Math Concepts Flashcards
What are integers?
Whole Numbers, positive, negative, and zero {…-3, -2, -1, 0, 1, 2, 3…}
The sum of two even integers is an ___
Even integer
The sum of two odd integers is an _____
Even integer
The sum of an even integer and an odd integer is an ____
Odd integer
The product of two even integer is an ___
Even integer
The product if two odd integers is an ___
Odd integer
The product of an even integer and an odd integer is an ____
Even integer
A prime number is an _____
Integer greater than 1 that has only two positive divisions: 1 and itself
What are the prime numbers up to 100?
2, 3, 5, 7, 13, 17, 19, 23, 29, 31, 37, 39, 41, 43, 47, 51, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97
An integer greater than 1 is not a prime number is called a ____
Composite Number
Another way to name a fraction is ___
Rational numbers
a^0 =
1
2^2 =
2^3 =
2^4 =
4
8
16
3^2=
3^3=
3^4 =
9
27
81
4^2 =
4^3 =
4^4 =
16
64
256
5^2=
5^3=
5^4=
25
125
625
6^2=
6^3=
6^4=
36
216
1,296
7^2=
7^3=
7^4=
49
343
2,401
8^2=
8^3=
8^4=
64
512
4,096
9^2=
9^3=
9^4=
81
729
6,561
10^2=
10^3=
10^4=
100
1,000
10,000
11^2=
11^3=
11^4=
121
1,331
14,641
12^2=
12^3=
12^4=
144
1,728
6,912
13^2=
169
14^2=
196
0^0 =
Undefined
All positive numbers have how many square roots?
Two, one pos, one neg
(√a)^2 =
a
(√ a^2 ) =
a
√ a √ b =
√ ab
√ a / √ b =
√ a/b
For odd order roots, there is exactly ____ root for ever number n, even when n is negative
One
(Ex. 3 √ 8 = 2 ; 3 √-8 = -2)
For even order roots, there is exactly ___ roots for every positive n and ___ roots for any negative number n
Two; No
(ex. 4 √ 8 and -4 √ 8; -8 does not have a fourth square root, since 8 is negative)
What are real numbers?
Real numbers consist of all rational and irrational numbers - which includes all integers, fractions, and decimals
R + S =
RS =
S + R
ST
Order doesn’t matter
(r + s) + t =
(rs)t =
r + (s + t)
t(rs)
With addition and multiplication, order doesn’t matter, sum and product stays the same
r(s + t) =
rs + rt
Factoring out r will give the following
Dividing by 0 is ___
Undefined
If both r and s are negative, the r + s is ____ and rs is ______
Negative ; positive
|r + s| < or =
|r| + |s|
Also know as the triangle inequality
|r||s| =
|rs|
If r > 1 then _____
If 0 < s then ____
r^2 > r
s^2 < s
A proportion is ___
An equation relating two ratios
(Ex. 9/12 = 3/4; to solve a problem using ratios, you can often write a proportion and solve it by cross multiplication)
How to find out if a number is divisible by 3?
Sun of the digits is divisible by 3
How to find out if a number is divisible by 4?
The last two digits of a number are divisible by 4
How to find out if a number is divisible by 5?
The last two digits is either a 5 or 0
How to find out if a number is divisible by 6?
The last digit is a Even number and the sum of the digits is divisible by 3
How to find out if a number is divisible by 8?
If the last three digits are divisible by 8
How to find out if a number is divisible by 9?
Sun of digits is divisible by 9
Percent change formula =
% change = change / original value
(X^a)(X^b) =
X^(a +b)
(X^a)/(X^b) =
X^(a-b)
(X^a)^b =
X^((a)(b))
X^0 =
1
X^(a/b) =
b√ X ^a
X^(-a) =
1/(X^a)
A negative number raised to an even power is _____
Positive
(Ex. (-2)^4 = 16
A negative number raised to an odd power is ____
Negative
(Ex. (-2)^5 = -32))
What is the Special rule concerning odd and even exponents?
