GMAT Quant
Powers and Roots
(0.04)^3 = 0.000064 --> (0.04)^3 has 2 places so 2 places x power of 3 = 6 places in the final answer (0.000000008) ^(1/3) = 0.002 --> (0.000000008)^(1/3) has 9 places so 9 places /power of 3 = 3 places in the final answer
Sum of Consecutive Integers
(1) Average the first and last term to find the middle of the set (2) Count the number of terms (3) Multiply the middle term by the number of terms to find the set
What the discriminant tells you
(1) If positive-->sq. root yields a positie number and it produces 2 roots (2 x-intercepts) (2) If it equals zero, sq. root yields zero. Only produces 1 root and the parabola has just 1 x-intercept (3) If negative-->sq. root cannot be performed and produces no roots, no x-intercepts
Properties of Intersecting Lines
(1) Interior angles formed by intersecting lines form a circle so the su, of the angles is 360 degrees (2) Interior angles that combine to form a line sum to 180 degrees (3) Angles found opposite each other where these two lines intersect are equal--vertical angles
Absolute value equations
(1) Isolate the expression within the absolute value brackets (2) Remove the absolute value brackets and solve for the equation in 2 cases. Case 1: x=a (x is positive). Case 2: x=-a (x is negative) (3) Check to see whether each solution is valid by putting each one back into the original equation and verifying that the two sides of the equation are in fact equal.
Properties of Evenly Spaced Sets
(1) Mean and the median are equal, (2) mean and median of the set are equal to the average of the FIRST and LAST terms, (3)the sum of the elements in the set equals the arithmetic mean x number of items in set
Angles of a Triangle
(1) The sum of the three angles of a triangle equals 180 degrees (2) Angles correspond to their opposite sides. The largest angle is opposite the longest side and vice versa. If two sides are equal, their opposite angles are also equal (isosceles).
When to Simplify Eponential Expressions
(1) You can only simplify expressions that are linked by multiplication or division (not addition or subtraction) (2) You can simplify expressions linked by multiplication or division if they have either a base or an exponent in common
Facts about sums and averages of evenly spaced sets
(1) the average of an ODD number of consecutive integers will always be an integer (2) the average of an EVEN number of consecutive integers will never be an integer
(f) area of a circle
(3.14) r^2 r = radius of the circle
Area of a Triangle
(Base x Height) / 2
Area of rhombus
(Diagonal 1 x Diagonal 2) / 2 Diagonals are always perpendicular bisectors (cut each other in half at a 90 degree angle) in a rhombus
Property of GCF and LCM
(GCF of m and n) x (LCM of m and n) = m x n
Counting Integers for consecutive multiples (how many integers from x to y?)
(Last - First) / Increment + 1
Area of a triangle?
(b*h)/2
(f) sum of polygon interior angles
(n-2)180 n = the number of sides of the polygon
(f) Area of a parallelogram
(the length of the altitude) x (the length of the base)
(f) Area of a triangle
(the length of the altitude) x (the length of the base) --------------------------------------------- 2
Factor x^2-y^2
(x+y)(x-y)
Factor x^2+2xy+y^2
(x+y)^2
How are coordinate plane points formatted?
(x,y)
Factor x^2-2xy+y^2
(x-y)^2
Common Right Triangles: 8-15-17
...
0/1 =
0 (any fraction with 0 on top is 0)
1^2
1
How do you determine the standard deviation for a set of numbers?
1 - Find the mean 2 - Find the differences between the mean and each of the n numbers 3 - Square each of the differences 4 - Find the average of the squared differences 5 - Take the nonnegative square root of the average
What is the ratio for a 45-45-90 triangle?
1 : 1 : sqrt of 2
What is the ratio for a 30-60-90 triangle?
1 : sqrt of 3 : 2
Simplifying Roots with Prime Factorization
1) Factor the number under the radical sign into primes 2) Pull out any pair of matching primes from under the randical sign, place one of them outside the root 3) Consolidate the expression
Estimating Decimal Equivalents
1) Make the denominator the nearest factor of 100 or another power of 10. 2) Change the numerator or denominator to make the fraction simplify easily. 3) Make percent adjustments by seeing how much you approximately changed the denominator and applying that percent to the final answer.
Exponents Strategy
1) Simplify or factor any additive or subtractive terms 2) Break every non-prime base down into prime factors 3) Distribute the exponents to every prime factor 4) Combine the exponents for each prime factor and simplify
When to try plugging in numbers
1) variables in the answer choices 2) percents in the answer choices (when they are percents of some unspecified amount) 3)Fractions or ratios in teh answer choices (when they are fractional parts or ratios of unspecified amounts)
Rules for "Plugging in" for cosmic problems
1)Pick numbers that make sense for the variable in the problem 2)Find an answer using those numbers to get the "target answer" 3)plug your #s into the answer choices to see which choice equals the target answer
Rules for "Plugging in" the answer choices
1)Start with choice "c" plug that into the problem to see if it works 2)If "c" is to small go larger 3) if "c" is too big try smaller 4)If you don't know try them all
No addition or subtraction shortcuts with fractions
1)find a common denominator 2) change each fraction so that it is expressed using this common denominator 3) add up the numerators only
=2^10
1,024
(f) Area of a trapezoid
1/2 (sum of bases)(height)
Area of a Triangle
1/2 base * height
Common Right Triangles: 5-12-13
10-24-26
=2^7
128
=2^4
16
=4^2
16
Step by Step from 2 points on a line
1: Find slope by calculating rise over run (if only one point and y-int are given, use y-int as a point (0,y) 2: Plug the slop into the equation 3: Use the given point to plug in for the values of y and x to find b 4: Write the complete equation
2^1
2
=2^1
2
(f) circumference of a circle
2 (3.14) r r = radius of the circle
Circumference
2 * π * radius
Divisibility Rules - 2
2 - If the last digit is even, the number is divisible by 2.
