Quant Flashcards
When you are multiplying a fraction by a variable and the variable is in the denominator - what do you do? e.e. a x (4/a)
They cancel out i.e. a / a , which leaves you with 4
How do you get ride of a square root?
Square the variable - i.e. root Y can be squared to give you Y
Two ways to solve simultaneous equations?
1) Solve by substitution i.e. isolate one variable pulg into the other equation get the output and replug into the first equation to get the second variable
2) Combination i.e. stack the two equations and multiple then add or subtract to remove one variable with the same coefficient - solve for Y then resubstitute into one equation
How do solve for absolute values
Remember the solution can be +/- thus |x| = +/- a thus when in this form set as as positive for one solution and negative for the second solution
When is b^3 < b
When b is a fraction between 0 and 1 or a negative number smaller than -1
When is b^2 > b
Allows for positive numbers greater than 1 or any negative number
When 4 is raised to an even power what is the units digit and when 4 is raised to an odd power what is the units digit
Even power = units digit of 6, odd power = units digit of 4
Raise 1 to any power you get ?
1
Raise 5 to any power and you get?
5 as the units digit
What is the algebra for consecutive integers?
(x-1)(x)(x+1)
1.25 or 125% is equal to what in fractions?
5/4
How to choose smart numbers?
DO NOT pick a number in the question stem, do not pick 0 and 1. Use odd / even
Exponent increases on a fraction between 0-1?
Value of the expression decreases - gets smaller!
What happens to decimals between 0 and 1 when their exponents increase?
They get smaller!!!
How do you solve (3y)^4?
Distribute to each i.e. 3^4 * y^4= 81y^4
Solve z^2*z^3 =?
2+3 = 5 therefore z^5 i.e. add exponents when multiplying the exponents with the same base
Solve z^6/ z^2
Same base therefore can subtract exponents i.e. z^4
How to deal with negative exponents?
1 over the exponent but make it positive i.e. y^-3 = 1/y^3
1/3^-3 = 3^3 if you have negatives just take the reciprocal
Solve (a^5)^4 =
a^20 i.e. raise and exponent to an exponent - multiply the exponents
Number greater than one with and odd / even exponent?
Gets bigger!
Number between 0 and 1 with an even or odd exponent?
Gets smaller
Number between -1 and 0 with an even or odd exponent
Gets bigger - i.e. -1/2^2 = 1/4 or -1/2^3 = -1/8 BOTH are bigger
Number less than -1 with an even or odd exponent?
Even means it gets bigger, odd means it keeps the negative sign so gets smaller!
Solve 11^3 +11^4
FACTOR - when adding or subtracting same bases factoring is the only option i.e. 11^3(11^0 +11^1) = 11^3(12)
What do to do when you have exponent expressions on both sides of the equation?
Make them the same bases and then solve using exponent rules - when bases are identical you can drop the bases and solve the equation algebraically
What can’t you do when you add or subtract roots
Cannot separate out into two separate roots and can’t combine into one root - ONLY when you multiply or divide roots can you separate or join
Fraction in an exponent means what?
Numerator tells you the power that the base should be raised to and denominator tells you the root to take…
Form of the standard quadratic? How to get the factors?
ax2+bx+c where a, b and c are integers. a on the gmat is usually 1 b will be the sum of the factors and c will be the product of the factors… and the factors then need to be put = 0 in order to get the solutions. i.e. (x+4) (x-1) = 0 thus x=-4 and +1
If you have a higher order equation like x^3+2x^2-3x=0 what is going on?
It is actually a quadratic - just got to factor out an x. One solution for x will therefore be 0 and then the other two factors in this case which are -3 and 1 here when set to 0
When do you not solve a quadratic by setting it to zero?
When the other side of the equation is a perfect square. You just take the square root of both sides which yields a clean equation and but remember you solve for the +/- solutions since you took the square root of the perfect square
Solve x^2+x-12/(x-2) =0
Notice that its a quadratic so factor numerator and make sure denominator doesn’t equal 0 so solutions are x=3 or x=-4 but x=2 is not a solution as denominator would become 0
x^2-y^2=?
