Math Special Flashcards
Can you determine whether a^n is even or odd given:
1) a
2) n
a^n has the SAME TYPE as a, the base (exponent DOES NOT MATTER)
> Only the base matters (so you can safely ignore exponents WHEN DETERMINING PROPERTIES Even or Odd)
1) Given the value of a, you can figure out whether a^n is even or odd AS LONG AS n > 0 AND integer
(n cannot be 0, n cannot be fractional value)
2) Given n, you cannot figure out whether a^n is even or odd (just signs)
*n must be a positive integer (NOT equal to 0 -> always equals 1 = odd)
Given n, are the following even or odd?
n + 4
n - 5
ODD/EVEN Concept:
Unknown variable +/- odd number = Opposite Type
Unknown variable +/- even number = Same Type
e.g., n + 1 = even if n is odd
n - 4 = odd if n is odd
What are the properties of three consecutive integers?
- at least one even integer, so the product is EVEN
- product is divisible by 3! –> 2, 3, and 6
- If the middle term is ODD, the product is divisible by 8 (two consecutive even integers)
If an integer has only 3 positive factors (including 1) or 2 positive factors other than 1, what does this tell you?
The integer is a PERFECT SQUARE of a PRIME NUMBER
> perfect square because odd # of factors
e.g., 2^2 = 4, 3^2 = 9, 5^2 = 25
How many unique prime factors does a^n have?
(1) a = 6
Concept:
a^n has the SAME unique prime factors as a
> exponent doesn’t matter
so if a = 6 = 2*3, 6^n has the same unique prime factors
ASSUMING n > 0
What are the properties of two consecutive even integers?
Product is a multiple of 8
Proof:
n*(n + 2) and n is even
(2n)(2n + 2)
4(n)(n + 1) —> n(n + 1) is also even
42 = 8
Common forms:
(n - 1)(n + 1)
How do you tell if the product of three integers is a multiple of 3, given an unknown variable?
e.g., n(n + 4)(n - 5)
Consecutive integers and Multiples:
n(n + 1)(n + 2) –> the product of ANY 3 consecutive integers is divisible by 3! and therefore a multiple of 3 (because one of the numbers MUST be a multiple of 3)
> cycle repeats every 3 numbers
Due to the cyclicality of multiples: If one of the numbers were a multiple of three, then the number +/- 3 would ALSO be a multiple of three
Strategy: Determine whether there is a COMPLETE set of three CONSECUTIVE INTEGERS, keeping in mind the cyclicality of multiples
n = n + 3 and n - 3 (same properties)
n + 1 = n + 4 and n - 2 and n - 5
n + 2 = n + 5 and n - 1
so rewrite n(n + 4)(n - 5) as:
n(n + 1)(n + 1) => we don’t have three consecutive integers so there is NO multiple of 3
How do you find the remainder when the divisor is 5?
Just like 10, the remainder of an integer divided by 5 is equal to the UNITS digit *
> *Small adjustment: Compare units digit to 0 and 5
e.g., 333^777 / 5
777/4 = 194 + remainder 1
[3, 9, 7, 1] –> units digit is 3 –> +3 from 0
So the remainder of 333^777 / 5 is 3
How do you find the remainder of large exponents divided by integers?
e.g., 2^5550 / 7 or 333^777 / 5
CONCEPT: Look for the PATTERN in the remainders when dividing different POWERS by the divisor
e.g., 333^777 / 5
3^1 / 5 –> R = 3
3^2 / 5 –> R = 4
3^3 / 5 –> R = 2
3^4 / 5 –> R = 1
3^5 / 5 –> R = 3 —> Repeats every cycle of 4
777/4 = 194 + remainder 1 –> remainder is 3
Also don’t forget:
> when the base is > 10, we care only about the UNIT DIGIT (e.g., 3 in 333)
> (For product or sum of integers): Units digit is influenced ONLY by the units digit of the BASE (drop any other digits)
a^4 + b^4
General rule: a^2 + b^2 = (a + b)^2 - 2ab
=> SUM of two EVEN powers or 1
= “sum of squares” (or other even powers)
Special Examples:
a^4 + b^4 = (a^2 + b^2)^2 - 2(a^2)(b^2)
a + b = (sqrta + sqrtb)^2 - 2(sqrta)(sqrtb)
a^8 - b^8
General rule: a^2 - b^2 = (a + b)(a - b)
=> difference of TWO EVEN POWERS or 1
= “difference of squares”
Special examples:
a^8 - b^8 = (a^4 + b^4)(a^4 - b^4) = (a^4 + b^4)(a^2 + b^2)(a^2 - b^2)= (a^4 + b^4)(a^2 + b^2)(a + b)(a - b)
a - b = (sqrta + sqrtb)(sqrta - sqrtb)
a^2 + 1/a^2
Application of sum of squares:
a^2 + 1/a^2 = (a + 1/a)^2 - 2
RECIPROCALS with EVEN EXPONENTS that are powers of 2
Typically, n variables require at least n different equations to solve. However, when is it generally sufficient to solve for two variables given one equation?
Exception #1) Linear Equations:
1) Sufficient to solve when the variables are INTEGERS
> e.g., quantities
Not sufficient to solve when the variables are DECIMALS
> e.g., prices
The “total” value of the equation is within ~10x of the sum of the coefficients (not too large)
**AFTER simplifying coefficients
e.g., 15a + 29b = 440
e.g., 23a + 21b = 130
NOT 2c + 3a = 1350
Typically also a good sign when you see multiples of 5 and/or 10 as coefficients
> units digit must be 0 or 5
** write out the multiples of each term and see if more than one valid combo works !
Exception #2) Quadratic or other equations (absolute value signs, squares, roots)
> these values must be 0 or positive
Sum of two nonnegative unknowns = 0
Positive unknown constraint –> subtraction adds a limit
e.g., a = 10 - 2b, a and b > 0
2 <= a <= 8
Exception #3) Combo questions
> e.g., what is 3A + 4B = ? (and you get a ratio of it)
e.g., what is ab = ?
Exception #4) Ratio questions (variable cancels out)
BEWARE of trap
–> identical equations are not sufficient!
–> Ratios of equations are not sufficient!
–> variable cancelled out completely is NOT sufficient!
If x1 and x2 are roots of a quadratic equation, what is x1 + x2?
Quadratic equations in the form: ax^2 + bx + c
x1 + x2 = - (b/a) —> MEMORIZE
e.g., x^2 - 6b + 9
x1 + x2 = - (-6/1) = 6
RECALL quadratic equation:
y = ax^2 + bx + c, opens up like a U if a >0
y = ax^2 + bx + c –> can factor to find two solutions for roots
Can also set y = 0 and take first derivative to solve for x coordinate of max or min
If x1 and x2 are roots of a quadratic equation, what is x1 * x2?
Quadratic equations in the form: ax^2 + bx + c
x1 * x2 = c/a
If the question gives you information about the number of solutions in a quadratic equation, what should you think about?
USE the Discriminant, subbing in values for a, b and c.
If 2 solutions: b^2 - 4ac > 0
If 1 solution: b^2 - 4ac = 0
If 0 solutions: b^2 - 4ac < 0
FULL quadratic equation: x = [-b +/- sqrt(b^2 - 4ac)]/2a
Given that x1 and x2 are solutions of a quadratic equation, create an equation
a(x - x1)(x - x2) = 0
*a is a CONSTANT that is NOT equal to 0
a^2 + b^2 = 0
How do you solve for the solution?
Quadratic equations with the sum of two squares:
a^2 and b^2 are BOTH positive or equal to 0. Therefore, the sum must be >= 0
So a^2 and b^2 must BOTH be equal to 0
PROPERTIES of this type of question:
> sum of TWO POSITIVE unknowns equal 0
Even though there is only one equation, two unknowns, you can STILL SOLVE (exception #2)
sqrt(a) + sqrt(b) = 0
How do you solve for the solution?
PROPERTIES of this type of question:
> sum of TWO NON-NEGATIVE unknowns equal 0
Even though there is only one equation, two unknowns, you can STILL SOLVE (exception #2)
Even roots (e.g., ^1/2, ^1/4, ^1/8)
The value under the root MUST BE POSITIVE or EQUAL to 0 (non-negative)
a^1/n if n is even, a >= 0
Also, the value of a^1/n is also >= 0
e.g., (16)^1/2 = 4
Arithmetic Sequences Formula for:
Term
Sum
ALWAYS START AT a1
n >= 1
Term:
An = a1 + (n - 1)*d
where d is the constant difference (+ or -) between any two consecutive terms
Sum:
Sum from a1 to an = (average * # of terms) —> equally spaced sequences
= (a1 + an)/2 * n
LINEAR growth problems can also be solved as an arithmetic sequence
> e.g., monthly info +/- constant amount
> e.g., height
Geometric Sequences Formula for:
Term
Sum
ALWAYS START AT a1
n >= 1
Term:
An = a1*r^(n - 1)
where r is the constant ratio (>1 or <1) between any two consecutive terms
Sum:
Sum from a1 to an = [a1 * (1 - r^n)]/(1 - r)
**special geometric sequences have a PRODUCT or division relationship (not +/-)
Examples:
1, x, x^2, x^3, x^4, x^5 —-> r = x, a1 = 1
If you are asked to solve for the possible VALUES of a variable, and you are given 1 variable inequality, what should you do?
e.g., x^2 + 6x + 9 > 0
1 Variable Inequality:
Use number line sewing approach to determine the SIGN of the product
Certain things to remember:
1. Move all the the terms to one side so that the other side is 0
2. Factor (product <>= 0)
3. Exponent on x must be 1 (Unless x^# alone is a factor or is always positive)
e.g., x^2 * (x + 1) < 0
e.g., (x^2 + 1) > 0
- Coefficients must be POSITIVE (inside factor and outside factor too)
e.g., (12 - x)(x + 1) > 0
(-x + 12)(x + 1) > 0
-(x - 12)(x + 1) > 0 **still have to get rid of -1 out front
(x - 12)*(x + 1) < 0 - Interchange signs at the roots, EXCEPT when the exponent on the factor is EVEN
- Keep track of inclusion or exclusion of the roots! (= or not)
e.g., (x + 2)^4(x + 1) < 0
If you are asked to solve for possible VALUES of a variable that is expressed in a FRACTIONAL inequality, what should you do?
e.g., (x + 2)/(x - 4) > -1
1 variable, fractional inequality:
Use number line sewing approach to determine the SIGN of the fraction
Certain things to remember:
1) Move all the terms to one side so that one side is 0
2) Factor the numerator and denominator of the fraction
3) Then assess the sign of the fraction as if it were a PRODUCT of factors
4) same rules apply as 1 variable inequality (coefficients must be positive, interchange signs at the roots except for even exponents, keep track of inclusion and exclusion of the roots)
5) Be mindful that roots in the denom CANNOT work !
x^2 < 3/2
Squares are treated similarly with absolute value sign
Think of positive case –> keep direction of the sign and square root it
Think of negative case –> switch direction of the sign and add negative and square root
x^2 < a means:
-sqrt(a) < x < sqrt(a)
so x^2 < 3/2
-sqrt(3/2) < x < sqrt(3/2)
x^2 > 3/2
Squares are treated similarly with absolute value sign
Think of positive case –> keep direction of the sign and square root it
Think of negative case –> switch direction of the sign and add negative and square root
x^2 > a means:
x > sqrt(a) or x < - sqrt(a)
so x^2 > 3/2 means:
x > sqrt(3/2) or x < - sqrt(3/2)
If you are asked to solve an inequality question with multiple variables, what should you do?
e.g., is (x - y)/(x+y) > 1 ?
e.g., is x < y < z?