The first twenty-six prime numbers are
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101 Note: only positive numbers can be primes
prime #
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 43,47
√5
2.236
15^2
225
=3^5
243
=2^8
256
=3^3
27
Where do the 4 quadrants appear? Diagram
2nd | 1st ------------------- 3rd | 4th
Circumference of a circle?
2pr
=3^1
3
Divisibility Rules - 3
3 - If the sum of the digits is divisible by 3, the number is also.
Right Triangle Frequent Combos
3 4 5 and 6 8 10 and 5 12 13
2^5
32
=2^5
32
6^2
36
Pythagorean Triples (lengths)?
3:4:5 -- 5:12:13 -- 7:24:25
2^2
4
=2^2
4
=4^1
4
2^9
512
=2^9
512
Another way to write cube root 5
5^(1/3)
=2^6
64
=4^3
64
=3^6
729
=2^3
8
=3^4
81
=3^2
9
(d) What is a circumscribed circle?
A circle containing a polygon where each vertex touches the circle
(d) What is an inscribed circle?
A circle that is contained by a polygon where each side of the polygon is tangent to the circle.
Cylinders and Surface Area
A cylinder is two circles and a wrapped up rectangle. The length of the rectangle is equal to the circumference of the circle (2pi r) and the width of the rectangle is equal to the height of the cylinder, h. Area of the rectangle is 2pi r x h Surface Area= 2 circles + rectangle SA= 2( pi r^2) + 2pi rh The only information needed to find SA is the radius of the cylinder and the height.
(d) What is a circumscribed polygon?
A polygon where each side is tangent to a circle. This means the circle is in the polygon.
convex polygon
A polygon with interior angles that measure less than 180 degrees.
(d) What type of triangle can be inscribed in a circle so that one of its sides is a diameter of the circle?
A right triangle
Pythagorean Theorem
A right triangle (with a 90 degree angle) is composed of two legs and a hypotenuse (side opposite the right angle). The sum of the square of two legs equals the square of the hypotenuse. a^2 + b^2 = c^2
(d) What is a proportion?
A statement that two ratios are equal. Example: 2/3 = 8/12
Probability that A and B will both happen?
A*B (Multiply both probabilities)
Probability that A or B will happen?
A+B (Add both probabilities)
Area of a circle
A= pi r^2
Counting Integers for consecutive integers (how many integers from x to y?)
Add one before you are done For example, how many integers between 6 and 10? Count 6, 7, 8, 9, 10 or just subtract 10-6 + 1 (Last - First + 1)
Adding or subtracting fractions with the same denominator
Add or subtract the numbers on top and keep the same denominator
Simultaneous Equations: solving by combination
Add or subtract the two equations to eliminate one of the variables. Use whenever it is easy to manipulate the equation so that the coefficients for one variable are the SAME or OPPOSITE.
Square
All angles are 90 degrees and all sides are equal
Rectangle
All angles are 90 degrees and opposite sides are equal
Factors of perfect squares
All perfect squares have an odd number of total factors. Vice versa so if a integer has an odd number of total factors it must be a perfect square
Evenly Spaced Sets
All sets of consecutive integers are sets of consecutive multiples. All sets of consecutive multiples are evenly spaced sets. All evenly spaced sets are fully defined if the 3 parameters are known: (1) first and last numbers in the set, (2) the increment, (3) the number of items in the set
Rhombus
All sides are equal. Opposite angles are equal.
Equilateral triangles
All three sides equal and therefore all angles equal to 60 degrees
Inscribed vs. central angles
An inscribed angle has a vertex on the circle itself. An inscribed angle is equal to half of the arc it intercepts
Maximum area of parallelogram or triangle
Another common optimization problem involves maximizing the area of a triangle or parallelogram with given side lengths. If you are given two sides of a triangle or parallelogram and you want to maximize the area, establish those sides as the base and height and make the angle between them 90 degrees. In other words, if you are given two sides of a triangle or parallelogram, you can maximize the area by placing those two sides perpendicular to each other.
Absolute value of a difference
Ansolute value of x-y can be interpreted as the distance between x and y. For example, rephrase the absolute value of x-3<4 as "the distance between x and 3 is less than 4"
An exponent of 0
Any nonzero base raised to the power of zero is equal to 1
Odds / evens with multiple variables
Approach by testing different odds/evens for each variable
Polygons and Area
Area refers to the space inside the polygon measured in square units
Numerator and denominator rules for positive numbers
As numerator goes up, the fraction increases. As denominator goes up, the fraction decreases. Adding the same number to both numerator and denominator brings the fraction closer to 1 regardless of the fraction's value. If the fraction is originally smaller than 1, the fraction increases in value as it approaches 1. if the fraction is originally larger than 1, the fraction decreases in value as it approaches 1.
Area of a parallelogram?
Base * Height
Area of Parallelogram
Base x Height
Factorials and Divisibility
Because it is a product of all the integers from 1 to N, any factorial N! must be divisible by all integers from 1 to N. N! is a multiple of integers from all the integers from 1 to N
Successive Percents
Best to solve by choosing real numbers and seeing what happens-- 100 is usually the easiest number
A combination is an unordered collection of k objects taken from a set of n distinct objects. The number of ways how we can choose k objects out of n distinct objects is denoted as:
C (n k) = n! / k! (n-k)!
Distance between 2 points
Calculate using the pythagorean theorem. Draw a right triangle between the two points.
Equilateral Triangle
Can be split ito two 30-60-90 triangles. Therefore it has a base of length S and a height of length S sq.rt.3 / 2. Area of an equilateral triangle therefore is S^2 sq.rt.3 / 4
Circles & cylinders
Circle is a set of points in a plane that are equidistant from a fixed center point. A line segment connecting the center point to a point on the circle is the radius. Any line segment connecting two points on a circle is a chord. Any chord that passes through the center of the circle is a diameter. A diameter is two times the radius. GMAT tests: (1) circumference and (2) area of whole and partial circles, (3) surface area and (4) volume of cylinders
What is the difference between a permutation or a combination?