(x+y)(x-y)
x^2+2xy+y^2=?
(x+y)(x+y) = (x+y)^2
x^2-2xy+y^2=?
(x-y)(x-y)= (x-y)^2
what is a^2-1=?
(a+1)(a-1)
solve (a+b)^2
a^2+2ab+b^2
How do you know an inscribed triangle in a circle has a 90 degree angle?
If one side is the diameter of the circle then the triangle is a right angle triangle
What has to be true for abc<0
1 negative and 2 positive or ALL negative
What has to be true for X/Y > 0
Both are negative or both are positive i.e. same sign
How do you calculate miles travelled?
miles per hour (rate) * hours
miles per gallon * gallons
Product of 3 consecutive integers?
Is divisible by 3
b^2 / ac^3 > 0 what has to be true?
a and c must have the same sign - both positive or both negative. b could be either sign, the square hides the sign
Absolute value square of fractions between 0-1 / 0 - -1
Absolute value square of either will be smaller than the starting fraction you take the absolute value of
ZONEF
Zero, one, negative, extremes, fractions
Total cost?
price * quantity
When is x^3 < x^2?
For all values of x < 1!!!! 0-1 gets smaller as exponent increases. Between 0 and -1 you have a sign issue making it negative when there is an odd power same goes for less than -1
If you have two equal angles in a triangle?
The sides opposite each angle are equal
What are the four quadrants of a plane?
Top right - positive x and positive y (x,y)
Top left - negative x and positive y (-x,y)
Bottom left - negative x and negative y (-x,-y)
Bottom right - positive x and negative y (x, -y)
What is the circumference of a circle?
2#r - sorry no symbol for pie - # is pie!
Total revenue
price * quantity
Value of an exterior angle?
is equal to the value of the two non-adjacent (opposite) interior angles…180 minus the adjacent angle on a line (c) = the value of the two opposite angles (a) and (b) so the value of the exterior angle x is equal to a & b - find flash card for visual
What is an equilateral triangle split in two?
30 / 60 / 90 triangle
How do you calculate the hypotenuse of an equilateral triangle?
2x
How do you calculate the height of an equilateral triangle?
x root 3
What are the primes between 20 and 50?
23, 29, 31, 37, 41, 43, 47
List the small primes?
2, 3, 5, 7, 11, 13, 17, 19
What do you do when you have multiple factors of x that share primes?
Combine and remove overlap - i.e. count the highest instance of a prime for a factor
(see pg 69/70 FOM)
How do you find prime factors?
Set up a factor tree
What do you do when you have multiple factors of x that DONT share primes?
Combine them all into one factor tree for that x
Total earnings?
wage rate per hour * hours worked
How do you find factors of a number?
set up a factor pair table - remember? small / large.. start with 1 then 2 then 3 and work your way down
How do you find the perimeter of a right angle triangle?
WALASI!!! Pythagorean theorem
a^2 + b^2 = c^2
where a and b are legs of the triangle - NOT equal
c is the hypotenuse
Area of an equilateral triangle?
((root 3 ) / 4) * a^2
Area of a trapezoid?
((base 1 + base 2)* h) / 2
Volume of a cube?
v= lwh where l=w=h i.e. 5^3 or 8^3 - cubed!
Area of a parallelogram?
b*h - but will have to draw in height that is parallel to the base creating a 90 degree angle
Area of a circle?
r^2
i.e. pie*r^2
Translate four pounds less than expected?
e - 4
Translate n persons have x beads each?
nx = total # of beads
When they use the words per, ratio, quotient or proportion what do you think?
division or that over something
Ratio of x to y means?
x over y
Translate - sum of 3 funds combined?
a+b+c
Profit =?