Positive Odd exponents only have one answer, even numbers have two
(Ex. X^3 = 8 -> X=2 ; X^4 = 16 -> X= 2 or X =-2)
How to easily find to raise 10 to any power ?
Just put tht many zeroes after the 1
(Ex. 10^5 = 100,000)
Can a square root be negative?
No because it is out of the scope of GRE, imaginary numbers
Quadratic Fomula
X = [ -b +- √ (b^2 - 4ac)] / 2a
What are The properties of a 30-60-90 right triangle
The 30-60-90 right triangle is is derived when an isosceles triangle so happens to have two 60degree angles, which will turn into an equilateral triangle. This equilateral triangle is then spilt in half giving rise to the 30-60-90 triangle.
Like the angles in an equilateral triangle, the sides are also equal. Meaning tht the length can be found by the ratio of X:X √ 3: 2X .
What are the Pythagorean triplets and their multiples ?
- 3-4-5 (6-8-9; 9-12-10; 12-16-15)
- 5-12-13 (10-24-26; 15-24-39; 20-36-52)
- 8-15-17 (16-30-34; 24-45-51)
- 7-24-25 ( 14-48-50; 21-72-75)
What are the properties of a 45-45-90 triangle ?
This right triangle is derived from a square split is half diagonally. The ratio of the sides is X:X:X√2
Area of a triangle
1/2 bh
In a triangle, the length of the longest side can never be ?
Greater than the sum of the two other sides
In a triangle, the length of the shortest side can never be?
Less than the positive difference of the other two sides
(X^a)(Y^a) =
XY^a
(X/Y)^a =
(X^a)/(Y^a)
When solving an inequality, when is the inequality preserved and when is it reversed?
The inequality is reversed when the constant is multiplied or divided by a negative number, every other case, the inequality is preserved
Simple interest formula
Value = Principal ( 1 + (rate)(time) / 100)
Compound interest formula for one year
Value = Principal (1 + (rate)/100)^time
Compound interest formula for multiple years
Value = Principal (1 + (rate)/(100number of years)) ^ (time)(n)
The shape of the graph where Y = |X|
A v-shape and consist of two linear pieces, y=X and Y = -X
Where in a graph is the Vertex of a parabola?
The vertex is the lowest point or highest point
What is the vertex of a triangle ?
Where two sides meet
In a graph, what equation forms a circle?
X^2 + Y^2
In a graph, what does the equation Y= X^2 form?
A parabola
In a graph, what does the equation Y= √ X form?
A half parabola
In a graph, what does the equation Y= -√X form?
A half parabola on its side
The graph of h(X) + c is the graph of h(X) is shifted _____ by c units
Upward
The graph of h(X) - c is the graph of h(X) is shifted _____ by c units
Downward
The graph of h(X + c) is the graph of h(X) is shifted _____ by c units
To the left
The graph of h(X - c) is the graph of h(X) is shifted _____ by c units
To the right
The graph of ch(X) is the graph of h(X) ____ vertically by a factor of c if c > 1
Stretched
The graph of ch(X) is the graph of h(X) ____ vertically by a factor of c if 0 < c < 1
Shrunk
How do I calculate the interior angle of an n-sided polygon?
180(n-2)
What is the side-side-side(sss) rule?
If the three sides of a triangle are congruent with another sides of a triangle then the triangles are congruent
What is the side-angle-side (sas) rule?
If two sides and an angle is congruent to another triangle, then those two triangles are said to be congruent
What is the angle-side-angle (asa) rule ?
If two angles and a side is congruent to another triangle, then it is said that the triangles are congruent to each other
What are similar triangles?
If the sides of two triangles have congruent angles and have the same ratio in lengths
(Ex. All 30-60-90 triangles are similar triangles of each other)
How do I find the ratio of the missing length in a similar triangle ?