Combinations don't care about the order of the problems
Comparing Fractions
Cross Multiply Cross multiply the fractions and put each answer by the corresponding numerator For example: 7/9 vs. 4/5 (7 x 5) = 35 (4 x 9) = 36 Put 35 next to corresponding 7/9 and 36 next to corresponding 4/5. Since 36 is larger than 35, 4/5 > 7/9
Two fractions that equal each other with one variable how do you solve for x?
Cross multiply
How do you find the slope of a line?
Difference in the y-coordinates over the difference in the x coodinates
What direction does a negative line slope slant?
Downward from left to right
(f) How do you determine the distance between 2 points in a coordinate plane?
Draw a triangle and use the Pythagorean theorem
EVEN+EVEN
EVEN
ODD+ODD
EVEN
ODD*EVEN
EVEN *
EVEN*EVEN
EVEN2
Comparing Fractions using Benchmark Values
Estimate values using benchmarks. For example: What is 10/22 of 5/18 of 2000? If you recognize that 10/22 is nearly 1/2 and 5/18 is approx. 1/4 then it is easier to determine. Try to make rounding errors cancel by rounding some numbers up and other numbers down
Divisible by 2 if
Even
Even exponent equations: 2 solutions
Even exponents hide the sign of the base and therefore can have positive and negative solutions. This is often the case even for equations with some odd exponents and some even exponents.
Representing Evens and Odds Algebraically
Even numbers can be written as 2n Odd numbers can be written as 2n+1 or 2n-1
Signs of Square Roots
Even roots only have a positive value. A root can only have a negative value if (1) it is an odd root and (2) the base of the root is negative
To convert a mixed-recurring decimal to fraction
Example #2: Convert 0.2512(12) to a fraction. 1. The number consisting with non-repeating digits and repeating digits is 2512; 2. Subtract 25 (non-repeating number) from above: 2512-25=2487; 3. Divide 2487 by 9900 (two 9's as there are two digits in 12 and 2 zeros as there are two digits in 25): 2487/9900=829/3300.
How do you unfactor a quadratic equation? (x+2)(x+5)
FOIL X^2+7x+10
Fewer Factors More Multiples
Factors divide into an integer and are therefore less than or equal to that integer. Positive multiples on the other hand multiply out from an integer and are therefore greater than or equal to that integer
Area of a sector
Find the fraction of the circle the sector represents and multiply by the area of the circle.
Surface Area
Find the sum of all the faces. Find the area of each face
Complex Absolute Value
For a problem with two different variables, generally without constants, are more easily solved using a conceptual approach rather than algebraic. Try picking and testing numbers, specifically positives, negatives and zero
Strategy for solving data sufficiency-test numbers
For a value question, try to find multiple answers. For a yes/no question try to find a maybe. Be sure to try a positive, negative, integera and fractional number unless explicitly told otherwise.
Sum of consecutive integers and divisibility
For any set of consecutive integers with an ODD number of items, the sum of all the integers is always a multiple of the number of items. For example: 4+5+6+7+8= 30 which is a multiple of 5 For any set of consecutive integers with an EVEN number of items, the sum of all the integers is never a multiple of the number of items.
Fraction Operations:Funky Results
For proper fractions: Adding Fractions --> Increases the value Subtracting Fractions --> Decreases the value Multiplying Fractions --> Decreases the value Dividing Fractions--> increases the value
Perpendicular bisectors of line segments
Forms a 90 degree angle with the segment and divides the segment in half. Has a negative reciprocal slope. In order to find a point on the perpendicular bisector, remember that it passes through the midpoint of the original line segment. Thus solve for the midpoint and that will provide you a point on the line of the perpendicular bisector. Use the midpoint as a point on the line to plug into the equation in order to find the y-intercept.
When to use Fractions vs. decimals
Fractions are good for cancelling factors in multiplications or expressing proportions that do not have clean decimal equivalents. Decimals are good for estimating results or for comparing sizes. Prefer fractions for doing multiplication/division but prefer decimals and percents for doing addition or subtraction, estimating numbers or comparing numbers
Parallel lines cut by a transversal
GMAT makes frequent use of diagrams that include parallel lines cut by a transversal All acute angles (less than 90 degrees) are equal All obtuse angles (greater than 90 degrees but less than 180 degrees) are equal Always be on the look out for parallel lines and extend lines and label the acute and obtuse angles. You might also label the parallel lines with arrows.
Combo problems: manipulations
GMAT often asks to solve for a combination of variables, for example: x+y In these cases do no isolate and solve for the individual variables until all other methods have been exhausted. Instead try to isolate the combination on one side. 4 manipulations (MADS): M: Multiply or divide the whole equation by a certain number A: Add or subtract a number on both sides of the equation D: Distribute or factor an expression on ONE side of the equation S: square or unsquare both sides of the equation Occur most frequently in data sufficiency: manipulate the equation so that the combo is one one side. If a value is on the other side, it's sufficient. If a variable expression is on the other side it is not sufficient.
Rate Problem
How far we have to go/ how fast we are getting there
(d) What is an inscribed polygon?
If each vertex of a polygon lies on a circle
Inscribed triangles
If one of the sides of an inscribed triangle is a diameter of the circle, then the triangle must be a right triangle. Vice versa.
Estimating Roots of Imperfect Squares
If there is no coefficient in front you may estimate by figuring the two closest perfect squares on either side of it. If you want to estimate a square root with a coefficient, simply estimate the square root and then multiply by the coefficient. Or combine the coefficient with the root.
Intersection of Two Lines
If two lines in a plane intersect in a single point, the coordinates of that point solve the equations of BOTH lines. If two lines in a plane do not intersect, then the lines are parallel and there is no pair of numbers that satisfies both equations at the same time.
Multiply Decimals
Ignore the decimal point until the end. Just multiply the numers as if they were whole numbers. Then count the total number of digits to the right of the decimal point in the factors. In the factors, count all the digits to the right of the decimal point--then put that many digits to the right of the decimal point in the product If multiplying a very large number and a very small number, move the decimals in the opposite direction the same number of places.