Rev - cost
53.0447 / 10^-2 i.e. negative power division
Makes number larger ! Negative power division makes number larger … i.e. move the decimal to the right by the # in the exponent soo this will be 5,304.47
How do you add or subtract decimals? that have different count
Line up the decimal points and add zeros were necessary after the decimal so they match up in terms
How do you multiply decimals? e.g. 0.02 * 1.4 =?
Remove decimals - multiply the whole numbers then count the digits to the right of both decimals then move the decimals the same number of times to the left i.e. 14*2 = 28 and there are 3 decimals therefore move 3 times to the left!
Multiply large number with very small decimal?
i.e. 40,000*0.0003
Move the decimals in both the same number of places but in opposite direction - i.e. 40,000 4 times to the left to give 4 and 0.0003 4 times to the right to give 3 = multiply which gives you 12
What is an equilateral triangle?
All 3 sides are equal and each angle is 60 degrees
When there is a horizontal line in a coordinate plane what does that indicate?
X could be anything but y = one number i.e. y=5 or y=-27
What does a vertical line on a coordinate plane indicate?
Y could be any number but X = one number i.e. x=2
What is the slope of a vertical line? i.e. when x=5
Undefined
What is the slope of a horizontal line? i.e. when y=3
ZERO!!!
How do you calculate slope of a line?
y2 - y1 / x2-x1 - remember the MINUS is in the formula so if you have a negative coordinate you’re adding!
Volume of a cylinder?
r^2h i.e. PIER^2*H
Area of a circle * Height
What are the 3 fraction that sectors represent?
1) central angle / 360
2) sector area / circle area
3) arc length / circumference
Each will give you a proportion for what the sector represents in terms of the circle
How do you find the third side of a triangle?
Sum of any two sides is greater than the 3rd side so the 3rd side has to be less than the sum of the two sides i.e. c < (a+b)
Has to be more than the difference of the two sides i.e. c > (a-b)
c is less than the sum of the two sides and more than the difference
if a = 5 and b = 3 what x? === x is between 2
Pythagorean triples? List all them!
3-4-5
6-8-10 - i.e. times 2
9-12-15 - i.e. times 3
12-16-20 - i.e. times 4
5-12-13
10-24-26 i.e. times 2
8-15-17
Definition of a circle?
Set of points in a plane that are equidistant from a fixed center point (360 degrees)
Relationship between sides and angles?
Largest side opposite largest angle, small side opposite small angle
What makes an Isosceles triangle?
Two equal sides and two equal angel
Relationship between factors and primes?
Each factor is made up of a specific set of primes
Area of a triangle?
1/2bh
Diameter of a circle?
2*r
How do you know if a number is divisible by 6?
Has to be divisible by 2 and 3
e.g. 48 = ends in 8 so divisible by 2
4+8 = 12 which is divisible by 3
so 48 is divisible by 6
How is it divisible by 4?
2 goes into it twice OR last two digits of the number are divisible by 4
Divisibility of multiples rule?
If you add two multiples of x then you get another multiple of x
If you subtract two multiples of y then you get another multiple of y
When you multiply by a negative power of 10?
You make it smaller - i.e. move to the left by the number in the exponent
so 6,782.01 * 10^-3 = 6.78201
When you divide by a positive power of 10?
Make it smaller - i.e. move the decimal to the left of the specified number of places i.e. 4,169.2 / 10^2 = 41.692
How to solve geometry problems? 5 steps!
1) re-draw the diagram
2) fill in all the given and needed info and identify the target
3) Recognize the relationship - write the equations needed
4) make inferences
5) solve
x^2 - x< 0
x is between 0 and 1
When you are adding or subtracting units?
Make the units the same first by converting then add or subtract
Rate =?
Distance / time
Total cost calculation w fixed cost?
Total cost = fixed cost + (p*additional q)
Parallel lines angles created?
Obtuse angles / opposite angles are equal
Acute angles / opposite angels are equal (smaller of the two)
Sum of interior angles of a polygon?
(n-2) * 180
5 sides = pentagon = 540 degrees
6 sides = hexagon = 720 degrees
How do plot a line?