By cross multiplying, I can obtain the other proportions
If one side of an inscribed triangle is a diameter of the circle, then the triangle is what type of triangle ?
Right triangle
If two inscribed angles hold the same chord of a circle, the two angles inscribed are ____?
Equal
Inscribed angles holding chords/arcs of equal length are said to have ____?
Equal angles
The surface area of a circular cylinder
2 πr^2 + 2 πrh = 2 πr (r + h)
The volume of a circular cylinder
Volume = r^2 π h
The diagonal length of a rectangular solid ?
Diagonal^2 = h^2 + w^2 + d^2
Diagonal = √ (h^2 + w^2 + d^2 )
The diagonal length of a cube
Diagonal ^2 = s^2 + s^2 + s^2
Diagonal ^2 = 3s^2
Diagonal = s √ 3
What are the measures of position?
L, Q1,Q2,Q3,G
What does the measure of position Q2 represent ?
The median
What does the measure of position Q1 and Q3 represent?
Q1 represents the median of the first half of a set of data
Q3 represents the median of the second half of a set of data
The standard deviation is how many points away from the mean?
-3 to 3, usually between 0-3
What does it mean when the standard deviation equal 0 ?
When the standard deviation equal the mean of a data set
How do you find the GCF of (x^8)(y^20) and (x^12)(y^15) ?
Because x^8 and x^12 have at least 8x’s therefore the GCF for x is x^8
Same goes for y the GCF is y^15
√ 2 =
Aprox 1.4
√ 3 =
Aprox 1.7
√ 5
Aprox 2.2
p - (-q) =
p + q
(-p) - q =
-(p + q)
p - Q =
-(Q - p)
Mental math: Addition and Subtraction
47 + 36 = (40+7) + (30 + 6) = 83
59 - 31 = (50 + 9) - (30 - 1)
= (50 - 30) + (9 - 1) = 28
(In subtraction the ones digit need to follow subtraction rules to remain pos; if the ones digit is bigger on right then use a diff trick)
a - b = (a + k) - (b + k) - > add k to reach a multiple of 10 then subtract
56 - 19 = (56+1) - (19+1) = 57 - 20 = 37
The sign rules for division
Same signs —> positive quotient
Different signs —> negative quotient
The combined sign rules
If we multiply or divide numbers with the same sign , the result is positive
If we multiply or divide numbers with different signs, the result is negative
Definition of an Absolute Value
The absolute value of a number gives the distance of the number from the origin, or zero
Mental Math: Dividing by 5
If I need to divide N by 5, I can :
(a) double N
(b) divide this by 10
I don’t need to do these steps in that order, I could do them in either order
Mental Math: Doubling and Halving
- I can double one factor and halve the other w/out changing the answer
- this trick can be applied multiple times in one problem
Mental Math: Squaring Shortcut
[A] squaring multiple of 10
- 40^ 2 = (4^2) x (10^2) = 16 x 100 = 1600
[B] squaring multiple of 5
- when squaring any # ending in 5, the squares number ends in 25. That’s always what will form the rightmost two digits
- 35^2 = 30^2 and 5^2 = 25 - 30^2 —> 30^2 < 35^2 < 40^2 —> 3 x 4 = 12 - no concaténate 12 and 25 to get 35^2 = 1225
[C] if I want to know the square of any value, n^2, then I can add tht value, n, and the next integer up, (n + 1), and this will result in the next square, (n+1)^2
- (n+1) (n+1) = (n+1)^2
41^2 —> 40^2 + 40 + 41 = 1600+40+41= 1681
69^2 —> 70^2 - 70 - 69 = 4900 -70-69= 4759
4^3=
64
The term “undefined” refers to the circumstance when the solution is a not real number. When could those circumstances occur?
(1) When a value would ultimately cause me to divide by 0
(2) when trying to take the square root of a negative number
What is the sequence formula?
a(subscript)n = a(subscript)1 + (n-1) x common difference (d)
What is the formula when given two “seed” values in a sequence problem ?
a(subscript)n = a(subscript) n-1 plus a(subscript) n-2
What is the definition of an arithmetic sequence ?