Multi-variable inequality
Often will be testing PROPERTIES (e.g., >0, <0)
So try to FACTOR and get one side equal to 0 (just like one variable inequality problems)
> sewing approach isn’t as effective here
Or try to use the positions on the number line, especially for questions that have compound inequalities
If you are given a question about absolute values with one variable, what should you do?
e.g., | 2x + 1 | > x + 1
One variable Absolute Value –> Find the range of VALUES of the unknown
THINK OF CASES for the sign of the inside of the absolute value sign:
Keep only VALID cases
> draw out a NUMBER LINE with markings representing the “roots” (what values of x makes the argument = 0)
> Determine the cases to test
> Don’t forget to add = to one of the cases for each root
WORKS FOR EMBEDDED ABSOLUTE VALUE SIGNS TOO –> for EACH absolute value sign, you have two cases
Example: | 2x + 1 | > x + 1
Root –> x = -1/2
1) x >= -1/2 –> 2x + 1 >= 0
2x + 1 > x + 1
x > 0 —> satisfies the test case and overrides it (“large choose large”)
2) x < -1/2 –> 2x + 1 < 0
-(2x + 1) > x + 1
x < -2/3 —> satisfies the test case and overrides it (“small choose small”)
so x > 0 or x < -2/3
x | —> x >= 0 –> | x | = x
| x | —> x < 0 —> | x | = -x
If you are given a question about absolute values with multiple variables, what should you do?
e.g., x - y = | y - z | + | z - x |
Multi variable Absolute Value –> Distances using Number Line
Characteristics:
> typically SUBTRACTION between the terms in the absolute value signs
Also | x - y | = | y - x |
Focus on the COMMON POINT
x | = distance that x is from 0
| x - 1 | = distance between x and 1
| x + 1 | = | x - (-1) | = distance between x and -1
If you are given a question about absolute values with multiple variables and a split, what should you do?
e.g., | x - y | > | |x| - |y| |
What if you have three or more variables?
e.g., | x - y - z | = x - y - z
Multi variable Absolute Value with Split –> SIGNS of the UNKNOWNS
= only happens when unknowns have the same sign (xy > 0)
| x - y | >= | x | - | y | —-> = only happens when x and y have the SAME SIGN AND when | x | > | y |
(otherwise, x - y will be larger)
Three or more variables:
> need to test cases (NOT the simple rule that if all the variables are the same sign, then =)
> x - y must be >= z
e.g., | 4 - 3 - 2 | =/ 4 - 3 - 2
e.g., | 10 - 2 - 1 | = 10 - 2 - 1 —> x - y - z >= 0
e.g., | 11 - 6 - 5 | = 0 = 11 - 6 - 5 —> x - y - z = 0
x + y | <= | x | + | y | —-> = only happens when x and y have the SAME SIGN (otherwise, x + y will be smaller)
What does it mean when two variables have the same sign?
xy > 0
or x/y > 0 (if x and y =/ 0)
How would you determine cases for the following:
x - 3 | + | x - 4 | < 2
Keep only VALID cases
> draw out a NUMBER LINE with markings representing the “roots” (what values of x makes the argument = 0)
> Determine the cases to test
> Don’t forget to add = to one of the cases for each root
Roots: x = 3, x = 4
1) x >= 4
–> (x - 3) + (x - 4) > 0
(x - 3) + (x - 4) < 2
2x - 7 < 2
2x < 9
x < 9/2
Therefore, 4 <= x < 9/2
2) 3 <= x < 4
–> (x - 3) > 0 and (x - 4) < 0
x - 3 + 4 - x < 2
1 < 2 (true)
Therefore, 3 <= x < 4
3) x < 3
–> (x - 3) < 0 and (x - 4) < 0
3 - x + 4 - x < 2
7 - 2x < 2
5 < 2x
5/2 < x
x > 5/2
Therefore 5/2 < x < 3
So in total (you can combine here):
5/2 < x < 9/2
What is the formula to calculate the sum of the interior angles in a polygon?
n sides
Sum of interior angles = (n - 2)*180
What are the constraints on every side of a triangle?
difference of the other two sides < side < sum of the other two sides
A triangle has the sides with values in this order: a < b < c. If the three angles are 60, 61, and 59 degrees, which side is opposite to which angle?
Angles correspond to sides
> Largest angle is opposite from the largest side
> Smallest angle is opposite from the smallest side
a is opposite from 59
b is opposite from 60
c is opposite from 61
What should you write down when you are dealing with an obtuse triangle?
Other indicators:
ONE angle is greater than 90 degrees
If a, b, c are the sides and a < b < c:
a^2 + b^2 < c^2
What should you write down when you are dealing with an acute triangle?
Other indicators:
ALL angles are less than 90 degrees
If a, b, c are the sides and a < b < c:
a^2 + b^2 > c^2
What are the properties of circle arcs?
Arcs are related to a circle’s CIRCUMFERENCE
Arcs are defined by: radius and central angle
The interior angle of one arc is EQUAL regardless of where the vertex is located
> can be used to determine the value of the central angle
Special example:
Inscribed angle’s vertex is on the same line as the center of the circle
What are the properties of circle sectors?
Sectors are related to a circle’s AREA
Sectors are defined by: radius and central angle
A sector can be broken into an isosceles triangle plus a curved part
When you see a circle question with chords, arcs, sectors and inscribed angles, what should you think about?
Radius is key
> isosceles or equilateral triangles (60 degrees!!)
Diameter’s inscribed angle = 90 degrees
Angle rules (line = 180 degrees, parallel lines)
Central angle = 2*inscribed angle sharing the same arc
*don’t be afraid to draw helping lines (based on connecting VERTICES on the circle)
Inscribed square inside a circle
Inscribed circle inside a square
Diagonal of the square = Diameter of the circle
Diameter of the circle = side of the square
> you can create a smaller square by drawing a line from the center of the circle to square’s vertex = diagonal of the smaller square
> circle’s radius becomes the sides
Triangle Angle Questions:
1) Triangles and circle questions about angles
Focus on the VALUE of angles, depending on which sides are equal (e.g., radius)
Use symbols for angles, such as alpha, beta
Triangle Angle Questions:
2) Triangles made up of sub-triangles or star shapes
Rely on EXTERIOR angles
Also might need to calculate the “perfect regular shape” using sum of interior angles = (n - 2)*180
Also FYI parallel lines –> draw aiding parallel lines
Data sufficiency questions about complex geometry tips
Don’t bother doing calculations. Start off by mapping out what info you need!
e.g., radius, angle
Inscribed angles sharing the same arc
Inscribed angle are all equal as long as they share the same arc
When do you know when two triangles are congruent?
SSS
SAS = side-angle-side are known and equal
ASA = angle-side=angle are known and equal
AAS = angle-angle-side are known and equal
(NOT ASS or SSA)
*with the above, the shape of the triangle is FIXED
FOR RIGHT TRIANGLES: a^2 + b^2 = c^2
> Hypotenuse + 1 legs
> 2 legs
How many faces does a cube or rectangular solid have?
How many edges does a cube or rectangular solid have?
How many vertices does a cube or rectangular solid have?
6 faces
12 edges
8 vertices
What is the formula to calculate distance between two points in a coordinate plane?
Distance = sqrt[ (x2 - x1)^2 + (y2 - y1)^2 ]
a^2 + b^2 = c , where c is an integer
What does this tell you?
Integer is the SUM of two perfect squares
It may be possible to test values if you know the value of c
Also in general if you are given a^2 + b^2 and asked to find ab –> likely a^2 - 2ab + b^2
Tips for simplifying algebra
Get into the habit of:
> Grouping variables (highest to lowest exponent)
factoring out negative (positive coefficients on variables)
removing roots from the denom of a fraction (by multiplying top and bottom by the conjugate or just itself)
** do this BEFORE combining into one fraction
factoring out fractions out of EVERY TERM first
in equations: moving terms to one side and factoring
e.g., ab = ac –> ab - ac = 0 —> a(b-c) = 0 (a = 0 OR b=c)
f(x) = ax^2 + c
Does the function open up like a U or down?
What is the max or min of this function?
Depends on sign of a:
a > 0 (positive) –> U shaped
a < 0 —> downward shaped
Max or Min depends on the sign of a
> U shaped –> min
> Downward shape –> max
*Tip: Quadratic functions are symmetrical about the axis of symmetry
> find the distance between the roots (if any) and divide by two to get the x coordinate of the max or min
In a coordinate plane, what is the reflection of (a, b) about the line y = x?
(a, b) is a reflection of (b, a)
Both points have the SAME distance from y = x (which acts as a perpendicular bisector of the line segment from (a,b) to (b,a))
REFLECTIONS deal with lines that are perpendicular bisectors of a line segment
In a coordinate plane, what is the reflection of (a, b) about the x axis?
(a, b) is a reflection of (a, -b)
Both points have the SAME distance from the x axis (which acts as a perpendicular bisector of the line segment from (a,b) to (a, -b))
REFLECTIONS deal with lines that are perpendicular bisectors of a line segment
In a coordinate plane, what is the reflection of (a, b) about the y axis?
(a, b) is a reflection of (-a, b)
Both points have the SAME distance from the y axis (which acts as a perpendicular bisector of the line segment from (a,b) to (-a, b))
REFLECTIONS deal with lines that are perpendicular bisectors of a line segment
If the slope of a line is negative, which quadrants in the xy plane MUST it cross?
MUST cross II and IV
If the slope of a line is positive, which quadrants in the xy plane MUST it cross?
MUST cross I and III
Circles in coordinate plane
Things to keep in mind?
Radius means EQUAL DISTANCE from every point on the circle to the center
If the center is the ORIGIN, you can create a right triangle to calculate x, y point on the circle and/or the radius
> just need two info
When do you write an unknown integer in its algebraic form?
e.g., n is a two digit integer with digits a and b in the tens and units place, respectively
n = 10a + b
e.g., abc - cba = a multiple of 7
Digit question
If you are given any info about the VALUE of the integer, you can write it in the algebraic form
**don’t forget decimals can also be written in algebraic form
e.g., 3.81 = 3 + 8/10 + 1/100
e.g. 1, if a and b were reversed, the resulting integer is 27 more than n.
10a + b + 27 = 10b + a
(COMBINE THE REMAINING TWO VARIABLES)
10b - b + a - 10a = 27
9b - 9a = 27
b - a = 3 —-> we set a < b
e.g., 2. abc - cba = a multiple of 7 —> clearly this is a digit/multiple hybrid question
100a + 10b + c - 100c - 10b - a = 7int
99a - 99c = 7int
99(a - c) = 7*int —> a - c must be a multiple of 7 (e.g., 9 - 2, 8 - 1)
e.g. 3, you can determine the value of the UNITS digit
196 = 100(a - b) + 10(c + d) + (z - c)
> know that z - c = 6 because the other terms are just multiples of 10.
List of units digit question types
Questions related to powers, multiplication and addition
(but you can have computations relating to subtraction, as long as there is tens column to borrow from)
e.g., 9^19 - 7^15
e.g., 15^9 - 16^5 = u 5 - u 6 = 9 (not -1 –> units digit must be positive. And there is a tens column to borrow from).
1) Large exponents - BOTH MULTIPLICATION AND ADDITION
e.g., units digit of 13^53 or product of large integers
> focus on the pattern of the units digit raised to different powers
e.g., 1^4 + 6^4 + x^4 + 4^4 = 16x4
e.g., (24^17 * 99^18)
2) Linear problems with two variables representing integers
e.g., 15A + 3B = 100
3) Digit questions given the value of the units digit
e.g., 2016 = 300A + 30B + a + b
e.g., 2016 = A(B + 1)
** make sure the units digit is POSITIVE
** carry over is allowed (e.g., 6a = 6 units digit, a = 1 or 6)
e.g., a + b > 0
e.g., z - q > 0
4) Remainder of an integer divided by 10 or 5
Questions that ask for “could be the value of” or “possible values of” might be asking for what?