Set x = 0 to get y-intercept
Use m (the slope) to get the second point on the line Numerator is change in y, denominator is change in x
Perimeter of a sector?
Arc length plus two radii
How do you find the distance between two points on a coordinate plane?
1) Draw a triangle
2) Calculate the rise - which is the height (difference in the y coordinates)
3) Calculate the run - which is the base (difference in the x coordinates)
4) Then use pythagorean theorem or triplets
If d has e and f as primes is d divisibly by (e*f)?
Yes
Surface area of a cube?
1) find the area of one side by finding one side and taking the square of it
2) multiply the area of w^2 i.e. the area of a square 6 since it has 6 sides.
So if w = 2 then one sides area is 2^2 = 4
and surface area of cube is then 46 = 24
Even integers end in?
0,2,4,6,8
How do you know if an integer is divisible by 2?
It is EVEN!
Define the law of divisibility?
If a is divisible by b and b is divisible by c then a is divisible by c as well!
How do you know if it is divisible by 9?
Sum of the integers is a multiple of 9 e.g. 288— 2+8+8=18 which is a multiple of 9
Special feature about primes?
1) Primes are always positive
2) 1 is NOT a prime number
3) is the only even prime - rest of primes are ODD!
What is a prime number?
Divisible by 1 and itself
How is a number divisible by 5?
Ends in 0 or 5
How do you know if it is divisible by 3?
Sum the digits and it yields a multiple of 3
i.e. 147 = 1+4+7 which equals 12 thus divisibly by 3
How do you know it is divisible by 8?
2 goes into it 3 times OR the last 3 digits are divisible by 8
Translate 6 lbs heavier than mike
M+6
If Patrick were 50 years older
P +50
What is the sum of a two primes (excluding 2)
All primes are odd so ODD + ODD = even
If the sum of two primes is odd what does that mean?
One of them is even as all primes other than 2 are odd… so odd prime plus 2 = odd prime
What is the area and perimeter of a square?
a= side^2 perimeter = 4 * side
what does this mean (x+y) (x-y)?
x^2 - y^2
What has to be true of xy > 0?
They have to have the same sign
x^2 - 2xy +y^2?
(x-y)^2 OR (x-y)(x-y)
What attribute does an isosceles triangle have?
2 sides are equal and 2 angles are equal
45 degree / 45 degrees / 90 degrees
x (leg) / x (leg) / x * root 2 (hypotenuse)
what are the attributes of a 30 /60 / 90 triangle?
30 / 60 / 90
x (leg) / x * root 3 (height) / 2x (hypotenuse)
EVEN + EVEN= ?
EVEN
ODD + ODD =?
EVEN
EVEN + ODD = ?
ODD think odd one out when they are different for the addition and subtraction set
Even * Even=?
Even
Odd * Odd = ?
Odd
Even * Odd = ?
Even
2n+1 means?
ODD
When you have an odd exponent what do you know about the sign?
it takes the sign of the base - i.e. doesn’t hide the sign
i.e only one solution +ve or -ve based on the sign of the base
Perimeter of a parallelogram
the lines parallel to each other are the same so 2(x+y)
Line in a coordinate plane formula?
Y = mx +b where m is the slope and b is the y-intercept
How can you split a rectangle?
Split into two right angle triangles with the diagonal of a triangle
Area of a rectangle?
a= l*w
No need to drop height as l & w are perpendicular
Perimeter of a rectangle?
2*(l+w)
If xy < 0 means?
They have different signs
If you have x^2 in DS what does that indicate?
Insufficiency! You have two possible solutions (+ve and -ve) for x thus insufficient unless they state it is positive in the stem
x^2 +2xy+y^2?
(x+y)^2 ==== (x+y) (x+y)
Place values of 452?