A sequence in which the same number is added or subtracted for each new term
What are the steps to find the sum of an arithmetic sequence?
Step 1: find the value of the missing terms
Step 2: find the avg between the first and final term
Step 3: multiply the avg by the number of terms
√ 2
Approx 1.4
√ 3
Approx 1.7
√ 5
Approx 2.2
√ 6
Approx 2.5
10^-3 =
.001
10^-5 =
.00001
0.00013 / 0.025 =
To make it easier to divide decimals, slide the decimal on the numerator and denominator to the left until I reach whole numbers.
0.00013 / 0.025 =
0.0013/ 0.25 =
Notice that the denominator is a quarter of 1, therefore, can I multiply the numerator and denominator by 4 to get a 1 in the denominator.
0.0013/ 0.25 =
0.0013(4)/ 0.25(4) =
0.0052 / 1 =
0.0052 is the answer
1/5 =
.20
1/6 =
0.17
1/7=
0.14
1/8 =
0.125
1/9 =
0.11
5/6 =
0.83
3/8 =
0.375
5/8=
0.625
7/8=
0.875
The product of any fraction with its reciprocal equals ___?
1
Ex) 4/17 * 17/4 = 1
One divided by any fraction equals ____?
The reciprocal of that fraction
(Ex) 1/(3/7) = 7/3
When will I use cross multiplication?
When I have an equation of the form:
Fraction = Fraction
Or a proportion
(Ex) 7/11 ?? 5/8
Cross multiply
78 ?? 115
56 > 55
What’s a quick and efficient way to compare “big ugly” fractions ?
Recognize when a big ugly fraction is close to a nice friendly one and use number sense.
(Ex) 83/240 ?? 199/601
83/240 can be estimated to 80/240
83/240 > 80/240 = 1/3
199/601 can be estimated to 200/600
199/601 < 200/600 = 1/3
Therefore, 83/240 > 199/601
449/150 ?? 20/7
Notice that :
3= 21/7 = 450/150
Therefore :
450/150 - 1/150 ?? 21/7 - 1/7
3 - 1/150 ?? 3 - 1/7
1/150 is way smaller than 1/7
Therefore: 449/150 > 20/7
What if we add the same number to both the numerator and the denominator?
Then the resultant fraction is closer to 1 than was the original fraction.
If the original fraction were less than 1 then the fraction will move up closer to one
If the original fraction were greater than 1 then the fraction will move down closer to one
Ex) 3/5
3/5= .60
(3+6)/(5+6) = 9/11
9/11=.81
Ex) 7/4
7/4= 1.75
(7+1) / (4+1) = 8/5
8/5=1.60
147/200 ?? 150/ 203
Notice Juan Ariel that if I add (147+3) /(200+3) I will get 150/203.
When I add to both denominator and numerator when the fraction is less than one, then I’m moving closer to one.
Therefore: 147/200< 150/203 because 150/203 is closer to 1
What happens to a fraction when adding p to the numerator and q to the denominator?
This is an advanced fraction comparison trick.
The fraction slides in the direction of p/q on the number line
If the original fraction is smaller than p/q, then the result is bigger. If the original fraction is larger than p/q, then the result is smaller.
What happens to a fraction when subtracting p to the numerator and q to the denominator?
This is an advanced fraction comparison trick.
The fraction slides in the direction of p/q on the number line
If the original fraction is bigger than p/q, then the result is smaller. If the original fraction is smaller than p/q, then the result is lager.
What happens if I add to the numerator of a fraction and subtract from the denominator, or vice versa?
Then which ever fraction has a larger numerator and a smaller denominator is bigger
37/60 ?? 43/69
Notice Juan Ariel, I could use an advanced comparison trick here.