Same base or same exponent
In problems that involve exponential expressions on BOTH sides or the equation, it is best to REWRITE the bases so that either the same base or the same exponent appears on both sides of the exponential equation. Be careful however if 0,1, or -1 is the base since the outcome of raising those bases to powers is not unique.
Divisible by 8 if
Integer is divisible by 2 three times or if the last three digits are divisible by 8
Dividing Fractions
Invert (reciprocal) the second fraction and multiply
Strategy for solving data sufficiency- type of problem
Is it a value question or a yes/no question? Never turn a yes/no question into a value question
If you raise a negative number to an even power, what happens to the number?
It becomes positive
If you raise a negative number to an odd power, what happens to the number?
It gets smaller
If you raise a positive fraction that is less than 1 to a power, what happens to the fraction?
It gets smaller
Mismatch Problems
Just because there are 3 variables does NOT mean 3 equations are necessary. For example, if they are only asking for one variable it may be possible to determine from only 2 equations. Follow through with the algebra to determine if the solution can be found. Be on the look out for exponents. 2 variables with 2 equations may not be solvable if there are exponents present. MASTER RULE for determining whether 2 equations involving 2 variables will be sufficient to solve: (1) If both of the equations are linear and there are no squared terms or xy terms, the equations will be sufficient unless two equations are mathematically identical (2) If there are any non-linear terms there will USUALLY be two or more different solutions for each of the variables and the equations will not be sufficient.
Common Right Triangles: 3-4-5
Key multiples: 6-8-10 9-12-15 12-16-20
Area of Rectangle
Length x Width
Adding and Subtracting Decimals
Line up the decimal points and proceed as usual
Add/Subtract Decimals
Line up the decimal points!
Defining rules for Sequences
Linear sequences are in the form kn+x, where k equals the difference between successive terms. Each term is equal to the previous term plus a constant k. Exponential sequences are in the form of x(k^n) where x and k are real numbers. Each term is equal to the previous term times a constant k.
Isosceles Triangles
Most popular is the 90 degree: 45-45-90 1:1: sq rt. 2 x:x:x sq rt 2 Important bc it is half of a square
Powers of 10- divide by positive power of 10
Move decimal backward (left) to make the positive number smaller. For example: 4169.2 / 10^2 = 41.692 (move backwards 2 spaces) 89.507 / 10 = 8.9507 (move backwards 1 space)
Powers of 10- multiply by positive power of 10
Move decimal forward (right) to make the positive number larger. For example: 3.9742 x 10^3 = 3974.2 (move decimal forward 3 spaces) 89.507 x 10 = 895.07 (move decimal forward 1 space) Add zeros if needed: 2.57 x 10^6 = 2,570,000 14.29 / 10^5 = 0.0001429
Dividing Decimals
Move the decimal to the right on both numbers until there is nothing to the right of the decimal
Adding or subtracting fractions with different denominators
Multiply both fractions to get a common denominator add or subtract as usual
Problem that asks you to choose a # of items to fill specific spots, when each spot is filled from a different source?
Multiply the # of choices for each of the spots
Permutations: single source, order matters
Multiply the # of choices for each of the spots--but the number of choices keeps getting smaller
Permutations: single source, order matters BUT only for a selection
Multiply the # of choices for each of the spots--but the number of choices keeps getting smaller, stop when all the places are taken
Exponents - Raising a power to a power
Multiply the exponents (4^3)^2 = 4^(3*2) = 4^6
Multiplying Fractions
Multiply the numerators and put the product over the product of the denominators
Does (x^2 + y^2 + z^2)/(x^2 + y^2) = z^2?
NO
Does x^2 + x^3 = x^5?
NO
Does x^6 - x^2 = x^4?
NO
Powers of 10-negative powers
Negative powers reverse the process. Ex: 6782.01 x 10^-3 = 6.78201 (Moving the decimal forward by negative 3 spaces means moving it backward by 3 spaces) 53.0447 / 10^-2 = 5304.47 (Moving the decimal backward by negative 2 spaces means moving it forward by 2 spaces)
ODD+EVEN
ODD +
Add/Subtract Odds & Evens
ODD +_ EVEN = ODD ODD + _ ODD = EVEN EVEN + _ EVEN = EVEN
Multiply/Divide Odds & Evens
ODD x ODD = ODD EVEN x EVEN = EVEN (and divisible by 4) ODD x EVEN = EVEN
ODD*ODD
ODD*
Trapezoid
One pair of opposite sides is parallel
Teminating Decimals
Only have prime factors of 2's and 5's. If there are other prime factors, it is not a terminating decimal.
Parallelogram
Opposite sides and opposite angles are equal
If P is a prime number and P is a factor of AB then
P is a factor of A or P is a factor of B.
probability of occurring event k times in n-time sequence could be expressed as:
P=C(n k) * (p^k)*(1-p)^(n-k)
Order of operations
PEMDAS (parenthesis, Exponents, Multiplication, Division, Addition, Subtraction)
Percents as Fractions
Part/Whole = Percent/100 Fill in the table, set up as a proportion, cancel cross-multiply and solve
Polygons and perimeter
Perimeter is the distance around a polygon and equals the sum of all sides
How do you tell if it's a permutation or combination?
Permutations ask for "arraingements" Combinations ask for "Groups"
How do you find the mode of a set of numbers?
Pick the # that occurs most frequently
Function Graphs and Quadratics
Quadratic functions look like: ax^2 +bx+c and take the form of a parabola. Positive value of a --> parabola curves upward Negative value of a--> parabola curves downward Large abs. value of a-->narrow curve Small abs. value of a-->wide curve Most likely questions asked are how many x-intercepts and what are they? The parabola touches the x-axis at those values of x that make f(x)=0 Sometimes you have to use the quadratic formula. If you do, the discriminant (b^2-4ac) located under the radical sign will tell you how many solutions the equation has.
Maximum Area of Quadrilateral
Question may be asked explicitly or implicitly (such as Is the area of rectangle ABCD less than 30?) Typically maximizing the area of a quadrilateral (usually a rectangle) with a fixed perimeter. Of all quadrilaterals with a given perimeter, the SQUARE has the largest area. Of all quadrilaterals with a given area, the SQUARE has the minimum perimeter.