4 is in the hundreds place
5 is in the thousands place
2 is in the units or ones place
Place value for 100.2534
2 is in the tenths place
5 is in the hundredths place
3 is in the thousandths place
4 is in the ten thousanthds place
Translate 1/2 of kids eat chocolate
1/2n = c
Translate 20% of kids like vanilla
0.2n= v
Multiplying by positive power of 10
Makes it bigger - move the decimal to the right by the number of times in the exponent
e.g 3.9742* 10^3 = 3,974.2
How do you solve word problems?
WANT - state what they give, i.e. what the target is
GIVE - what info do they give you to work with
EVALUATE - put into equations
GET - solve
If ppl work at the same time side by side then what is the rate?
Add their individual rate
FOM page 383
Work?
Rate * time
Examples of evil twins?
1) Reciprocals of each other
2) % adding up to 100
3) fractions that add to one
When testing cases what is the number line from left to right?
1) Even / odds
2) Fractions between 0 and - 1
3) 0
4) Fractions between 0 and 1
5) Primes, even / odd
21 + 18 = WHAT?
A multiple of 3 - i.e 39 is a multiple of 3!
A multiple of 3 plus a multiple of 3 gives you a multiple of 3
Equilateral triangles are made up of what?
They are made up of two 30/60/90 triangles
Deconstruct a hexagon?
Made up of 6 equilateral triangles
Interior angles equal 720 degrees
What to notice if a square is inscribed inside a circle?
Is the diagonal of the square the diameter of the circle? passes through the center…If so you have two right angle triangles with the diameter as the hypotenuse which is really a 45 / 45 / 90 triangle with hypotenuse measured by x*root 2
What to notice when there is a circle inside a square?
The side length of the square is the diameter of the circle
If the unit of work = 1? then what is the rate and time?
Rate and time are reciprocals
How to estimate time taken working together?
Has to be between half of one persons time and half of the other persons time i.e.
x/2 < t < y/2 (x and y are the two separate times taken)
t is the time taken together!
quick way to solve fraction with 1 as the numerator?
Sum of the denominator is the numerator and product of the denominator is the denominator
i.e. 1/3 + 1/4 = 7/12
What does finding a root mean?
Means the solution to a quadratic
How do you calculate the probability of something happening?
The number of times it could happen / the total occurrences in the pot (with and without your situation, ie. all situations combined)
How do you calculate combined probability ? Using “AND”
Probability 1 x probability 2 = combined probability
What does “or” mean when doing combinatorics or probability?
Add
What does the word “and” mean when doing combinatorics or probability?
Multiply
What does n! mean?
Product of all integers from 1 to n inclusive
1! =?
1
2!=?
2
3!=?
6
4!=?
24
5!=?
120
6!=?
720
Probability equations?
Number of desired outcomes / total number of possible outcomes
Two ways to calculate probability?
P(A) or 1-P(Not A) i.e. 1 minus the prob of it not happening…
Indicator words that signal use of 1-P(notA)
Prob that “at least” or “at most” X happens
How do you calculate the sum of a set?
sum of set = (mean of set) * (number of terms in the set)
How do you calculate the mean of a consecutive set?
first + last / 2
How do you calculate the number of terms in a set?
Last (last - first) + 1
What is the remainder if an integer is divided by 5?
0, if the integer’s units digit is 0 or 5
1, if the integer’s units digit is 1 or 6
2, if the integer’s units digit is 2 or 7
3, if the integer’s units digit is 3 or 8
4, if the integer’s units digit is 4 or 9
How do you solve X^5/ X^2 =?
When dividing exponents with the same base - subtract the exponents…i.e. the answer is X^3
x^3x^5 = ?
When multiplying numbers with the same base add the exponents = X^8
How do you simplify x^2y^2 =?
(xy)^2
Simplify (x^2)^3 =?
When raising a number to an exponent and then another exponent you multiply the exponents = x^6
x^-6 =?
You can remove a negative exponent by taking the reciprocal of the base so = 1/ x^6
When you have addition under a square root?
Cannot split or combine - if the numbers are known you can factor out a perfect square or add them or subtract them under the root and then take the root of the resulting number