A RANGE of values
> need to find the MAX and MIN
> combine into one inequality (compound inequality)
CONSIDER whether the end points are INCLUSIVE or not
> typically if you solve for the max or min it will be inclusive
How do you determine whether a decimal is a terminating decimal?
e.g., 25/144
Write the fraction’s numerator and denom as prime factors
e.g., (5^2)/(2^4*3^2)
Simplify as much as possible
e.g., (5^2)/(2^4*3^2)
If the denominator has only powers of 2 or 5, then the decimal is terminating
e.g., (5^2)/(2^4*3^2) —> Not a terminating decimal
*integers are terminating decimals too
**DS problems:
> if you KNOW the denominator contains only 2’s and/or 5’s, then it is sufficient
e.g., t/s, s = 4
However if the denominator contains anything other than 2’s or 5’s, then it is not sufficient
e.g., t/s, s = 15 (we don’t know if 3 is cancelled out)
Probability questions with “at least”
e.g., P(at least 1 person successfully decodes)
P(at least __) = 1 - P(NOT_ and NOT_)
Triple venn diagram questions
> three groups: A, B and C
Formula to compute the total # of people
Formula to compute # of people ONLY 2 criteria
Formula to compute # of people ONLY 1 criterion
Total = A + B + C - (AandB + BandC + AandC) + center + other **
(could write them as %: 100% = A% + B% etc.)
> then multiply the final percent by total
A = all of A
B = all of B
C = all of C
AndB = only AandB AND center
…
Use the formulaic approach when the usual numbers approach (i.e., starting inside out) doesn’t work
To get # of people ONLY 2 (without center) –> AandB + BandC + AandC - 3*center
To get # of people ONLY 1 criterion:
Total - ONLY 2 - center
Questions about mean
What should you always do?
What about for questions about consecutive integers and mean vs median?
Write out the formula for mean
x bar = sum/# of terms
> often times there will be valuable info from the equation!
For questions about consecutive integers and mean vs median?
> mean = median ONLY for evenly spaced sets
> If Sum1 = Sum2, then mean1 = mean2 only equals if the sums = 0 or if the two sets have equal number of terms.
(BUT the medians DON’T have to be equal to each other!!
e.g., {-5, 2, 3} vs {-1, 0, 1}
Generally:
> Info about the relationship between AVERAGES is NOT sufficient to know about the relationship between medians
> Info about the relationship between MEDIANS is NOT sufficient to know about the relationship between averages
> info about the relationship between MEDIANS is NOT sufficient to know about the relationship between # of TERMS
If every term of a set changes by a constant difference, what happens to:
> the mean of the set
the median of the set
the range
the standard deviation of the set
Statistics - set transformations
If EVERY term of the set increases/decreases by delta (e.g., +3, -4)
Mean changes by delta
Median changes by delta
Range DOES NOT CHANGE
Standard deviation DOES NOT CHANGE
If every term of a set changes by a constant factor, what happens to:
> the mean of the set
the median of the set
the range
the standard deviation of the set
Statistics - set transformations
If EVERY term of the set increases/decreases by a factor of k (e.g., *3, /4, *-2)
Mean changes by *k
Median changes by *k
Range CHANGES by *k (keep the sign)
Standard Deviation CHANGES by *k
** standard deviation is ALWAYS POSITIVE (so multiply std by | k |)
What should you think about every time you divide unknowns or have a product of integers?
e.g., (a + 2)(4) = (t + 3)(7)
e.g., product of 5 integers
Determine whether the unknowns can equal 0
0 is a multiple of ANY number
Prime numbers properties (algebraically)
If p is a prime number, then p has no factors n such that 1 < n < p
In other words, the only factors (2) of p are 1 and p
How many factors does 6x have, if x is a prime number?
of factors = (exponent on prime + 1)(exponent on prime + 1) etc.
Prime Factor Properties
We NEED to know the value of x, otherwise, inconclusive
> x could already be a prime factor (e.g., 2 and 3 for 6)
> if x is NOT already accounted for, then the # of factors would DOUBLE
For complex geometry DS problems where you cannot exactly compute for an answer, what should you do?
Rubber Band Geometry
> once you FIXATE the shape, there must be one solution (it is possible to solve –> sufficient)
You don’t have to know how to solve the shape, as long as you know the shape is FIXED
e.g., SSS, SAS, AAS, ASA for triangles
e.g., radius and angle for sectors and arcs
e.g., length of a side in regular polygons
Formula to calculate TOTAL Simple Interest after t periods and TOTAL investment amount
Total Simple Interest = Initial Inv * (rate per period) * # of periods
**simple interest does not build on accumulated interest
Total Inv Amount = Initial Inv + Total Simple Interest
Formula to calculate TOTAL compound interest after t periods
Total Compound Interest = Initial inv * (1 + r per period)^n periods - Initial Inv
= P(1 + r/m)^mt - P
(where m = # of periods in a year, t = # of years)
Total Inv Amount = Initial inv * (1 + r per period)^n periods
a is an even number
b is an odd number
How would you express these algebraically?
a = 2n
b = 2n + 1
Application:
> PS questions about odd/even properties
> DS questions about remainders and divisibility
e.g., p = odd = a^2 + b^2
> a and b have opposite types
What is the remainder of p/4?
(2n)^2 + (2n + 1)^2 = 4n^2 + 4n^2 + 4n + 1 —> remainder always = 1
Can integers be terminating decimals?
Yes
Probability of event M occurring or event M not occurring
1 = P(m) + P(not m)
Rate questions
Get into the habit of doing what?
Writing it out in terms of DISTANCE (rather than a fraction):
Distance = rate * time
Is Even / Even an even or odd number?
Even or Odd
E/E = O —> E = O * E
E/E = E —> E = E * E
e.g., 4/2 = 2
e.g., 14/2 = 7
Is Even / Odd an even or odd number?
Even number
E/O = E –> E = E * O
E/O = O –> E = O * O (Doesn’t work)
e.g., 10/5 = 2
Memorize: EOE
Is Odd / Odd an even or odd number?
Odd number
O/O = O –> O = O * O
O/O = E –> O = O * E (Doesn’t work)
e.g., 21/3 = 7
Memorize: OOO
Is Odd / Even an even or odd number?
NOT POSSIBLE
Tip for solving decimal-rounding problems
e.g., d is a decimal. Is d >= 0.5?
Write out the place values and decimal
e.g., d = _ . _ _ _ = _ . a b c
is a >= 5?
Write as a fraction:
10^-2a+2
Don’t add brackets that aren’t there - assume default
> must factor out a -1
10^(-2a+2) = 10^-(2a - 2) = 1/[10^(2a - 2)]
Minimum value of exponent on 10 to make the product an integer
What should you keep in mind?
e.g., (0.0025)(0.002)*10^k = integer
Check to see if the non-10 integers can be multiplied into an integer ending with 0
> if yes –> reduce the power on 10
> if no –> keep power on 10 the same
e.g., (0.0025)(0.002)10^k = integer
(25 * 10^-4)(210^-3)10^k = integer
50(10^-7)(10^k) = integer —> 50 ends with 0
510(10^-7)(10^k) = integer
5(10^-6)*(10^k) = integer
k = 6 (not 7)
For complex geometry or “Must be True” PS problems where NO measurements are given, what is a helpful tip?
Since it is PS, there must be a solution.
You can assume one valid case is a REGULAR polygon –> with equal sides and solve
OR test any valid case
**question must not have given you ANY measurements
Mixture problems - how to set up (if WA doesn’t work)
What if you are given RATIOS in a mixture problem?
e.g., a 10L drink contains juice and water in the ratio 3:2.
Quantity of Ingredient Before the Mixture = Quantity of Ingredient After the Mixture
e.g., % alcohol * volume = amount of alcohol after mixture
> don’t overcomplicate variables (e.g., removing P from original V)
removed liquid shares same characteristics as the entire mixture (e.g., if 2/3 of the mixture is alcohol, then 2/3 of the removed volume is alcohol)
also Total Q of mixture + Total Q of mixture = Final Q of mixture
e.g., If you mix 1 ton of A with 2 tons of B, you get 3 tons of a combined mixture
**also keep in mind ratios can be converted to PERCENTS
e.g., Juice to Water -> 3:2 –> 3/5 = Wj, 2/5 = Ww
Machine X takes x hours to complete on job and machine Y takes y hours to complete the same job. If x < y, what does this tell us about the time to complete the job if both machines are working at their respective rates?
> total t relative to x and y?
Work problems:
x < y means that Machine x is more efficient (takes less time, has a faster rate)
1/x > 1/y
The TOTAL combined time (t) is always LESS than each machine’s individual time to complete
t < x < y
The RANGE of total time (t):
> consider what t would be if you had TWO machine x’s or TWO machine y’s
> t is smaller than the time it would take two machine y’s and greater than the time it would take two machine x’s
x/2 < t < y/2
Proof:
(1/x + 1/x)*t = 1
t = x/2
(1/y + 1/y)*t = 1
t = y/2
x/2 < y/2
What is the solution for these linear equations, given:
2A + 3B = 15
2A + 3B = 17
No solution
Parallel lines
What is the solution for these linear equations, given:
2A + 3B = 15
6A + 9B = 45
Not solvable because of infinite number of solutions
> SAME LINE
n is an integer that is both a square of an integer and a cube of an integer.
Write a general expression for n.
n = a^6
n has an exponent that is both divisible by 2 and 3.
Derived from:
n = (a^3)^2 = a^6
n = (a^2)^3 = a^6
If 3x = 5y and x and y both don’t equal to zero, what does this tell you about x and y?
What about (x)(x - 1) = y*m (all integers > 1)
Multiple and Factors:
x must be a multiple of 5
y must be a multiple of 3
**if you know either x or y, you FIXATE the value of the other!!
e.g., x = 5, then y must = 3
x = 15, then y must = 9
In the second example, we are NOT SURE if x is a multiple of y or m, or if x-1 is a multiple of y or m.
*x and x-1 have zero common factors other than 1
*also applies to x and x+1 (have zero common factors other than 1)
ANOTHER APPLICATION: h(100) and h(100) + 1 have zero common factors other than 1
> so the LEAST common prime factor must be LARGER than the current possible primes by one of the two numbers
e.g., h(100) = 50!, so the smallest prime factor of h(100) + 1 must be greater than 50 (such as 53)
> multiple + non-multiple = non-multiple
(n - 1)(n + 1)
What could the question be testing?
1) Possibly testing product of consecutive even or odd integers
(n - 1)(n)(n + 1)
2) Possibly testing difference of squares
3) possibly testing FACTORS (if given # of the other side of an equation)
Two people start moving towards each other at the same time and then meet together
What do they share in common?
Share common time
Dist travelled by A + Dist travelled by B = Total distance between them
Two people move towards each other, one starting before the other.
What do they share in common?
Share common time AFTER the second person starts moving
Dist Travelled by First mover at beginning + Dist travelled by first mover when other moves + Dist travelled by Second mover = Total Distance
One person (faster speed) chases another person. There is an initial gap.
What do they share in common?
Share common time after the faster person starts moving.
Share common distance once faster person catches up
Initial Gap + Dist travelled by slower person = Dist travelled by faster person
Two people move in the same direction. Faster person waits for the other person to catch up.
What do they share in common?
Share common distance once slower person catches up.
Share common time when both are moving
Dist travelled by faster person = Dist travelled by slower person when faster person is still moving + Dist travelled by slower person while faster person is waiting
Two people run around a circular field. One person is faster than the other. The faster person eventually meets up with the slower person.
What do they share in common?
Share common time
Gap between two people = 2pir
Dis travelled by faster - Dist travelled by slower = 2pir
f(x) = a^x properties
Exponential function, has negative and positive values for x
If a > 1 –> upward sloping –> can drop the base in algebra
If 0 < a < 1 –> downward sloping
Running list of ODD/EVEN testing points
1) Odd/Even addition, subtraction, multiplication and division
O +/- O = even
E +/- E = even
O +/- E = odd
OO = odd
Oe = even
e*e = even
E/E = Odd or even
E/O = Even
O/O = odd
O/E = not possible (not an integer)
(2) Unknown +/- even or odd integer
x + 1 –> opposite type of x
x + 2 –> same type as x
(3) Identifying if 2 is a factor of unknown
(4) Consecutive integers
(n)(n + 1) –> one is even
Factor (a + b)^3
(a + b)^3
= a^3 + 3a^2b + 3ab^2 + b^3
(Exponent on “a” decreases from 3 to 0; exponent on “b” increases from 0 to 3)
Middle terms have 3 as coefficient
If there are 10 symbols and 5 are chosen to form a code, how many different codes are there?