Negative Exponents
Raising a number to a negative exponent is the same as raising the number's reciprocal to the equivalent positive exponent
Distance Formula?
Rate * Time = Distance (Hint: Draw a chart)
Stategies to solve data sufficiency - rephrase
Rephrase: take the given information and reduce to its simplest form then focus on how the piece of info relates to the question
Data Sufficiency strategy
Rephrasing: You should be able to rephrase the equation to have one equation with one variable. If you are unable to do this, the stem is not sufficient.
Chemical Mixtures
Set up a mixture chart with the substance labels in rows and "original," "change" and "new" in the columns. This way you can keep track of various components and their changes.
Direct Proportion Functionality Types
Set up ratios fir the before and after cases and then set the ratios equal to each other
Right Isosceles triangle (lengths)?
Side, Side, Side^2
Q: If Jessica rolls a die, what is the probability of getting at least a "3"?
Solution: There are 4 outcomes that satisfy our condition (at least 3): {3, 4, 5, 6}. The probability of each outcome is 1/6. The probability of getting at least a "3" is: 1/6 * 4 = 2/3
Factor Counting
Solve factor counting problems by writing the prime factorization in exponential form, adding 1 to all of the exponents and multiplying
Sequences and Rules
Sometimes a rule is too difficult, one can also look for patterns and apply those patterns.
Combinations: single source, order doesn't matter
Start with the number of permutations, that is the numerator(top). The Denominator is the number of combinations (eg 4*3*2*1)
Polygons and Interior Angles
Sum of Interior Angles of a polygon = (n-2) x 180 Triangle has 3 sides and 180 degrees Quadrilateral has 4 sides and 360 degrees Pentagon has 5 sides and 540 degrees Hexagon has 6 sides and 720 degrees Another way to find the sum is to divide the polygon into triangles
Divisible by 3 if
Sum of the integer's digits are divisble by 3
Property of GCF and LCM
The GCF of m and n cannot be larger than the difference between m and n. For exmaple, assume the GCF of m and n is 12. Thus m and n are both multiples of 12. Consecutive multiples of 12 are 12 units apart on the number line and therefore cannot be less than 12 units apart
Perimeter of a sector
The boundaries of a sector of a circle are formed by the arc and two radii (like a slice of pizza--the arc is the crust and the center of the circle is the tip of the slice). If you know the length of the radius and the central angle, you can find the perimeter of the sector
(f) Formula for slope of a line
The difference in the y-coordinates ----------------------------------------------- The difference in the x-coordinates
What happens if you multiply or divide both sides of an inequality by a negative number?
The direction of the inequality symbol changes
Circumference of a circle
The distance around a circle. C=2 pi r pi= approx 3.14 C=d pi A full revolution of a wheel is equivalent to the wheel going around once
Greatest Common Factor (Divisior) - GCF (GCD)
The greatest common divisor (gcd), also known as the greatest common factor (gcf), or highest common factor (hcf), of two or more non-zero integers, is the largest positive integer that divides the numbers without a remainder.
(d) What is the range for a series of numbers?
The greatest value in the set minus the least value.
Percent problems
The key is to make them concrete by picking real numbers with which to work
Never split the denominator
The numerator may be split, but never split the denominator.
(d) What is the x-intercept?
The point where a line intersects the x-axis.
(d) What is the y-intercept?
The point where a line intersects the y-axis.
Any number to the negative power (y) is equal to?
The reciprocal of the same number to the positive power (y). (eg 3^-2 = 1/(3^2) = 1/9)
Sides of a triangle
The sum of any two sides of a triangle must be GREATER THAN the third side. If you are given two sides of a triangle, the length of the third side must lie between the difference and the sum of the two given sides.
Sums of Consecutive Integers and Divisibility
The sum of n consecutive integers is divisible by n if n is odd, but it is NOT divisible by n if n is even
Beware of even exponents
They hide the original sign of the base. Any base raised to an even power will result in a positive answer
Types of slopes
Think of slope as walking from left to right. If you walk along a line with a positive slope, you would walk up. A horizonal line has a zero slope and a vertical line has an undefined slope
Benchmark Values: 10%
To find 10% of any number, just move the decimal point to the left one place
Eliminating Roots: Square both sides
To solve variable square roots, square both sides of the equation. Be sure to check the solution in the original equation since squaring can introduce an extraneous solution.
4/0 =
Undefined
Sum of two primes
Unless one of the numbers is 2, it will result in an even number
Unspecified Number Amounts
Use Smart numbers. To make computation easier, choose numbers equal to common multiples of the denominators of the fraction in the problem.
Simultaneous Equations: solving by substitution
Use substitution whenever one variable can be easily expressed in terms of the other.
Percent Increase and decrease
Use the percent table however adjust it for change instead of part. Change/Original = Percent/100 Also can do so with following equations: ORIGINAL x (1+% increase/100) = NEW ORIGINAL x (1-% decrease/100) = NEW
Heavy Division Shortcut
Used for large numbers. Example: What is 1,530,794 / (31.49 x 10^4) to the nearest whole number? Step 1: Set up the division problem in fraction form Step 2: Rewrite the problem, eliminating powers of 10: 1,530,794 / 314,900 Step 3: The goal is to get a single digit to the left of the decimal in the denominator. Just remember whatever you do to the denominator, to do to the numerator. 1,530,794 / 314,900 = 15.30794 / 3.14900 Now you have the single digit 3 to the left of the decimal in the denominator Step 4: Focus only on the whole number parts and solve. 15.30794 / 3.14900 is approx 15/3 which = 5
Cylinders and volume
V=pi r^2h Only need the radius of the cylinder and height of the cylinder Two cylinders can have the same volume but different shapes
The Last digit Shortcut
When asked to find the units digit, just look at the units digit of the product. For example: What is the units digit of (7^2)(9^2)(3^3)? Step 1: 7 x 7 = 49 - Drop all except units digit - 9 Step 2: 9 x 9 = 81 - Drop all except units digit - 1 Step 3: 3 x 3 x 3 = 27 - Drop all except units digit - 7 Step 4: 9 x 1 x 7 = 63 The units digit of the final product is 3
Subtracting Exponents
When dividing two terms with the same base, combine exponents by subtracting
Adding Exponents
When multiplying two terms with the same base, combine exponents by adding
Nested Exponents
When raising a power to a power, combne exponents by multiplying
Distributing Exponents
When several #s are inside parenthesis, the exponent outside the parentheses must be distributed to all the #s within (4y)^2 = ((4)^2)((y)^2) = (4^2) * (y^2) = 16y^2
Percent shortcuts
When trying to find a more complicated percentage, break it into easy to find chunks. For example: 23% of 400: 10% of 400 is 40 therefore 20% is 2 x 40 = 80. 1% of 400 is 4 and 3% is 3 x 4 or 12. Putting it together, we get 80+12=92
Range of possible remainders
When you divide an integer by 7, the remainder could be any number between 0 and 6 inclusive. Notice that you cannot have a negative remainder or remainders larger than 7. There are exactly 7 possible remainders.
