ?

A

z + 3 = + or - 5

z = 2 or z = -8

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23
Q

What are the cycle of powers (the unit digit cycle when we add an infinite exponent x) of:

2

5

7

A

2: 2, 4, 8, 6
5: 5 always
7: 7, 9, 3, 1

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24
Q

What is the range of value for x?

x2 - x < 0

A

x2 < x

0 < x < 1

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25
What is a *midsegment* of a triangle and what are its properties?
Midsegment is a line that connects the midpoints of two different bases of a triangle. - The midsegment is parallel to the 3rd side. - The midsegment is half the length of the 3rd side. - The smaller triangle formed (BED) is 1/4 of the big triangle (ABC)
26
f(x) = x2 + 1 g(x) = 2x For what value of x does f(g(x)) = g(f(x))?
Square root of 1/2 Check Algebra page 172
27
Solve: 3x2 - 3 = 8x
3x2 ​- 3 = 8x 3x2 - 8x - 3 = 0 (3x + 1)(x - 3) = 0 3x + 1 = 0, x = -(1/3) or x = 3
28
Two lines are perpendicular on the coordiante plane when \_\_\_\_\_.
When there slopes are negative reciprocals. For example: x's reciprocal = 2/3 y's reciprocal = - 3/2
29
How many 0s are at the end of the 42! ?
Formula: n/5 + n/52 + n/53 ... + n/5k, where 5k \< n 42/5 + 42/52 = 8 + 1 = 9 (number of 0s) (take notice, the remainders are excluded)
30
Find the number of factors of 450. Explain the process.
1. First we have to do the prime factorization of 450. We find out that 450 = 52 \* 21 \* 32 2. We isolate the exponents, add 1 to each of them and multiply them. (2 + 1)(2 + 1)(1 + 1) = 3 \* 3 \* 2 = **18** (the number of factors)
31
The amount of electrical currect that flows through a wire is inversely proportional to the resistance in that wire. If a wire currently carries 4 amperes, but the resistance is then cut of to one third of its original value, how many amperes of electrical current will flow through the wire?
We have a inverse proportionality. When x is multiplied by 2, y is multiplied by 1/2. 4 \* r1 = x \* r1 \* (1/3) For this problem we should use smart numbers. r1 = 3 4 \* 3 = x \* 1 x = 12
32
Is x2 \> y2? (1) x \> y (2) x \> 0
(E) Check Algebra page 190
33
Number of powers of a prime in a factorial. What is the number of powers of 3 in 49! ?
Formula: n/p + n/p2 + n/p3 ... n/pk, where pk \< n 49/3 + 49/9 + 49/27 = 16 + 5 + 1 = 22 322
34
Find the remainder of (32 \* 26) / 23 the quickest way possible. What is the formula? Explain.
Rof (32 \* 26) / 23 = Rof (9 \* 3) / 23 = 27/23 = 4 (remainder) Formula: Rof (x \* y)/n = Rof ((Rof x/n) \* (Rof y/n)/n
35
Any triangle, whos sides are in the ratio of 3:4:5 is a _____ triangle.
**right triangle**
36
6b = 2a Is a \> b? Explain.
Cannot be determined. If b = -2, than a = -6 and b\>a If b = 3, than a = 9 and b
37
What is the *origin* of the coordinate plane?
The origin is the point (0,0)
38
What is the number of factors of 1372? Explain the process.
1. First we have to do the prime factorization of 1372 We find out that 1372 = 22 \* 73 2. We isolate the exponents, add 1 to each of them and multiply them. (3 + 1)(2 + 1) = 4 \* 3 = **12** (the number of factors)
39
w(3w - 6) = 0 w = ?
**w = 0** or 3w - 6 = 0 3w = 6 **w = 2**
40
Solve for x2: 9x4 – 4y4 = 3x2 + 2y2
((3x2 + 2y2)(3x2 - 2y2​)) / 3x2 + 2y2 = 1 3x2 - 2y2 = 1 3x2 = 2y2 + 1 x2 = (2y2 + 1) / 3
41
Ben has less than twice as many apples as bananas in inventory. Nr of apples = a Nr of bananas = b Express algebraicaly.
a\<2b
42
Jane's speed is directly proportional with 52 and inversely poportional with 23. Represent Jane's speed using the knowledge of direct and inverse proportionality.
Let the unknown *j* represent Jane's speed. When x is directly proportional with *j* this = as bigger gets x, that bigger gets *j.* j \* x When x is inversely proportional with *j* this = as bigger gets x, as smaller gets *j.* j/x Now we know that Jane's speed can be represented as: j\*52/23 = 25j/8
43
Is 2161/3 (or cube root of 216) an integer? How can we check.
We do the prime factorization, for the cube root of any number to be an integer, the number of every prime factor must be divisible by 3. It must have at least 3 (or 6, or 9....) of every prime factor. In our case 216 = 3 \* 3 \* 3 \* 2 \* 2 \* 2 = 63 It has three 3s and three 2s, so its cube is an integer.
44