Permutation
Pm,n = From m choose n
= m! / (m - n)!
**assumes NO duplicates!
> if there ARE duplicates, then you have to think about TIERS
e.g., PEPPER –> _ _ _ , how many unique arrangements?
P P P = 1 way
P P other = 2(3!/2!)
E E other = 2(3!/2!)
P E R = 3!
Total = 1 + 6 + 6 + 6 = 19
LONG WAY to think about it:
> First determine the # of ways to choose 5 symbols from 10
> Then, multiply the # of ways by the # of ways to arrange 5 chosen symbols
= (m!)/[(m-n)!*(n!)] * n!
If there are 10 students in a class, how many different groups of 5 can be formed?
Combination
Cm,n = From m choose n
= m! / (n! * (m - n)!)
** often part of complex probability questions
e.g., 100 students, 20 girls, 80 boys. A group of three is formed. What is the probability that the group comprises only of girls?
_ _ _ = G G G
Approach 1: C20,3 / C100,3
Approach 2: P(G)P(G)P(G) = 20/100 * 19/99 * 18/98
What is the approach for solving inequality word problems in DS?
Characteristics:
> Different number of integer values
> Sum given
> Asked if one value > or < a number
Test two cases: Yes and No case
> if both work, NS
See if it is possible for the sum of the remaining numbers to meet the remaining sum
Helpful to find the average of the remaining numbers and adjust each number so that the sum of the gaps = 0
If a question asks about factors of an integer, what should you think about?
e.g., x is a factor of 20.
e.g., (n - k)(n + k) = 48
e.g., (x + 1)(y + 1) = 9
Factor PAIRS (doesn’t have to be a prime factor!!)
Start with 1 * n
You can also evaluate the INSIDE of factors too:
(common for # of positive factor questions)
e.g., (x + 1)*(y + 1) = 9
1 * 9 = 9 —–> x + 1 = 1 and y + 1 = 9 —-> x = 0, y = 8
3 * 3 = 9 —–> x + 1 = 3 and y + 1 = 3 —–> x = 2, y = 2
** might have integer constraints on the variables
Remainders: Is this sufficient to know the remainder?
[7*sqrt(C)]/7 , where sqrt(C) is an integer
Yes, remainder is 0, regardless of the value of sqrt(C)
If there are n terms in a set and n is ODD and x is the median, how many terms are smaller than x and larger than x (assuming all numbers are different)
of terms smaller than x = (n - 1)/2
ODD # of Terms:
# of terms other than x = n - 1
# of terms larger than x = (n - 1)/2
Max VALUE (consecutive integers) = x + (n-1)/2
MIN value (consecutive integers) = x - (n - 1)/2
Median = 50th percentile (an equal number of terms are smaller and larger than the median, not including the median itself)
> if it were 90th percentile, the value is at position 90% (position/n = 90%)
e.g., 1 2 3 4 5 –> median is 3, 5 terms
There are 5 - 1 = 4 terms other than the median.
And there are 2 terms BELOW 3 and 2 terms ABOVE 3
If there are n terms in a set and n is EVEN and x the median, how many terms are smaller than x and larger than x (assuming all the numbers are different)
If y is at the 60th percentile, how many terms are smaller than y and larger than y (assuming all numbers are different)
e.g., 10 terms, y is at 60th percentile
of terms smaller than x: (n - 1)/2 —> will be a fraction with denom 2 (ROUND UP)
EVEN # of terms: Same as odd # of terms:
# of terms larger than x: (n - 1)/2
Max VALUE (consecutive integers) = x + (n-1)/2
MIN value (consecutive integers) = x - (n - 1)/2
Means the position of y is 60%
–> Position of y / n = 60%
e.g., 60%*10 = 6 (y is located at position 6)
> There are 5 terms below y
> There are 4 terms above y
1 2 3 4 5 6 7 8 9 10
y = position 6 = 6
6/10 = 60th percentile
Is it possible to solve for the difference of two unknowns (like profit = rev - costs) given only their ratio?
No
e.g., Py - Px = ?, is not sufficient to solve given Py = 2Px
—> 2Px - Px = Px = ?
This is different from ratios –> sometimes the unknowns cancel out
Profit Percentage Formula (for GMAT)
(Price - Cost)/Cost = positive or negative percentage
Round table permutation question - formula for the # of different arrangements?
(n!)/(n)
= (n - 1)!
ABCDE = EABCD …
> you don’t need to divide by n factorial because the factorial accounts for DIFFERENT orders of the seats (we just want to account for the SAME ORDER, but moving around the circle)
When can you square both sides of an inequality?
If you KNOW the signs of both sides
> flip sign if both sides are negative
> keep sign if both sides are positive
> Cannot square if sides have opposite signs
HAVE NO FEAR If both sides are POSITIVE
e.g., If x > y > 0, then x^2 > y^2 > 0
e.g., sqrt(x) > sqrt(y)
x > y
e.g., | x | > | y |
x^2 > y^2
e.g., x > 3
x^2 > 9
e.g., sqrt( 63 + sqrt(3)) > x + y*sqrt(3), where x and y > 0
When can you divide both sides of an inequality by an unknown without changing the direction of the inequality?
e.g., r^2 < | r |
If both sides are POSITIVE
For what values does this hold?
What about | x - 3 | <= -y (y >= 0)
x | = -| q |
x and q both equal 0
Any other number would make the statement false
x - 3 = 0 to make the second statement hold true
| x - 3 | + y <= 0
x | + | q | = 0
Probability questions
When do you multiply by cases ?
Always reword the question with probability stem:
P( ) = ?
If P( different items) AND not fixed already, you need to multiply by cases.
e.g., P(A and B and C) = P(ABC) * 3!
P(A and A and C) = P(AAC) * (3! / 2!)
P( G and G and B and B) = P(GGBB) * (4!/(2!*2!))
If P( identical items), you don’t need to multiply by cases
e.g., P(AA)
e.g., P(MM)
If P(all different pairs method), you don’t need to multiply by cases
P(4 different numbers) = P(diff and diff and diff and diff)
How do you tell if a permutation question has duplicates or not?
“Identical” except colour –> duplicates (e.g., 3 red balls and 4 green balls. All the red balls are duplicates of one another).
Think about if order matters among the same “type”
e.g., order of performances where there are 5 songs and 3 dances –> yes, order matters (d1 then d2 is different from d2 then d1)
e.g., order of MMM and FFF
So differentiate between SAME TYPE and IDENTICAL
> “same type” e.g., 6 junior partners and 3 senior partners —> need a group with two J and 1 S
e.g., 5 perennial flowers and 2 annual flowers —> need a group with 4 P and 1 A
How would you express this:
What is the probability that neither of the two numbers is prime?
Or How many people belong to neither A or B?
P(not prime and not prime)
NOT
1 - P(prime and prime) —> misses case where one of the numbers is prime
Sets variation: Neither A or B = Total - (A + B - AandB)
How would you simplify this?
(n+1)!/(3!(n-2)!) - n!/(3!(n-3)!) = 150
Factorials
Recall that you can expand the factorial:
(n+1)! = (n+1)(n)! = (n+1)(n)*(n-1)!
Therefore:
(n+1)(n)(n-1)(n-2)! / (3!(n-2)!) - (n)(n-1)(n-2)(n-3)! / (3!(n-3)!)
(n+1)(n)(n-1)/3! - (n)(n-1)(n-2)/3! = 150
(n)(n - 1)*(3) / 3! = 150
If you are given a NUMBER, it is generally SUFFICIENT TO SOLVE (simplifies into a quadratic function)
> one variable, one equation (and negative in front of bx and c)
**
If you have two statements in a DS problem that give you two possible roots of a quadratic function each, can you figure out what the solution is?
Yes if there is one overlap
e.g. 1) x = 5 or -2
2) x = 5 or -4
Together, x must be 5 to satisfy BOTH
PS problem involving a RANGE of possible values
e.g., if x^4 + y^4 = 100, then the greatest possible value of x is between:
0 and 3
3 and 6
6 and 9
9 and 12
12 and 15
TYPE 1: Solve for the range of values (max and min)
> if you are asked what the VALUE of something falls in
> keep track of end points
Type 2: Solve for the max (or min) and then see which answer choice reflects the correct range
> if you are asked for EITHER the MAX or the MIN (so you cannot solve for the “range”)
e.g., x max = sqrt(10) = 3.xx
So 3.xx falls between 3 and 6.
What factors do x and x-1 or x+1 have in common?
No factors in common (other than 1)
or
| a - c | = | b - c |
c - a | = | c - b |
Distance on number line
> c is in the MIDDLE between a and b (if a =/ b)
> or a = b (a and b are the same point)
Also | c - a | = | a - c |
When drawing scenarios, focus on the common point, c:
c +/- #
Whenever you see male and female in a set question, what should you do?
M + F = total —> might be sufficient to solve algebraically
What is the least possible value of n?
n^3 = 450*y
Concept: a PRIME of n^3 MUST BE a PRIME of n
Prime factorization:
n^3 = 23^25^2 * y= nnn
> three identical factor sets of n
n must contain the MINIMUM number of integers such that n^3 = 450*y
To guarantee 450 is a factor of n, n contains at least:
> one 2
> one 3
> one 5
n = 235 (least possible value of n)
n^3 = 2^3 * 3^3 * 5^3
= 450 * (2^2 * 3 * 5)*m where m is another possible factor that is a cube
y = (2^2 * 3 * 5)*m
> y is a multiple of 60
Max and min problems:
a < b < c
a <= b <= c
Assume a, b, c are positive integers
Maximize C?
Minimize C?
Maximize A?
Maximize any value by MINIMIZING the other values
> think of a number line and what the the max and min value a number can take
a < b < c
Maximize C: Minimize a and b (a = 1, b = 2)
1 < 2 < c
Minimize C: Maximize a and b (b is 1 less c, a is 2 less c)
c - 2 < c - 1 < c
Maximize A: Minimize b and c (b is 1 more a, c is 2 more a)
a < a + 1 < a + 2
a <= b <= c
Maximize C: Minimize a and b (a = b = 1)
1 <= 1 <= c
Minimize C: Maximize a and b (a = b = c)
c <= c <= c
Maximize A: Minimize b and c (b = c = a)
a <= a <= a
*often a constraint on c too (c is dependent on a –> then you just minimize b)
Watch out for wording e.g., each integer can appear in the list at most twice (one OR two times!!)
Weighted Average Formula
WA = weight% * value + weight% * value
WA = [(value# of items) + (value# of items)] / (total # of items)
Application: mixture question, multiple averages
e.g., Avg total = [(avgA * # of A) + (avgB * # of B)]/(A + B)
> if you know the individual averages and total avg, you can solve for the ratio of # of A to # of B
> avgA * # of A = Sum of A
**Can also work for MORE THAN THREE ITEMS
(you can still use visual approach or mathematical approach).
WA(QA + QB + QC) = PriceAQA + PriceBQB + PriceCQC
22QA + 7QB = 5QC
AND:
5QA + 5QB < 22QA + 7QB = 5QC
QA + QB < QC —> Max is C
What is x - y?
Given: y - x
SPOT THE PATTERN:
y - x = - (x - y)
represents + - * or /
Symbol questions:
Is (6#2)#4 = 6#(2#4)?
(1) 3#2 > 3
(2) 3#1 = 3
In each statement, test each symbol to see if the statement remains True.