An Exponent of 1
When you see a base without an exponent, write in an exponent of 1
Disguised + or - questions
Whenever you see >0 or <0, think Positives & Negatives
Fractional Base Exponents
While most positive numbers increase when raised to a higher exponent, numbers between 0 and 1 decrease
Odds that at least one thing will happen?
Will = 1 - Won't
Fractional Exponents
Within the exponent fraction, the numerator tells us what power to raise the base to, and the denominator tells us which root to take
Odds that something doesn't happen?
Won't = 1 - Will
Arithmetic with remainders
You can add and subtract remainders as long as you correct excess or negative remainders
Multiplication with remainders
You can multiply remainders as long as you correct excess remainders at the end. For example, if x has a remainder of 4 upon division by 7 and z has a remainder of 5 upon division by 7, then 4 x 5 gives 20. Two additional 7's can be taken out of this remainder, so xz will have remainder 6 upon division by 7.
Simplifying Square Roots
You may only seperate or combine the product or quotient of two roots. You cannot seperate or combine the sum or difference of two roots
The solvability rule
You must have at least as many distinct equations as you have variables
Area of Trapezoid
[(Base 1 + Base 2) x Height] / 2 Average of two bases multiplied by the height
smaller than the original fraction
a fraction between 0 and 1 raised to a positive integer is...
What is a chord of a circle?
a line segment that has its endpoints on the circle
Intercepts of a line
a point where a line interescts a coordinate axis is called an intercept. X-intercept is where the line intersects the x-axis and y-intercept is where the line crosses the y-axis. The x-intercept is the point on the line in which y=0 The y-intercept is the point on the line in which x=0. To find the x-intercept, plug in 0 for y. Do the same technique to find the y-intercept.
gcd(a, b)*lcm(a, b)
a*b
(f) Pythagorean theorem
a^2 + b^2 = c^2 Only applies to right triangles
Exponents - Multiplying numbers with same base
add their exponents (6^2 *6^3 = 6^5)
Key to percent increase/decrease
amount of increase/original amount = x/100 (Then cross multiply to find x)
Complex Abs Value with more than one expression
an absolute value with more than one expression but only one variable and one or more constants is usually easier solved with an algebraic approach
Circumference and arc length
an arc is a portion of the circle rather than the whole. Arc length can be determined by what fraction the arc is of the entire circumference
Exterior angles of a triangle
an exterior angle of a triangle is equal in measure to the sum of the two non-adjacent (opposite) interior angles of the triangle. Tested frequently on the GMAT In particular look for exterior angles within more complicated diagrams. Perhaps try isolating the triangle.
How do you find the median of a set of numbers?
arrainge #s from least to greatest and take the middle number or the average of the two middle #s
a*broot(r*s)
a√r * b√s =...
Delux Pythagorean Theorem
d^2=x^2+y^2+z^2 where x, y and z are the sides of the rectangular solid and d is the main diagonal
Repeating Decimals
divide any number by 9 and it becomes a repeating decimal. For example: 4/9 = 0.44444444 forever. 3/11 = 27/99 = 0.272727272727 forever
even + even =
even
even x even =
even
even x odd =
even
odd + odd =
even
(f) Quadratic polynomial function
f (x) = x^2 - 1 Typically used to find coordinates (x, f(x))
Quadrilaterals
four sided shapes trapezoids, parallelograms and special parallelograms such as rhombuses, rectangles and squares
Property of GCF-consecutive multiples
have a GCF of n. For example, 8 and 12 are consecutive multiples of 4. Thus 4 is a common factor of both numbers. But 8 and 12 are exactly 4 units apart. Thus 4 is the greatest possible common factor of 8 and 12
Volume of Cylinder
height * π *r^2
name of the side of a triangle opposite the right angle
hypotenuse
When is a number divisible by 6?
if it is divisible by both 2 and 3 (318 is because it is even and the sum of 3+1+8 is divisible by 3)
When is a number divisible by 4?
if the number formed by it's last 2 digits is divisible by 4 (3,028 is because 28 is divisible by 4)
When is a number divisible by 3?
if the sum of its can be divided evenly by 3 (216, is because 2+1+6 is divisible by 3)
multiplying decimals
ignor decimal points and multiply the two numbers, then count the digits to the right of the decimal points in the original numbers and place the decimal so there are the same number of digits to the right of the decimal
Divisible by 10 if
integer ends in 0
Divisible by 5 if
integer ends in 5 or 0
Divisible by 4 if
integer is divisible by 2 twice or the last two digits are divisible by 4
Divisible by 6 if
integer is divisible by both 2 and 3
Any number to the first power is?
itself
A permutation is an ordered collection of k objects taken from a set of n distinct objects. The number of ways how we can choose k objects out of n distinct objects is denoted
n! / (n-k)!
n*(n-1)*(n-2)*...*1
n! = ...