**Triangles** What is the center of an inscribed circle?
The center of an inscribed circle is the intersection of *angle* *bisectors*.
45
(-4)2 =
16
46
(x + y)2 - (x - y)2 =
4xy Source: GMAT Club book.
47
**Quadratic formula** What can tell you the *discriminant* (the value under radical of the quadratic formula) about the number of posible solutions?
The *discriminant* \> 0 = two solutions The discriminant \< 0 = no solutions The discriminant = 0 = one solutions
48
An exterior angle of a triangle is equal to the sum of \_\_\_\_\_\_\_.
Is equal to the sum of opposite angles.
49
a2 = 25 a =
a = 5 or a = -5
50
What does it mean: 1. That x is directly proportional with y. 2. That x is inversely proportional with z.
1. As bigger gets y, that bigger gets x. y \* x 2. As bigger gets z, that smaller get x. x/z
51
Find the unknowns: 3x + 4y = 17 6x + 8y = 34 Explain.
The unknowns cannot be found because the equations are proportionatey equivalent.
52
Inequalities Given that *ab \< 0* and *a \> b* which of the following must be true? I. a \> 0 II. b \> 0 III. 1/a \> 1/b
I. and III. are true.
53
(1/8)-4/3 =
84/3 = 24 = 16 Check Manhattan Algebra p. 72
54
What is the *altitude* of a triangle? What is the *median* of a triangle? What are the differences?
The altitude of the triangle is a perpendicular from the base to the opposite vertex. Perpendicular line forms two 90 degree angles with the base line. The median of a triangle is a line from the midpoint of the base to the opposite vertex. Triangles have 3 altitudes and 3 medians. The difference between median and altitude is that the altitude line has to be perpendicular to the base, while the median connects to the midpoint.
55
even \* even = even \* odd = odd \* odd =
even \* even = even; even \* odd = even; odd \* odd = odd. Division of two integers can result into an even/odd integer or a fraction.
56
(x + y + z)2 =
= x2 + y2 + z2 + 2(xy + yz + zx)
57
What is the sum of factors of 2484?
2484 = 22 \* 33 \* 231 ((22+1 -1)(33+1 -1)(231+1 -1)) / ((2-1)(3-1)(23-1) = = (7 \* 80 \* 528) / (2 \* 22) = 6720
58
Solve: (3xy - 9y) / (x - 3)
(3xy - 9y) / (x - 3) = (3y(x - 3)) / (x - 3) = 3y
59
Explain the difference of the location of a *hundreth* *digit* and a *hundreds digit* of a number.
* Hundreths* are located **right of the decimal dot.** * Hundreds* are located **left of the decimal dot.**
60
What is the range of possible values of: |x2 - 4|
|x2 - 4| must be greater then or equal to 0, because whether x is positive or negative, x2 will always be positive.
61
What is the sum of factors of 450?
450 = 52 \* 32 \* 21 ((52+1 -1)(32+1 -1)(21+1 -1)) / ((5-1)(3-1)(2-1)) = = (124 \* 26 \* 3) / (4 \* 2) = 1209
62
Simplify: 4 / (3 - square root of 2)
1. First we multiply both sides by 3 + square root of 2 to create a quadratic in the denominator. 2. Simplify and you will get **(12 + 4 \* square root of 2) / 7**
63
What is an *angle bisector* of a triangle? What is the difference between a *median* and an *angle bisector* of a triangle? What is the angle bisector theorem?
A line that bisects the vertex of the triangle and goes to the opposite side. A median bisects a side and does not necessarily bisects a vertex. A angle bisector bisects a vertex and does not necessarily bisects a side. The angle bisector theorem:
64
If each number in a sequence is 3 more than the previous number and the 6th number = 32, what is the 100th?
From the 6th to the 100th jump there are 94 jumps (100 - 6 = 94). So, there are 94 jumps of 3, which equals to 94 \* 3 = 282. To find the 100th term we will add 282 with 32. Thus the **100th term = 314.**
65
x3 + 2x2 - 3x = 0 x = ?
x = 0 or x = -3 or x = 1
66
Solve for x:
(x + 4)2 = 9 x + 4 = 3 or x + 4 = -3 Thus, x = -7 or x = -1
67
**True / False** |x| = square root of x2
True
68
(5 + 7)(x2 - 4) = z - 17 What can be the range of possible results of "z - 17"?
Since x2 is a square, x2 - 4 can only be equal with 0 or greater than 0, y - 17 can be equal with 0 or to another infinite positive term
69
Simplify: (x/5)-2
(5/x)2 = 25/x2
70
Simplify: (3x)4
81x4
71
Can 20x be the greatest divisor of 20y and 35x? Explain.
**No.** First we have to make the factorization of the numbers: 20x = 4 \* 5 \* x 20y = 4 \* 5 \* y 35x = 7 \* 5 \* x It is clear that 20x could easily be the greatest divisor of 20y, if for example y = 8 and x = 4, the hard part is to find out if it is a divisor of 35x. It can be noticed that 20x is missing a 7 to have common factors with 35, however if x = 7, 35x will have two sevens as prime factors and thus will not be divisible by 20x.
72