Keep only VALID cases and then evaluate the question.
e.g.,
(1) 3#2 > 3:
3 + 2 = 5 > 3 (valid)
3 - 2 = 1 > 3 (invalid)
3 * 2 = 6 > 3 (valid)
3/2 = 1.5 > 3 (invalid)
so # can either be + or *:
Does (6+2) + 4 = 6 + (2+4)? —> Yes
Does (62)4 = 624 ? —-> yes
Always Yes –> sufficient
(2) 3#1 = 3:
3+1 = 4 = 3 (invalid)
3-1 = 2 = 3 (invalid)
3*1 = 3 (valid)
3/1 = 3 (valid)
So # can either be * or /:
Does (6+2) + 4 = 6 + (2+4)? —> Yes
Does (6/2)/4 = 6/(2/4)? —> No
NS
Perfect squares and divisibility
All perfect squares are in the form:
4a or 4a + 1
> (even int)^2 —> divisible by 4
> (odd int)^2 –> remainder 1 when divided by 4
3a or 3a + 1
> (multiple of 3)^2 –> divisible by 3
> (non-multiple of 3)^2 –> remainder 1 when divided by 3
Divisibility questions
e.g., There are 15 white roses and 85 red roses. All of them must be used to create a number of bouquets. Each bouquet contains an equal number of white and red roses. What is the max number of bouquets that can be created?
Max # of bouquets = GCF = B
15 = # white roses per bouquet * B = 3 * 5
85 = # of red roses per bouquet * B = 17 * 5
B = 5
Can you take reciprocals of inequalities?
e.g., If x < y, can I make it into 1/x and 1/y?
If you DON’T know the sign of x and y, you CANNOT take reciprocals
Otherwise, the sign of x and y determine whether to flip the inequality or not.
> FLIP the sign UNLESS x and y have DIFFERENT signs (+- or -+)
e.g., -6 < 2 (different signs, don’t flip the inequality)
1/-6 < 1/2
RULE:
Same sign: Flip
Opposite signs: Do not flip
of terms?
Evenly spaced sets
Mean and median?
Sum?
of terms = (Last - First)/increment + 1
Mean = Median = (First + Last)/2
Sum: Avg * # of terms
ALL INCLUSIVE end points
> e.g., if trying to find multiples of 7 and the end points are [10,80] –> change to [14, 77]
APPLIES TO:
- consecutive integers
- any evenly spaced set (arithmetic sequence)
Integer
Positive integer
What is 0
Integer = + whole number or - whole number or 0
Positive Integer: > 0 int
Negative Integer: < 0 int
0 is NEITHER positive NOR negative
a + b = -1
ab = -6
or 2a - b = 1
ab = 4
Is this sufficient to find the values of a and b?
Yes – we still have two different equations
> eventually will get a quadratic function where ax^2 - bx - c (one positive answer)
(negative in front of bx term and c, or just in front of c term - especially important for positive integer constraint questions!!)
Long way proof:
a = -1 - b
(-1 - b)*b = -6
-b - b^2 = -6
b^2 + b - 6 = 0
(b + 3)(b - 2) = 0
Therefore, b = -3 or b = 2
If b = -3, a = 2
If b = 2, a = -3
We can tell that a =/ b. And we can see there are only TWO unique roots
Translate the following into equations:
1) A is 1/4 taller than B.
2) A is 1/4 meters taller than B.
What if you are given “fraction” instead?
3) B increased by a fraction to get A
4) What about: A is 5/6 times B
5) A is three times greater than B
6) There is twice as much A as B
7) A is 10% greater than B
PAY ATTENTION TO WORDING:
A is 1/4 taller than B: A > B by a factor of 1/4
A = (1.25)B
CHANGE = A - B = (1/4)B
IN VARIABLE FORM:
A = (1 + fraction)B (if fraction < 1) or A = (fraction)B (if fraction > 1)
A is 1/4 meters taller than B: A > B by a difference of 1/4
A = B + 0.25
TIMES = MULTIPLICATION ALWAYS (ignore other stuff):
A is 5/6 times B:
A = (5/6)*B
A is three times greater than B:
> same as A is 3 times B
A = 3*B
There is twice as much A as B:
A = 2B
A is 10% greater than B
A = (1 + 10%)*B
A = 1.1B
GCF and LCM DS questions
Pros and Cons of being given GCF and LCM of two unknowns?
GCF = product of the LEAST POWERS in each “column”
> represents the DIVISORS of unknown
> circle the number in each column that dictates the GCF
> use descriptive exponents e.g., 1+ to indicate that there could be more
> typically might have powers of 0
> COULD EXCLUDE FACTORS (but provides info about common divisors = multiples)
LCM = product of the GREATEST POWERS in each “column”
> the unknown is the divisor of the LCM
> INCLUDES ALL FACTORS (but hides which factor belongs to which number)
> if you are given LCM of two unknowns and asked to find LCM of the two unknowns plus one more integer, IT IS SUFFICIENT
Ratios
A) how do you express actual amounts?
e.g., M/F = 2/1
e.g., the length and width of a rectangle are in the ratio 3 to 2.
B) What if you are given the MAX amount available (e.g., max amount of yellow paint is 10 quarts)?
C) What about “doubling” the ratio or “halving” the ratio?
D) Can you combine ratios?
e.g., A/B = 2/1 and B/C = 3/5
E) Part vs whole
e.g., # of men = 2/3 # of women
A) Use “x” as the multiplier (could have an integer constraint)
M: 2x
F: 1x
L/W = 3/2 –> L = 3x and W = 2x
*you can perform operations on the “actual” amounts
e.g., 2x - 10 –> subtract 10 from the number of men
B) Max amount simply constrains the ACTUAL value:
e.g., Blue, Yellow, and Red paint are in the ratio 2:3:1
Actual Yellow Paint = 3x
> If max amount available is 10, then Actual <= 10, or
3x <= 10
C) Doubling ratio = 2ratio
Halving ratio = 0.5ratio
D) Yes, you can combine ratios –> make sure the individual ratios still work in the combined ratio
E) Part vs whole –> you can FIGURE OUT THE RATIO of part to whole (but not know the actual amount)
> recall that ratios and percentages are interchangeable
e.g., # of men = 2/3 # of women
M = 2/3*W
M/W = 2/3
THEREFORE M/Total = 2/5
Ratios - how flexible are they?
e.g., M/F = 2/1 —> can you express as F/M = 1/2?
Ratios are flexible - you can invert them (because of cross multiplication)
PS: Tips for approximations?
A) Closest to a number (i.e., 0.5)
B) Approximations of roots (i.e., sqrt(569000), (4)^1/3)
C) Approximation of decimals and fractions (i.e., 0.99999999/1.0001)
A) Choose the answer that has the SMALLEST GAP from the desired number (i.e., 0.5)
> calculate all the gaps, then choose the smallest one
B) Set up the appropriate RANGE of values based on easy numbers.
> for Square roots: try turning square roots into POWERS
e.g., sqrt(569000) = x
569000 = x^2
700^2 < 569000 < 800^2 —–> 700 < x < 800
e.g., 4^1/3
> we know 8^1/3 = 2; 1^1/3 = 1
1 < 4^1/3 < 2
C) Remember DIFFERENCE of squares
e.g., 0.99999999/1.0001
= (1 - 0.00000001)/(1 + 0.0001)
= (1 - 10^-8)/(1 + 10^-4)
= (1 - 10^-4)*(1 + 10^-4) / (1 + 10^-4)
= 1 - 10^-4
**don’t forget that you can WORK BACKWARDS from the answers (B, D)
** # of zeros (including before the decimal) = value of the exponent
e.g., 10^3 = 1 with 3 zeros = 1000
10^-3 = 1 with 3 zeros = 0.001
How do you find the remainder given a decimal?
e.g., s/4= 3.75
e.g., s/t = 96.12
RECALL: a/b = int + remainder/b
a = b*int + remainder
Multiply the decimal by the divisor to get an INTEGER REMAINDER
e.g.,
s/4 = 3 + 0.75
s = 12 + 4(0.75)
s = 12 + 3 —-> remainder is 3 (and s = 15)
e.g.,
s/t = 96 + 0.12
s = 96t + 0.12t
0.12t must be an INTEGER –> (3/25)*t must be an integer, so t must be a multiple of 25.
If you are given the REMAINDER, then you can SOLVE FOR the divisor.
If a rubber band is wrapped tightly around two circles, what should you think about?
Tangents - the rubber band leaves the circles at the SAME symmetrical spot
When you see graphs asking about percent change, what should you do?
Write out: (P2 - P1)/P1 or P2/P1 - 1
Ratios and Percents: When can you declare something SUFFICIENT (two variables)?
1) Given RATIO (part to part relationship), you can calculate the PERCENT (or weights) or FRACION (part to whole)
e.g., M/F = 2/1 or M = 2F
M/(M + F) = (2F)/(2F + F) —> F cancels out
(M + F)/M = (2F + F)/2F —> F cancels out
2) Given PERCENT (or weights), you can calculate the RATIO (via cross multiplication)
e.g., M/(M + F) = 2/3
3M = 2M + 2F
M = 2F
MUST BE FOR TWO VARIABLES
IN EITHER CASE ABOVE, you DON’T KNOW the QUANTITIES
Right triangle with height drawn, creating two mini triangles –> what are the special properties of this triangle?
3 triangles (two smaller, one larger) are SIMILAR
Match up sides based on the angle
> Larger triangle: 90 degrees, angle a, angle b
> each mini triangle has 90 degrees and either angle a or angle b. So the third angle must be either angle b or angle a
Review the proportions
If h = height:
h/x = y/h
Double Matrix problems - two types and strategies
1) Value question
> Indicators: given quantities
> often needs at least two info in each column or row to solve
2) Ratio question
> Indicators: given percents or ratios
> often can solve using just one equation that gives you the necessary ratios (read the question carefully!)
Two lines both have positive slopes - which slope is larger?
Two lines both have negative slopes - which slope is larger?
When two lines have slopes that are the SAME sign, the line with the LARGER ANGLE with respect to the positive x axis has the larger slope.
Xy plane geometry - asked to find the area of a triangle (which may or may not be a right triangle) and you are given coordinates of each vertex
Strategy?
*you could see if the triangle is a right triangle by comparing the slopes and seeing if there are negative reciprocals
Safer approach that works with ANY triangle: Trapezoid or Square or rectangle
> Create a trapezoid or square comprised of the triangle in question and other smaller triangles or rectangles
> draw helping lines to the closest axis (x or y) - depends on where the vertex lies
Area of trapezoid = 1/2(base1 + base2) * height
If one of the vertices is on the axis –> can create right triangles
Area of the triangle = Area of trapezoid - area of smaller shapes
Questions about rectangular border or rectangular solids with some “thickness”
Concept: Must take into account thickness*2 into each length and width (and height)
Length = inner length + 2thickness
Width = inner width + 2thickness
Height = inner height + 2*thickness
AREA of BORDER = Area of larger rectangle - Area of smaller rectangle ***
Approach to solving ugly numerical calculations (with large numbers, decimals)
e.g., 0.999999/1.001 or 2495*2505
Recognize factor patterns
Difference of Squares
DS Word problems with inequalities - Best approach?
e.g., is r > 8%
e.g., is x < 0.8?
e.g.,, is c + d > 140?
e.g., is Friday > 11?
e.g., is Fx/Px > (Fx + Fy)/(Px + Py)?
1) ONE VARIABLE - Solve for the value of the unknown (using statements) or Solve for an Expression of the unknown
e.g., r = 10% (Sufficient) —-> as long as you know it is solvable, you can declare S
e.g., x < 0.6 (Sufficient)
or
2) ONE VARIABLE - Test two cases around the boundary (see if the statements remain valid or not)
e.g., test case r = 8% and r = 9% –> both work, so Not Sufficient
e.g., test c + d < 140 –> invalid and never possible –> Sufficient to know that c + d > 140
e.g., test F = 11 and F = 12 –> both work, so Not Sufficient
or
3) MULTI VARIABLE - move all the variables to one side –> evaluate properties (>0 < 0)
What is the meaning of Proportional “AND” Inversely Proportional
e.g., Cost is proportional to the square of the length and also proportional to the thickness.