1. How many objects we can put at 1st place? n. 2. How many objects we can put at 2nd place? n - 1. We can't put the object that already placed at 1st place. ..... n. How nany objects we can put at n-th place? 1. Only one object remains. Therefore, the total number of arrangements of n different objects in a raw is
n*(n-1)*(n-2)....2*1 = n!
reciprocal (1/x^y)
negative exponents show... (x ^ -y =...)
Basic probability formula?
number of outcomes you want/total number of possible outcomes
even + odd =
odd
odd x odd =
odd
30-60-90
often formed from equilateral triangles 1: sq rt 3:2
Any number to the 0 power is?
one (1)
Two events are mutually exclusive if they cannot occur at the same time. For n mutually exclusive events the probability is the sum of all probabilities of events:
p = p1 + p2 + ... + pn-1 + pn
A number expressing the probability (p) that a specific event will occur, expressed as the ratio of the number of actual occurrences (n) to the number of possible occurrences (N).
p=n/N
POSITIVE AND NEGATIVE NUMBERS: Multiplication
positive * positive = positive positive * negative = negative negative * negative = positive
Area of a circle?
pr^2
Add/Subtract a multiple with a nonmultiple
result is a nonmultiple Multiple of 3 + Nonmultiple of 3 = Nonmultiple of 3
Add/Subtract two nonmultiples
results can be multiple or nonmultiple
Add/Subtract two multiples
results is a multiple Multiple of 3 + Multiple of 3 = Multiple of 3
Slope of a line
rise/run or change in y/change in x
Work Problems?
simmilar to distance but ... Think how much of the job can be done in one hour (as a fration of the whole).
Exponents - Dividing numbers with same base
subtract the bottom exponents from the top exponents (3^6 / 3^2 = 3^4)
how do you find the range of a set of numbers?
subtract the smallest number from the greatest number
Divisible by 9 if
sum of the integer's digits are divisble by 9
(f) perimeter of a polygon
sum of the lengths of its sides
(d) A line that has exactly one point in common with circle
tangent
Volume
the amount of stuff a shape can hold Volume = Length x Width x Height essentially it is equal to the area of the base multiplied by the height Remember when fitting 3D objects into other 3D opjects, knowing the volumes is not enough. We must know the specific dimensions (length, width and height) of the object to determine whether it will fit
Prime Factorization Factors
the prime factorization of a perfect square contains only even powers of primes. Vice versa.
Products of Consecutive Integers and Divisibility
the product of n consecutive integers is divisible by n!
Multiplying & Dividing Signed Number
the result will be positive if you have an EVEN number of negative numbers in the collection. The result will be negative if you have an ODD number of negative numbers.
greater than the original fraction
the square root of a fraction between 0 and 1 is...
What is a quadrilateral with two sides that are parallel?
trapezoid
Similar Triangles
triangles with equal corresponding angles and proportional corresponding sides. If 2 triangles have 2 pairs of equal angles you know they are similar triangles. If two similar triangles have corresponding side lengths in ratio of a:b, then their areas will be in ratio a^2:b^2--this holds true for any similar figures. For similar solids with corresponding sides in ratio a:b, their volumes will be in ratio a^3:b^3
subtract exponents (x^3)
when dividing... (x^8 / x^5=...)
add exponents (x^7)
when multiplying exponents... (x^4 * x^3= ...)
multiply exponents (x^6)
when raising to a power... ( (x^3)^2 =...)
(f) length of an arc of a circle
x / 360 x = angle between two circle radius
30:60:90 Right Triangle (Lenghts)
x:x^(1/3):2x
1
x^0 = ...
Unfactor (x+y)^2
x^2+2xy+y^2
Unfactor (x-y)^2
x^2-2xy+y^2
Unfactor (x+y)(x-y)
x^2-y^2
Factoring: three Special Products
x^2-y^2 =(x+y)(x-y) x^2+2xy+y^2= (x+y)(x+y) = (x+y)^2 x^2-2xy+y^2 = (x-y)(x-y) = (x-y)^2
Slope-intercept equations
y=mx+b m represents the slope of the line and b represents the y-intercept of the line. Vertical lines take the form x=some number Horizontal lines take the form y=some number Linear equations take the shape Ax+By=C and never have sq. roots, squares or xy.
Equation for a line?
y=mx+b where b is the y-intercept (the point which crosses the y-axis) m is the slope of the line, and x & y are the coordinates of some point on that line
Area of a circle
π * r^2
Length of diagonal for square
√2 * side
Height in Equil Triangle
√3/2 * side
Problem: 1<x<9. What inequality represents this condition?
Solution: 10sec. Traditional 3-steps method is too time-consume technique. First of all we find length (9-1)=8 and center (1+8/2=5) of the segment represented by 1<x<9. Now, let's look at our options. Only B and D has 8/2=4 on the right side and D had left site 0 at x=5. Therefore, answer is D.
Sum of n first positive even numbers
a1+a2+a3+...+aN = n(n+1)
Sum of n first odd numbers
a1+a2+a3+....+aN = n^2
abs rule
abs(X) + abs(Y) >= abs(X+Y)
If the first term is a1 and the common difference of successive members is d, then the nth term of the sequence is given by:
an=a1+d(n-1)
If n is even, the sum of consecutive integers is never divisible
by n
If n is odd, the sum of consecutive integers is always divisible
by n
Even and Odd Numbers: Multiplication
even * even = even; even * odd = even; odd * odd = odd.
Even and Odd Numbers: Addition / Subtraction
even +/- even = even; even +/- odd = odd; odd +/- odd = even.
Every common divisor of a and b is a divisor of
gcd(a, b).
Divisibility Rules - 11
11 - If you sum every second digit and then subtract all other digits and the answer is: 0, or is divisible by 11, then the number is divisible by 11. Example: to see whether 9,488,699 is divisible by 11, sum every second digit: 4+8+9=21, then subtract the sum of other digits: 21-(9+8+6+9)=-11, -11 is divisible by 11, hence 9,488,699 is divisible by 11.
Divisibility Rules - 12
12 - If the number is divisible by both 3 and 4, it is also divisible by 12.