I can add 6 the denominator and 9 to denominator, or 2/3, so the resultan fraction will be closer to 6/9 =2/3.
I know that 37/60 < 40/60 =2/3
Therefore, 37/60 had to get slightly bigger, meaning that
37/60 < 43/69
At a certain school, in December 2013, there were 6 teachers and 200 students. On Jan 2,2014, 1 teacher and 35 students joined the school. No one left the school.
QA: the teacher-student ratio in December
QB : the teacher-student ratio in Jan
Old ratio = 6/200 = 3/100 = 1/33.3
The i am adding 1 to the numerator and 35 to the denominator, which is going to move everything closer to one-thirty fifth
One thirty fifth is smaller than 1 over 33.3, bigger denominator = smaller fraction.
The fraction 1/35 is smaller than this, so adding 1 to the numerator and 35 to the denominator will decrease the ratio.
Therefore the old ratio is > the new ratio
5/14 * 7/15 =
Remember Juan Ariel that when multiplying fractions I can cancel out numerator of one fraction with the denominator of the other fraction.
5/14 * 7/15 =
1/14 * 7/3 =
1/2 * 1/3 =
1/6
When is using a mixed numerals useful versus an improper fraction?
Mixed numerals are useful if I need to locate the fraction on a number line. Mixed numerals can also be useful in comparing the fraction in size to another number
Improper fractions are useful when mult, divi, and raising to power
5/7 = 3/X find X
Remember Juan Ariel that in proportion questions, I can cross multiply
Therefore,
5X = (3)(7)
5x = 21
X = 21/5
In proportion problems what are the two legal ways that I can cancel factors and what is illegal way?
I can cancel horizontally (both numerators or both denominators on both sides) and vertically (the numerator and denominator of the same fraction)
I can NOT cancel diagonally( cross cancellation can only be done in multiplication problems )
In a word problem the word “is” means ___?
Equal
In a word problem the word “of” means ___?
Multiply
Cathy’s salary is 3/7 of Nora salary and is 5/4 of Teresa salary. Nora’s salary is what fraction of Teresa’s salary?
Remember Juan Ariel tht of = multiply and is = equal
C = 3/7 * N = 5/4 * T
The question is asking to focus on Norah’s and Teresa’s salary, so we can ignore Cathy’s.
3/7* N = 5/4 * T
And we are looking N = (fraction) * T
N = 7/3 * 5/4 * T
N = 35/12 * T
Answer is 35/12
What is a complex fraction and how do I solve them ?
A complex fraction is a fraction that has fractions in them.
The most efficient way to solve them is by finding the LCM of the “little” denominators
(Ex) simplify
(x^2/7) - (10x/7) + 3 / (x/6) -(1/2)
Find the LCM, which is 42.
42{ (x^2/7) - (10x/7) + 3} / 42{(x/6) -(1/2) }
I’m going to separate 42= 6*7
67{ (x^2/7) - (10x/7) + 3 }/ 76{(x/6) -(1/2) }
Cancel 7 and 6s
6(x^2 -10x+21) / 7(x-3)
Simplify the quadratic
6(X -3)(X-7) / 7(x-3)
Cancel x-3
Answer is 6/7 (X-7)
B^ (1/2) =
√ b
(Ex) 2^(1/2) =
because there’s no other half of the equation, I’m going to substitute K in
2^(1/2) = K
2^(1/2) ^2= K ^2
2= K^2
√ 2= K or √ 2 = 2^(1/2)
B^(1/m)=
^m √ B
B^(m/n) =
(B^m)^(1/n) = (B^n)^m
If the denominator of the exponent -fraction is odd, then the base B can be negative as well
X^a = X^b —>
a = b
√ X =
X ^(1/2)
How do I know a number is divisible by 11?
Take the value and subtract then add every digit. If the final value is a divisible of 11 then the original value is a factor if 11.
Ex) 54,879 is divisible by 11
In regards to exponent and roots, when do I know to consider pos and negative answers?