Combine into ONE equation
Proportional: y = kx
Inversely Proportional: y = k/x
*k is a constant
e.g., c = k(L^2)(T)
Questions dealing with a time range
e.g., between 8am and 10am
MAKE SURE you don’t forget the inclusive range, even if the data set starts after the start of the range or stops before the end of the range.
e.g., [8am, 10am]
When I see x + y > 0 …
AT LEAST ONE is positive (Recall: x PLUS y > < 0 have “at least one”)
+ + > 0
+ - > 0
- + > 0
Often shows up on positive/negative questions
> test a few cases if you are unsure
> don’t forget that you can COMBINE inequalities (addition if the signs are in the same direction) –> evaluating two of these together
When I see x - y < 0 …
Difference must be negative but…
- < 0
+ + < 0 (| y | > | x |)
- < 0
- < 0 (| x | > | y |)
Often shows up on positive/negative questions
> test a few cases if you are unsure
> don’t forget that you can COMBINE inequalities (addition if the signs are in the same direction) –> evaluating two of these together
When I see x - y > 0 …
Difference must be positive but…
+ + > 0
+ - > 0
- - > 0 (| y | > | x | such as -1 - (-2) = 1)
Often shows up on positive/negative questions
> test a few cases if you are unsure
> don’t forget that you can COMBINE inequalities (addition if the signs are in the same direction) –> evaluating two of these together
if x^2 is an integer, what does this tell you about x?
What about x^3?
x is an integer OR an irrational number (sqrt(2) - cannot be expressed as a simple fraction)
TYPICALLY: Only an integer * integer = integer
When I see x + y < 0 …
AT LEAST ONE is negative (Recall: x PLUS y > < 0 have “at least one”)
- < 0
+ - < 0
- < 0
- < 0
e.g., 2x + 3y < 0
Often shows up on positive/negative questions
> test a few cases if you are unsure
> don’t forget that you can COMBINE inequalities (addition if the signs are in the same direction) –> evaluating two of these together
Inequality MUST be true question types (multi-select)
How do you approach them?
I) single variable, one inequality sign
(x^2 - x^3 > x^4 - x^5)
II) Single variable, multiple inequality signs and exponents
(1/x < 2x < x^2)
III) Multi variable, multiple inequality signs
(x > y^2 > z^4)
I) single variable, one inequality sign
> Sewing approach –> see if the answer must be true in light of the fact
II) Single variable, multiple inequality signs and exponents
> Functions –> draw it out very accurately (compare y values for A GIVEN value of x)
III) Multi variable, multiple inequality signs
> Number properties (fractions, positive/negative) – test cases
List prime numbers from 1 to 50
First 10 primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29
Full list of primes from 1 to 50: 15 primes
2, 3, 5, 7, 11, 13, 17, 19, 23, 29
31, 37
41, 43, 47
Formula for diagonals in a regular polygon
e.g., octagon, heptagon, nonagon, decagon
Diagonals = [(n - 3)*n]/2
Logic:
> each vertex can connect with n - 3 vertices
> divide by 2 to get rid of duplicates
Heptagon = (7-3)70.5 = 14
Octagon = (8-3)80.5 = 20
Nonagon = (9-3)90.5 = 27
Decagon = (10-3)100.5 = 35
**also the same approach for shaking hand questions
# of shakes = [n*(n - # of ppl cannot shake)] / 2
e.g., 24 ppl need to shake hands, but you cannot shake hands with the people on your team (4 ppl per team)
**Could also be viewed as a COMBINATION question
Cm,r - restriction
Factorial questions
Often about factors (i.e., combining into powers of 2, 5, multiples)
**any factorial greater than 1 is EVEN
e.g., 200! = p*10^q
What is the max value of q?
e.g., Is k a composite number?
13! + 2 <= k <= 13! + 13
13! + 2, 13! + 3, … 13! + 13 ALL are MULTIPLES of an integer => composite!
Percent questions - what should you think of?
e.g., Mrs. Lee’s May income was 60% of her family’s total income
Immediately write out the “other” percent
e.g., Mrs. Lee’s May income was 60% of her family’s total income
THEREFORE, her other family member’s income was 40%
What are the units digit of 2^(8r + 6) ?
Units digit of 2 has a cycle of 4:
[2, 4, 8, 6]
Every EXPONENT that is a multiple of 4 has a units digit of 6
Exponent in this question: 8r + 6
> exponent is always 6 more than a multiple of 8
> or always two more than a multiple of 4
> so the remainder is FIXED regardless of the value of r
(8r + 6)/4
= 2r + 6/4
Exponent is remainder 2, so the units digit is 4
Questions about quadrants
e.g., If ab =/ 0 and points (-a, b) and (-b, a) are in the same quadrant of the xy-plane, is point (-x, y) in this same quadrant?
Either a question about:
1) Slopes
2) SIGN of x and y coordinates
> to be in the same quadrant, the x coordinates must have the same sign AND the y coordinates must have the same sign
Special type of venn diagrams:
Some overlap
Complete overlap (i.e., B is inside of A)
of ppl = A + B - overlap + neither
Some overlap = A + B - overlap
Complete overlap means that every member of B is ALWAYS a member of A
> however, not every member of A is a member of B
Total # of ppl = A + B - B = A (outside circle)
**DON’T FORGET THE OUTSIDE PPL
> always draw a box
If you are trying to minimize the # of ppl –> maximize overlap
Work problem - typical variables?
Time (therefore rate = output/time)
or
Rate (so rate*time = output)
CONNECT RATE AND TIME in DS:
rate = job/time
Whenever I see unknown percents, how should I write them?
e.g., The population in 1998 increased by x% from 1997 levels, and the population in 1999 fell by x% from 1998 levels.
x% = x/100, where x is an value greater than 1 (e.g., 2%)
1-x% = (100 - x)/100 = 1 - (x/100)
E.g.,
1998 = 1997_amount * (1 + x/100)
1999 = 1997_amount * (1 + x/100) * (100 - x)/100
Common conversions:
A) Seconds to hours
B) Mins to day
C) m^3 to cm^3
A) 3600 seconds in an hour
B) 1440 minutes in a day
C) VOLUME
1m = 100cm (start with the basic unit conversion, then transform)
1m^3 = 100^3 cm^3
Percents combined with statistics question
e.g., 67% of the expenditure was paid by the top 6 highest-contributing countries
e.g., 25% of the classes had more than 20 students in their class
TOTAL absolute value is IRRELEVANT
Type 1) Percent represents VALUE
> sum of individual percents adds to the total percent
e.g., _ _ _ (n = 3)
25% , 25%, 50% (percents are individual TERMS)
Type 2) Percent represents POSITION (e.g., percentile)
> Focus in on PLACEMENT or ORDER of these values
e.g., _ _ _
(less than 20), (20), (over 20)
Hard permutation and combination question tips
e.g., Coordinate plane geometry shapes
e.g., Set A has 6 subsets with exactly 2 terms in them each.
e.g., What is the least number of letters that can be used to create enough codes for 12 employees?
> one or two letter code (letters must be arranged in alphabetical order)
What is the formula for total number of possible groupings (anywhere from 0 in a set to all terms in one set)?
e.g., How many different ways can a group of 8 people be divided into 4 teams of 2 people each?
1) Draw out a few possible combos or perms first
> if the answer choices are quite small, you can probably just list them out
other times, do in terms of GROUPS of options
e.g., Set A has 6 subsets with exactly 2 terms in them each.
6 = Cm,2
e.g., What is the least number of letters that can be used to create enough codes for 12 employees?
12 <= Cm,1 + Cm,2
> 2^n - 1 is for at least one term chosen
often needed when you are given the total number of SUBSETS
> e.g., digit questions (sometimes it is helpful to organize by digit – 7 in 000s, 8 in 000s, then 9 in 000s) —> “paths”
e.g., Once you fixate point P (based on # of possible x * # of possible y), the other two points are fixed either vertically or horizontally
(The number of ways in which n different items can be divided equally into x groups, each containing y objects and the order of the groups is NOT important)
C8,2 * C6,2 * C4,2 * C2,2 = # of possible teams of four with 2 people each, and ORDER of TEAMS matters (e.g., [1,2]-[3,4]-[5,6]-[7,8] is currently treated differently from [3,4]-[5,6]-[7,8]-[1,2])
> need to DIVIDE by # of groups! (e.g.., 4!)
Special numbers to consider when testing
0 - multiplication (pos/neg), exponent (a^0 = 1), multiple (0 is a multiple of every number)
1 - factor (1 is a factor of every number!!), exponents (1^n = 1)
> e.g., 81/x
2 - even/odd, only even prime number
What do these decimals signal?
1.44
14.4
1.69
1.21
2.25
Perfect squares of DECIMALS
1.44 = (1.21.2)
14.4 = (1.21.2)10
1.69 = (1.31.3)
1.21 = (1.11.1)
2.25 = (1.51.5)
How do you express rounding?
e.g., When distance is rounded to the nearest integer,, Sally travelled 12 m
e.g., when rounded to the nearest multiple of 100, 500 is closest to the integer x
Using compound inequalities
e.g., When distance is rounded to the nearest integer,, Sally travelled 12 m
11.5 <= Distance < 12.5
e.g., when rounded to the nearest multiple of 100, 500 is closest to x
450 <= x <= 549
If x is a decimal between 0 and 1, and 9x is an integer, what are the possible values of x?
What are the possible values of k if (10x + 20y)/(x + y) = k (x, y and k are all positive integers, x < y)
Integer * decimal (between 0 and 1) = Integer
> rewrite decimal as a FRACTION
> DENOM of the fraction must be entirely cancelled out
Ex: 9x = integer
x = (1/9), (2/9), (3/9), (4/9), (5/9)….(8/9)
Ex: 10(x + 2y)/(x + y) = k
x + y = 10 to cancel out the 10 in the numerator
x < y:
1, 9 —> k = 19
2, 8 —> k = 18
3, 7 —> k = 17
4, 6 —> k = 16
Semi-circle Tunnel Question - things to keep in mind?
Max height of the tunnel = radius
Max Height of a Truck < Radius (because the truck has a WIDTH)
Max Height of a Truck (assuming 0 space between height of the truck and the tunnel) = calculate using Pythagorean Theorem
> hypotenuse = Radius
> smaller leg = half the width of the truck
> longer leg = max height
Formula for Volume of a Cylinder and SA of a cylinder?
Formula for Volume of a Cone and SA of a cone?
Formula for Volume and SA of a sphere?
Formula for Hemisphere SA, Volume and properties
Cylinders: both need height
V = (pir^2) * h ——-> area of base * height
SA: 2(pir^2) + 2pir*h —-> 2 circles + rectangle
Cone:
V: 1/3volume of a cylinder
= (1/3)(pi*r^2) * h
SA: circle + sector = pir^2 + pir*L
(symmetry)
**Length is the hypotenuse in a right triangle with Height and Radius
Sphere:
V: (4/3)(pir^3)
SA: 4pir^2
(no height for sphere)
> both have “4”
Hemisphere: 1/2 of a sphere
V = (2/3)(pir^3)
****SA: 2pir^2 + pir^2 = 3pir^2
= area of a sphere/2 + base
Radius = Height
**Differentiate between “curved part” and flat parts
Whenever you get questions about “similar” polygons or solids? (typically after transformations)
e.g., Cylinder –> increase both radius and height by the same factor
e.g., triangle –> reduce all sides by the same factor
e.g., Cone –> increase both radius and height by the same factor
RATIOSSSS
(Ratio of sides) = constant factor = k
> typically do large side/small side
2D: Ratio of Areas = (Ratio of Sides)^2 = (k)^2
3D: Ratio of Surface Areas = (Ratio of Sides)^2 = (k)^2
3D: Ratio of Volumes = (Ratio of Sides)^3 = (k)^3
What do you do if you see this:
a^2 - a + 6 = 0
n^2 - 4 = 0
w^2 - 4w = 0
x - 11 = sqrt(x) - sqrt(11)
FACTOR
First one: a^2 - a + 6 —> Quadratic factoring
> Same method is used to factor square of a sum and square of a difference
Second one: Difference of Squares
Third one: factor out w
Fourth one: Difference of squares pattern
x - 11 = [ sqrt(x) + sqrt(11) ] * [sqrt(x) - sqrt(11)] = sqrt(x) - sqrt(11)
Quadratic functions (one variable)
Tips for solving DS?
ax^2 +/- bx MINUS C —> will have ONE positive and one negative solution
> sufficient for positive constraint questions
ax^2 +/- bx PLUS C –> will have to factor out to see if you get:
> One solution (e.g., (a - b)^2)
> Some other constraint in the question (e.g., n > 5)
Integer/x = integer
versus
x/integer = integer
versus
x/y = integer
Integer/x = integer —–> x is a divisor or FACTOR (list out factor pairs, INCLUDING 1)
> x could also be a fraction
x/integer = integer —–> x is a MULTIPLE of something
x/y = integer —-> x is a multiple of y (y can be cancelled out)
Tips for Solving Multi Variable Linear Equations in DS?
e.g., 3 unknowns, given two equations
1) Try to SOLVE as much as possible for you to confidently declare Sufficient or Not Sufficient
> e.g., 3 variables, 2 equations –> might be sufficient if THINGS CANCEL OUT
When I see:
“Is $10 enough money to spend on 3 pens and 2 pencils”
“Enough codes for 12 people”
ENOUGH = Inequality !!!