11^2
121
2^7
128
12^2
144
2^4
16
4^2
16
13^2
169
14^2
196
5^2
25
Divisibility Rules - 25
25 - Numbers ending with 00, 25, 50, or 75 represent numbers divisible by 25.
2^8
256
Finding the Sum of the Factors of an Integer
(a^(p+1) - 1)*(b^(q+1) - 1)*(c^(r+1) - 1) / (a-1)(b-1)(c-1)
Finding the Number of Factors of an Integer
(p+1)(q+1)(r+1)....(z+1)
2^0
1
What is the last digit of 127^39?
1. 7^1=7 (last digit is 7) 2. 7^2=9 (last digit is 9) 3. 7^3=3 (last digit is 3) 4. 7^4=1 (last digit is 1) 5. 7^5=7 (last digit is 7 again!) ... So, the cyclisity of 7 is 4. Now divide 39 (power) by 4 (cyclisity), remainder is 3.So, the last digit of is the 127^39 same as that of the last digit of , is the same as that of the last digit of 7^39 , which is 7^3, which is 3.
To convert a recurring decimal to fraction
1. Separate the recurring number from the decimal fraction 2. Annex denominator with "9" as many times as the length of the recurring number 3. Reduce the fraction to its lowest terms
√2
1.414
√3
1.732
Area of Trapezoid
1/2 (long base+short base) * height
Divisibility Rules - 10
10 - If the number ends in 0, it is divisible by 10.
10^2
100
Divisibility Rules - 4
4 - If the last two digits form a number divisible by 4, the number is also
Volume of sphere
4/3 * π *r^3
7^2
49
Divisibility Rules - 5
5 - If the last digit is a 5 or a 0, the number is divisible by 5.
Divisibility Rules - 6
6 - If the number is divisible by both 3 and 2, it is also divisible by 6.
25^2
625
2^6
64
8^2
64
all prime numbers above 3 are of the form
6n - 1 or 6n + 1
Divisibility Rules - 7
7 - Take the last digit, double it, and subtract it from the rest of the number, if the answer is divisible by 7 (including 0), then the number is divisible by 7.
2^3
8
Divisibility Rules - 8
8 - If the last three digits of a number are divisible by 8, then so is the whole number
9^2
81
3^2
9
Divisibility Rules - 9
9 - If the sum of the digits is divisible by 9, so is the number
Perfect Square
A perfect square, is an integer that can be written as the square of some other integer. For example 16=4^2, is an perfect square. There are some tips about the perfect square: • The number of distinct factors of a perfect square is ALWAYS ODD. • The sum of distinct factors of a perfect square is ALWAYS ODD. • A perfect square ALWAYS has an ODD number of Odd-factors, and EVEN number of Even-factors. • Perfect square always has even number of powers of prime factors.
If a number equals the sum of its proper divisors, it is said to be a perfect number.
Example: The proper divisors of 6 are 1, 2, and 3: 1+2+3=6, hence 6 is a perfect number.
II. Converting inequalities with modulus into range expression. In many cases, especially in DS problems, it helps avoid silly mistakes.
For example, |x|<5 is equal to x e (-5,5). |x+3|>3 is equal to x e (-inf,-6)&(0,+inf)
Last n digits of a product of integers are last n digits of the product of last n digits of these integers.
For instance last 2 digits of 845*9512*408*613 would be the last 2 digits of 45*12*8*13=540*104=40*4=160=60 Example: The last digit of 85945*89*58307=5*9*7=45*7=35=5?
If is a positive integer greater than 1, then there is always a prime number
P whth N<P<2N
Lowest Common Multiple - LCM
The lowest common multiple or lowest common multiple (lcm) or smallest common multiple of two integers a and b is the smallest positive integer that is a multiple both of a and of b. Since it is a multiple, it can be divided by a and b without a remainder. If either a or b is 0, so that there is no such positive integer, then lcm(a, b) is defined to be zero. To find the LCM, you will need to do prime-factorization. Then multiply all the factors (pick the highest power of the common factors).
Problem: A<X<Y<B. Is |A-X| <|X-B|?
We can think about absolute values here as distance between points. Statement 1 means than distance between Y and A is less than Y and B. Because X is between A and Y, distance between |X-A| < |Y-A| and at the same time distance between X and B will be larger than that between Y and B (|B-Y|<|B-X|). Therefore, statement 1 is sufficient.
one important concept in problems with marbles/cards/balls
When the first marble is removed from a jar and not replaced, the probability for the second marble differs ( 9/99 vs. 10/100 ). Whereas in case of a coin or dice the probabilities are always the same ( 1/6 and 1/2). Usually, a problem explicitly states: it is a problem with replacement or without replacement.
a^n - b^n
is always divisible by a-b
a^n - b^n
is divisible by a+b if n is even
a^n +b^n
is divisible by a+b if n is odd, and not divisible by a+b if n is even
In any evenly spaced set the arithmetic mean (average) is equal to the median and can be calculated by the formula
mean = median = (a1+aN) / 2
The product of n consecutive integers is always divisible by
n!
The number of trailing zeros in the decimal representation of n!, the factorial of a nonnegative integer n, can be determined with this formula:
n/5 + n/5^2 + n/5^3 + ... + n/5^k
Finding the number of powers of a prime number P, in the N!.
n/p + n/p^2 + n/p^3... till p^x < n
Two events are independent if occurrence of one event does not influence occurrence of other events. For n independent events the probability is the product of all probabilities of independent events:
p = p1 * p2 * ... * pn-1 * pn
POSITIVE AND NEGATIVE NUMBERS: Division
positive / positive = positive positive / negative = negative negative / negative = positive
Compound Interest
principal* (1 + interest / C)^ time*C
A number expressing the probability (q) that a specific event will not occur:
q=1-p = 1-n/N
The sum of the elements in any evenly spaced set is given by:
sum = (a1+aN)/2 * N or (2a1 + d(n-1)) / 2 * n
Sum of n consecutive integers equals
the mean multiplied by the number of terms,
