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