Budget (up to $x to spend) = inequality
Example:
Spending <= Budget
# of codes >= 12
What should you think about when you see a question about SETS
- What types of terms?
> integers, positive integers, decimals - Can the terms be the same or must they all be different?
a < b < c or a <= b <= c
P(A or B)
or equivalently, P(A union B)
or “probability that event A or event B or both”
= P(A) + P(B) - P(AandB)
*** don’t forget to subtract the SHARED COMPONENTS IF non-mutually exclusive
(BUT you have to know WHAT is the RELATIONSHIP between A and B
> 0 overlap
> independent events: P(A)*P(B)
> complete overlap: one circle inside the other (smaller probability)
= 1 - P(not A and not B) —-> 1 - neither A nor B
= 1 - (1 - P(A)) * (1 - P(B))
Therefore:
1 = P(A) + P(B) - P(AandB) + P(notA and notB)
Think of two circles with possible overlap (intersection)
MIN intersection is 0
MAX intersection is if one circle is INSIDE the other (choose the smaller probability)
Questions that give you a finite number of options
e.g., ab = 6 (a and b are integers)
e.g., m is a factor of 2^6
e.g., x^2 + y^2 = 10 (x and y are integers)
LIST THEM OUT!
> easier for you to visualize properties
Three different numbers are in a list. What are the possible values of the sum?
a < b < c
3a < a + b + c < 3c
Generalized:
Smallest termn < Range of sum for n terms < Largest termn
Special Diagram: Square with overlapping sectors of circles inside
What is the fastest way to solve the area of the overlapping regions?
What about a large circle with four overlapping smaller circles inside? How would you find the area of the shaded region (four outside parts)
Utilize your knowledge about the area of the SQUARE
Area of the Overlap = (Area of each sector * # of sectors) - Area of the Square
What about a large circle with four overlapping smaller circles inside?
Area shaded = Area of Larger Circle - Area of Center Square - 4(area of half circles)
What is the value of the hundreds digit when two two-digit integers are added together?
ab + cd = efg
Hundreds digit must be 1
At most, you carry over 1 into the next column
Ratio Mixture Problems
> Characteristics
> Logic
Characteristics of Ratio Mixture Problems:
> Group 1 and Group 2 have similar “features” that can be expressed as a RATIO or PERCENT (e.g., male/female, male/total, female/total, part-time/full-time)
> Group 1 and Group 2 are COMBINED to form a larger group
> The combined group has a related RATIO on the same features
> Group 2 is often the NEW members added
E.g., M/F
Ratio of Group 1: M1/F1
Ratio of Group 2: M2/F2
If M1/F1 < M2/F2:
M1/F1 < (M1 + M2)/(F1 + F2) < M2/F2
If M1/F1 = M2/F2:
M1/F1 = (M1 + M2)/(F1 + F2) = M2/F2
In other words:
If (M1 + M2)/(F1 + F2) > M1/F1, then M2/F2 MUST BE larger than M1/F1
Xy plane geometry with shaded regions
e.g., 3x + 4y <= 60
e.g., y > x^2 - 4x
Differentiate between “manufactured y” (output from the function given a value of x) and actual y (point you want to compare)
> manufactured y is also ON THE FUNCTION
It is helpful to express the equation of the line in the form: y = x
> sub in a value of x on RHS
> RHS gives you output (manufactured y, on the function)
> LHS y is the ACTUAL y you want to compare
Ex: 3x + 4y <= 60 Region
4y <= -3x + 60
y <= (-3x + 60)/4 —> Means actual y must be equal to or less than manufactured y to fall in the acceptable region
Therefore, whatever point (s, t), (r, s) etc. satisfies this inequality, the point falls in the region
Must be true questions
If there is a Yes/No test case –> NOT ALWAYS TRUE
Strategy
- try to find a NO case to INVALIDATE the option
- or solve algebraically to prove MUST BE TRUE
Tricky wording: A contractor charges a total of r dollars for the first 4 hours plus 0.2r dollars for each additional hour or fraction of an hour
What would the charge be for a 7.5 hour job?
0.2r dollars for each additional hour OR fraction of an hour = round UP
e.g., 7 hours –> Charge = 0.2r * (7 - 4)
e.g., 7.5 hours —> Charge = 0.2r * ( 7.5 - 4) = 0.2r * (8 - 4)
Which of the following CANNOT be a factor of 7x^6 + 310x^5 + 511x^4 + 716x^3 + 323x^2 + x + 30?
A: x + 1
B: x + 2
C: x + 3
D: x + 4
E: x + 5
Complicated PS question with UNKNOWN – plug in any value of x (e.g., 0 or 1) to evaluate
> Use Smart Numbers when asked about UNKNOWNS and not given any information about the unknowns
e.g., x = 0
then the sum simplifies to 30
A 1
B 2
C 3
D 4 —-> not a factor of 30 **
E 5
When I see: (x + y)*(x - y) = Integer….
1) Even/Odd question
x + y and x - y have the SAME TYPE –> EE or OO
2) Factor question
> List out the factors of the integer
> then solve the system of equations for x and y (might have an integer constraint)
Tricky wording pt 2:
1) No integer appears more than twice in the list
2) X is divisible by exactly 4 positive integers less than X
or
How many positive integers less than X are divisors of X?
1) No integer appears more than twice in the list
=> max # of duplicates is 2 (up to 2 repeats)
2) X is divisible by exactly 4 positive integers less than X
=> X has 4 factors OTHER THAN itself
of revolutions –> what can this be used for?
rev * time per rev = Total Time
Geometry circle questions
> # of rev is often combined with RATES
When I see square root on either one or both sides…
e.g., sqrt(63 + sqrt(3)) = x + y*sqrt(3)
e.g., sqrt(10) = sqrt( (x - 1)^2 + (y - 1)^2)
e.g., b = sqrt(a)
For equations (=) —> SQUARE BOTH SIDES
For inequalities –> if both sides are POSITIVE, you can SQUARE BOTH SIDES
In general, whenever you see a square root:
> rationalize with conjugate
> simplify and solve
> or square both sides
Tricky probability sets:
Not A
Not B
Not A = Only B + NeitherANorB
Not B = Only A + NeitherANorB
1 - P(A or Not B) = Only B
1 - P(B or Not A) = Only A
What is the remainder when the three digit integer abc is divided by each of the following:
a + b + c = 22
1) abc/3
2) abc/9
Divisibility Rule for 3 –> divisible if the sum of the digits is a MULTIPLE of 3
> otherwise, Remainder is FIXED = Sum of Digits - closest multiple of 3
e.g., 22 - 21 = R 1
e.g., 787/3 = 262 R1
Divisibility Rule for 9 –> divisible if the sum of the digits is a MULTIPLE of 9
> otherwise, Remainder is FIXED = Sum of Digits - closest multiple of 9
e.g., 22 - 18 = R 4
e.g., 787 / 9 = 87 R4
Is abc/3, is the remainder fixed here?
100(a + 1) + 10(b + 5) + c + 1 = 3*int
YES
Regardless of carry over
> look at sum of the digits
a + 1 + b + 5 + c + 1 = 3int
a + b + c + 7 = 3int
a + b + c = 3*int - 7
Pattern in Remainder: 0 1 2 0 1 2 0 1 2 0
3int - 7 –> start from 0, move left 7 times.
R = 2
Closing Price of Stock Question
Characteristics
Characteristics:
> typically a RANGE PS problem (need to set up inequality, max and min)
> typically relative to ONE PRICE
> then either use range or percent change (+ or -) to calculate the range of values for the closing stock price
READ CAREFULLY (relative to what price)
How d you factor this?
10z^3 - 10z^2 - 90z + 90 = 0
Try different groupings:
e.g., 10z^2(z - 1) - 90(z - 1) = 0
(z - 1)(10z^2 - 90) = 0
10(z - 1)(z^2 - 9) = 0
10(z - 1)(z + 3)(z - 3) = 0
z = 1, -3, 3
Factorials properties
What can be a factorial?
Nonnegative integers
0! = 1
1! = 1
2! = 2*1
Any factorial greater than or equal to 2 is EVEN
Tip for solving DS involving value of an unknown
e.g., what is the value of j?
Given: | j | = 1/j
Easier to view as a PRODUCT
e.g., | j | * j = 1 —-> j must be > 0
j^2 = 1
j = + 1
Triangle ABC is a right triangle. If one of the legs is 6, is the triangle fixed?
NO
> side lengths DO NOT have to be integers (6-8-10)
Could be anything that satisfies the Pythagorean theorem:
6^2 + b^2 = c^2
e.g., 6^2 + 1^2 = (sqrt(37))^2
When is it sufficient to know a sequence of consecutive integers?
If you know:
- n terms and either the A) First term, B) Last term, C) Average or Median
NS if you just know:
- range and n terms (gap between first and last term)
What is the “median” of a triangle?
What is the median of a right triangle equal to? Proof?
Special application?
Line drawn from one vertex to the midpoint of the opposite side
Each median divides the triangle into smaller triangles with the SAME AREA
Median of a right triangle’s length (drawn from the 90 degree to hypotenuse) = 1/2 * hypotenuse
Proof:
> Circle method –> diameter represents the hypotenuse, 90 degree angle is on the circle
> each segment represents the radius
Special application: Right Isosceles triangle’s median = splits triangle into two congruent right isosceles triangles (follows isosceles altitude property)
AND median’s length = 1/2 hypotenuse
> half of a square
Triangle diagram: 8
Opposite angles are equal
Sum of other two angles in Triangle 1 = Sum of other two angles in Triangle 2
Triangle diagram: Arrow
Concave angle = sum of three angles
Multi-variable factoring (3+)
e.g., Factor x^2 - 6xy + 9y^2 - 5xz + 15yz + 6z^2 = 0
Choose one variable to be “x”, then make all other variables the coefficient.
Factor using methods:
- criss cross
- difference of squares
- simple factoring/grouping
e.g., let x be “x”
x^2 - (6y)x - (5z)x + 9y^2 + 15yz + 6z^2
x^2 - x(6y + 5z) + (9y^2 + 15yz + 6z^2) —> factor last term using criss cross method
x^2 -x(6y + 5z) + 3(3y^2 + 5yz + 2z^2)
x^2 - x(6y + 5z) + 3(y + z)(3y + 2z) —> factor using criss cross method
(x - 3y - 3z)*(x - 3y - 2z) = 0
Recognize geometry relationship:
Square with four right triangles formed by drawing a diagonal and two line segments perpendicular to the diagonal (however, the segments are not joined at the center of the square)
Two angles at the corner = sum to 90 degrees
Four congruent triangles (ASA)
How do you prove two parallel lines?
Prove the certain angles are equal (angle rules)
1) Corresponding angles (F shape) are equal
2) Alternate angles (Z shape) are equal
3) Interior angles are (C shape) complements (add to 180 degrees)
Likewise, if you KNOW two lines are parallel, angle rules apply
Isosceles triangle property: Altitude/Median and proof
Special applications?