What are the cycle of powers (the unit digit cycle when we add an infinite exponent x) of: 3 6 8
3: 3, 9, 7, 1 6: 6 always 8: 8, 4, 2, 6
73
Explain *positive* and *negative* correlation.
* Positive correlation* - X and Y are positively correlated means that when X grows, Y grows too, and viceversa. * Negative correlation* - X and Y are negatively correlated means that when X grows, Y shrinks and when Y grows X shrinks.
74
We have an equilateral triangle and an isosceles triangle (with two sides equal) with the same perimeter, which has bigger area?
For a given perimeter **equilateral triangle** has the largest area.
75
Is b \< 0 if b3 \< b
Not necessary. It is not said that b is an integer. Test cases. If b = - 2, the answer is yes. - 8 \< - 2 If b = 1/2 , the answer is no. 1/8 \< 1/2, thus b might not be negative.
76
What is (10 times cube root of 12) divided by (2 times cube root of 3)
**10**
77
What is the graph of the function f(x) = -2x + 1? Explain.
The input of a function = x coordinate. The output = y coordinate. Try to input a serie of numbers (usually -1; 0 and 1) Input -1 = (-1; 3) Input 0 = (0; 1) Input 1 = (1; -1)
78
How to calculate the area of a *hexagon*?
Hexagon is composed of six equilateral triangles and its area = the total are of the triangles. Find the area of a triangle and multiply by 6.
79
A quadratic equation has no solutions if \_\_\_\_\_. A quadratic equation has exactly one solution if \_\_\_\_\_.
b2 \< 4ac b2 = 4ac
80
How can you tell, the fastest way possible, how many solutions a quadratic equation has?
Use quadratic formula. The value of discriminant (value under radical b2 - 4ac) will tell you. If it is positive = 2 solutions, negative = 0, and if its 0 = one solution
81
What is the 4th root of cube root of x?
= 8th root of x
82
even +/- even = even +/- odd = odd +/- odd =
even +/- even = even; even +/- odd = odd; odd +/- odd = even.
83
What is the diagonal of a cube with a side 5?
5 \* square root of 3
84
10 + x2 \> 19 What is the possible value of x?
x2 - 9 \> 0 (we have a quadratic) (x - 3)(x + 3) \> 0 |x| \> 3 If x is positive, then x \> 3 If x is negative, then x \< -3
85
c = 4b / 9b c = ?
c = (4/9) \* (b/b) = 4/9
86
We have an equilateral triangle and an isosceles triangle (with two sides equal) with the same area, which has bigger perimeter?
The isosceles triangle. For a given area, **equilateral triangle** has the smallest perimeter.
87
If x and y are integers, x does not equal 0 and 2y - x = 2xy, which of the following could equal y? (a) 2 (b) 1 (c) 0 (d) -1 (e) -2
Test cases. Try to input every option to see when x = an integer. Correct answer: **D**
88
The maximum height reached by an object thrown directly upwards is directly proportional to the square of velocity with which the object is thrown. If an object thrown upward at 16 feet per second reaches a maximum height of 4 feet, with what speed must the object be thrown upward to reach a maximum height of 9 feet?
We know that the height is directly proportional with velocity *h/v* 4/162 Now we have to find what is the velocity of a 9 feet high throw. 4/162 = 9/x2 x2 = 9 \* 64 = 576 x = 24
89
Is x \> y? We know that 1/x \< 1/y. Explain.
We cannot say whether x \> y, knowing only that 1/x \< 1/y. If x and y are both positive or both negative, then the statement is true. The sign of an inequality changes when it involves the reciprocals of the terms. However, if x an y have different signs it can be both cases. For example if x is negative and y is positive the sign flips, but when y is negative and x is positive the sign remains the same.
90
Can an integer less than 10 yield an integer when divided by 10?
Yes. -10 for example
91
Is a sum of 3 consecutive integers divisible by 10?
Yes. 9, 10 and 11 for example.
92
A certain company that sells only cars and trucks reported that revenues from car sales in 1997 were down 11 percent from 1996 and revenues from truck sales were up 7 percent from 1996. If total revenues from car sales and truck sales in 1997 were up 1 percent from 1996, what is the ratio of revenue from car sales in 1996 to revenue from truck sales in 1996?
1:2

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