Squaring any number, I have to consider the pos and neg numbers (ex. X^2 = -/+ )
When the √ sign is written in the question stem then only consider the positive root only. To consider the negative root, it would have to be written (ex. - √ 2).
So if I have to initiate the square root then I have to consider both neg and pos .
If the GRE says: “ solve the equation x^2 = 5”, then how many solutions are there?
Two ; X = √ 5 And x = -√ 5
(Positive)^3 =
Positive
(Negative)^3 =
Negative
True or false:
With square roots, we can find the square roots only of positives, and cannot take the square root of a negative
True
True or false:
We can take the cube root of any number on the number line, positive or zero or negative
True
What’s a quick way to know if a fraction is repeating decimal or a terminating decimal?
If the simplified denominator has any prime number beside 2 and/or 5, then the fraction is a repeating decimal.
If the simplified denominator only has the prime number 2 and/or 5 then it is a terminating decimal.
Odd number divided by even number is ____?
Never an integer
What is the difference between “as much as” and “more” when it comes to percentage question problems?
“As much as “ will be the equal of the value
“More” will be additional
For example: if a questions states 100% more than a value, tht means the new value will be 2x as much.
If the questions stem states 150% more than a value, then the new value will be 2.5x as much.
What is the work Equation?
A = RT
A= amount of work done
R= the work rate
T= time
What does
A = RT
Mean?
The work equation;
A - amount of work done
R - the work rate
T - time
What are the two types of work problems?
- Questions involving proportions (work rate is a ratio)
- Multiple machines or workers working at different rates, and figuring out how much work these get done together
What is the secret to Work problems involving different machines or workers at different work rates?
The combined work rate of people/machines working together is the SUM of the individual work rates
In identical work rates problem, what is the general shortcut to these types of problems?
M - multiply (workers X time)
A - adjust (if needed)
D - divide (by time or workers)
12 factory robots, working at the same constant rate, take y hours to sort z items. How many factory robots, in terms of y and z, would it take to sort 100 items in 5 hours?
A) 240y/z
B) 1200y/z
C) 240z/y
D) 1200z/y
E)1200yz
Use the identical work rate shortcut MAD
Multiply (workers X times)
Adjust (if needed)
Divide (by worker or times)
12 robots X y hours = 12y robot-hours = z items
12y/z robot-hours = 1 ítem
1200y/z robot-hours = 100 items
1200y/z robot-hours divided by 5 hours = 240y/z
Philip walked at 4mph on his way to work but took a route that was twice as long on the way back from home. If his overall average speed across both journeys was 8mph, what was his speed back on the way back?
What type of question is this?
This is a average speed question and I could use the avg speed shortcut.
*remember: Distance = Time X Speed *
The question stem states that P walked at 4miles/1 hour and came back at double the distance, 8 miles.
The question is asking for the speed on his way back home.
Therefore, the avg speed trick is to find a distance that’s a derivative of 4 and 8, let’s say P walked 40 miles going toward his location at an avg pace of 4 miles per hour, - divide 40 miles / 4mph = 10 hours, P took 10 hours to drive 40 Miles.
So if it him 40 miles to go and 80 miles to come back, then he traveled a total distance of 120 miles, with and average speed of 8miles/1hour.
D =TS —> T=D/S
T = 120miles / 8mph = 15 hours total time walking
Therefore, it took P 15 hours total to walk back and Forth: 40/10 + 80/5 = 8/1
80miles / 5 hr = 16 miles /1hr
16mph was P returning speed
What are the 6 facts about standard deviation?
- SD can only be positive or zero, never negative
- If all the numbers on a list are identical, then the SD = 0
- If all the numbers on a list are the same distance from the mean, SD = that distance
- Lots of points close to the mean -> small SD
Lots of points away from from the mean -> large SD - Add/subtract a same value from all numbers in a list, SD does NOT change
- Multiply all numbers from a list with a value x, then the NEW SD will be old SD times X