Altitude of an isosceles triangle with the incongruent side as the base = Median
> splits the triangle into two congruent triangles
> incongruent angle is split into half
PROOF:
> Median bisects the base (incongruent side) so that SAS = two triangles are congruent
> To prove 90 degrees, use exterior angles
Angle 1 + 2 + alpha = 180 degrees
Angle 3 + 4 + beta = 180 degrees
Angle 1 + 3 + 2 + 4 = 180 degrees
Therefore: 180 - alpha + 180 - beta = 180
alpha + beta = 180
AND alpha = beta (because of congruency)
Therefore alpha = beta = 90 degrees
Special applications:
> Equilateral triangle –> altitude splits into two congruent triangles
> all 3 medians/altitudes are the SAME
Triangles: Angle Bisector Theorem
Proof?
Lines drawn from the same point on an angle bisector and are perpendicular to each ray are EQUIDISTANT
Proof:
Two congruent triangles –> AAS (90-angle-shared side)
Three angle bisectors in a triangle intersect at a single point
Proof
Use the Angle Bisector Theorem - Lines drawn from the same point on an angle bisector and are perpendicular to each ray are EQUIDISTANT
Prove that the third segment bisects the angle
Incenter of a triangle
Properties?
Applicable triangle rule to find the radius of a circle?
INcenter = inscribed circle in a triangle
Property: Angle Bisectors
Rule: Angle Bisector Theorem
> Lines drawn from the same point on an angle bisector and are perpendicular to each ray are EQUIDISTANT
(AAS proof)
> shared point = incenter = center of the circle
> equidistant segments = radius
Circumcenter of a triangle
Properties?
Applicable triangle rule to find the radius of a circle?
CIRCUMcenter = Circle is outside the triangle (inscribed triangle)
Property: Perpendicular bisectors (from the circumcenter to each side of the triangle)
Rule: Perpendicular Bisector Theorem
> Each line drawn from a point on the perpendicular bisector to an endpoint of the segment IS EQUAL in length
> creates isosceles triangles
Triangles: Perpendicular Bisector Theorem
Each line drawn from a point on the perpendicular bisector to an endpoint of the segment IS EQUAL in length
You have a linear line through two points, A and B. If point C is equidistant from A and B, where is point C?
> Can be on the line connecting A and B (midpoint)
Can be on ANY POINT on the line that is a perpendicular bisector of the line (recall perpendicular bisector theorem)
Are prime numbers ever negative?
No
> Smallest prime number is 2
What is: sqrt( x^2 )
What is sqrt ( (x - 3)^2 )
| x - 3 |
x |
What is a fraction is equal to its reciprocal? What does that mean?
e.g., x/y = y/x
Cross multiply:
x^2 = y^2 or | x | = | y | or x = +/- y and y = +/- x
Law of Exponents:
What is (x^a)^b?
E.g., (5^sqrt2)^2
What is (x^a * y^b)^c?
What is (x / y)^a?
Power rule - sprinkle (multiply) outside exponent by inner exponents
(x^a)^b = x^(ab) = (x^b)^a
Product of a power rule - sprinkle (multiply) outside exponent by inner exponents
(x^a * y^b)^c = x^ac * y^bc
Power of a fraction rule - sprinkle (multiply) outside exponent by inner exponents
(x / y)^a = x^a / y^a
Long way - multiply the argument in the brackets b number of times
E.g., (5^sqrt2)^2 becomes (5^sqrt2)*(5^sqrt2)
> then you can PULL THE EXPONENT OUT
= (5*5)^sqrt2 (because the exponent sprinkles onto the stuff in the brackets
NOTE:
> REVERSE RULES APPLY TOO
> with multiplication and powers, you can rearrange order (as long as the outcome is the same)
e.g., 5^(2sqrt2) = (5^2)^sqrt2 = 25^sqrt2
Law of Exponents:
What is x^a * x^b?
What is x^a / x^b?
Product rule –> same base, add exponents
x^a * x^b
= x^(a + b)
Quotient rule –> same base, subtract exponents in order
x^a / x^b
= x^(a - b)
Therefore, REVERSE RULES APPLY TOO
Law of Exponents:
Simplify x^ac * y^bc?
Product of a power rule - sprinkle (multiply) outside exponent by inner exponents
(x^a * y^b)^c = x^ac * y^bc
What values could be possible solutions to this expression?
(assume a and b > 0)
a^2 - b^2 = 23
a and b can be integers or fractions
> don’t be afraid to test square roots given the square
e.g., a = sqrt(24), b = 1
e.g., a = sqrt(27), b = 2
Six points are in the XY plane, where any 3 points do not lie on the same line - how many unique triangles can be formed?
This is a COMBINATION QUESTION
> any 3 points chosen can create a triangle
C6,3 = 20
For more complicated XY plane questions (e.g., right angled triangles, dots can be on the same line), start with the overall number (combination), then subtract lines and other invalid shapes
If K^2/L^2 = integer, does this mean that K/L is an integer?
Assume L =/ 0
No = cannot deduce that fraction squared = int means that the fraction is an int, UNLESS we know K and L themselves are integers
K and L could be irrational numbers
e.g., K = sqrt(2) and L = 1
Organized way of solving 3 linear equations?
Choose one equation as the main one (e.g., Equation #1)
Then subtract remaining two equations from Equation #1
Then see what the results are and solve for each variable
Independent and Conditional probability formulas
Independent events: P(A AND B) = P(A) * P(B)
Conditional (dependent) events: P(A AND B) = P(A) * P(B | A)
Tip: it is helpful to think of DECISION TREES (or branches) to visualize the different outcomes
> each branch represents a different outcome (“or”)
> Each branch has several events that must happen for the final outcome to occur (“and”)
There is at least one viper and at least one cobra in Pandora’s box. How many cobras are there?
(1) From any two snakes from Pandora’s box at least one is a viper.
(2) The total number of snakes in Pandora’s box is 99.
Logic question
Ans is sufficient with option A (only statement 1)
Think about the options if you pick ANY 2 snakes: {V,V} or {V,C} or {C,C}
> since there is AT LEAST one viper, {C,C} is NOT a valid option
> and we know C>=1 so {V,V} is NOT an option
> therefore, C = 1
Lesson: If you think the answer is C, double check each statement individually especially as you learn more
> or just always definitely prove the statement is insufficient
Approximately, what is pi as a fraction
22/7
Order the following fractions from least to greatest?
1/2
111/112
11/12
2/3
1) Rule for POSITIVE FRACTIONS (proper and improper):
If you ADD the exact same number to both the numerator and denominator, the resulting fraction gets closer to 1
(could increase or decrease towards 1)
Order from least to greatest:
1/2 < 2/3 < 11/12 < 111/112
2) Another Rule: when COMPARING fractions, CROSS MULTIPLY and compare the products
> this works because we are essentially finding the common denominator and comparing numerators
e.g.,
Is 2/3 < 11/12?
Is 24 < 33 (Yes, so 2/3 < 11/12)
Triple venn diagram: Max # of people only 1?
e.g.: Among 200 people, 56% like strawberry jam, 44% like apple jam, and 40% like raspberry jam. If 30% of the people like both strawberry and apple jam, what is the largest possible number of people who like raspberry jam but do not like either strawberry or apple jam?
Max # people only 1 (only raspberry) –> minimize # of overlap
Formula: 100% = A + B + C - (AB + BC + CA) + center + other
100% = 56% + 44% + 40% - 30% - AR - SR + center + other
100% = 110% - AR - SR + center + other
BECOMES:
100% = 110% - ARonly - center - SRonly - center + center + other
100% = 110% - ARonly - SRonly - center + other
To minimize the overlap with R, we want AR only, SRonly, and center to be as small as possible:
ARonly + SRonly + center = 10% + other —-> make other = 0%
Therefore we know R total = 40%
So R only = R total - SRonly - ARonly - center = 40% - 10% = 30%
Arithmetic Operations: What is the communicative vs associative property?
Both ADDITION and MULTIPLICATION have the commutative and associative property
Commutative Property: Order does not matter
a + b = b + a
a * b = b * a
Associative Property: Bracket placement does not matter
(a + b) + c = a + (b + c)
(a * b) * c = a * (b * c)
Subtraction and division both LACK commutative and associative property
Converting fractions to decimals
e.g., 53/5000
What approaches can you use?
(1) long division
(2) convert fraction to a denominator that is a power of 10
> allows you to move the decimal place
e.g., 53/5000 = 106/10,000 = 0.0106
x^2 + y^2 = 100
What should you think of?
What could the MAX value of x and y be?
> set other variable equal to 0
Max value = 10
Large exponents, how do you compare them?
e.g., Is 2^300 > 3^200?
Either convert to the SAME base or SAME exponent
e.g., 2^300 = 8^100
3^200 = 9^100
So 3^200 > 2^300
OR scale down exponents (taking root where n is GCF)
e.g., GCF of exponents is 100
so becomes 2^3 vs 3^2 —> 3^2 and therefore 3^200 is bigger
Count of digit questions
e.g., how many times does the digit 7 appear from 1 to 1000?
Approach using TIERS
> consider whether to count duplicates (i.e., if we are counting DIGITS) or exclude them (i.e., if we are counting INTEGERS)
Example: how many times does the digit 7 appear from 1 to 1000?
HUNDREDS digit: [700, 799] = 100 numbers with 7 in hundreds digit
TENS digit: [70, 79] for all numbers between 1 to 1000
= 10 numbers with 7 in tens digit per each hundred range * 10 hundred ranges
= 100
ONES digit [07, 17, 27, 37, 47, 57, 67, 77, 87, 97]
= 10 * 10
= 100
TOTAL = 300 times 7 appears as a digit
Converting square cm to cm
e.g., what is the difference between 1cm and 1cm^2
Treat like algebra, works for rectangles and squares
e.g., 1 cm = 4m –> square both sides to get area relationship
1 cm^2 = 16m^2
PROOF:
> if you use that linear relationship to get two cm and multiply the two together to find area of rectangle, you will get same area relationship
a cm = b m
a * x = b * x
a * m = b * m
therefore: a^2 * x * m = b^2 * x * m
a^2 = b^2 (same as if you squared a = b)
Another example:
2 cm = 8m
3 cm = 12m
area if 6 cm^2 = 96m^2
Geometry: Exterior angles
Extend the side (visually with dotted line)
Exterior angle = 180 - interior angle
Fun fact: sum of exterior angles of a quadrilateral also equals 360
The sum of exterior angles of a triangle is equal to 360°
In rectangles, does the “length” have to always be longer than “width”?
No (at least for GMAT no)
Manipulating inequalities
e.g., Is the length of rectangular field F greater than the length of rectangular field G ?
The area of F is greater than the area of G.
The width of F is less than the width of G.
Remember that you can MULTIPLY positive values to both side without changing sign
Case 1) Assume Lf < Lg
Multiply both sides by positive value Wf to get LfWf < LgWf
Then separately we know Wf < Wg
Multiply both sides by positive value Lg to get LgWf < LgWg
COMBINE inequalities together:
LfWf < LgWf < LgWg
LfWf < Lg*Wg
Area F < Area G (False)
Case 2) Assume Lf > Lg
Perform same steps to arrive at Area F > Area G (True)
Sufficient
Ratios and units
e.g., 1 cm on the map = 5 meters in real life
How do you express this ratio?
Express units as ratio and numbers as ratio too
cm (C) to meters (M)
or as a fraction
C / M = 1 / 5
Cross multiply to get 5C = M
If C = 1, M = 5
If C = 2, M = 10
Formula for diagonal of a cube
x*sqrt(3)
Proof:
Each face is a square that follows special right angle x-x-x*sqrt(2)
Diagonal is also a right triangle with x and x*sqrt(2) as legs
More broadly:
> if you know surface area (6*x^2), you know volume (tied by knowing side)
What does it mean to be a “similar triangle”?
How do you prove similar triangles?
Concept: Similar triangles mean the triangles have PROPORTIONAL side lengths and heights, and EQUAL ANGLES
Just prove that TWO angles are equal (AA rule)
Ex: think of overlapping triangles with a parallel line through the triangle
