GRE Quantitative Section, GRE math
Which is greater? 27^(-4) or 9^(-8)
27^(-4)
Which of these fractions is impossible? 0/5 5/0
5/0 is impossible Zero can be a numerator but never a denominator
#3 What are the important properties of a 45-45-90 triangle?
• The ratio of the lengths of the three sides is x:x:x√2.
#2 What are the important properties of a 45-45-90 triangle?
• The triangle is isosceles (AC=BC).
(a^-1)/a^5
1/a^6
The negative exponent x⁻ⁿ is equivalent to what?
1/xⁿ i.e. 5^-3 = 1/(5^3) = 1/ 125 = .008
Legs 6, 8. Hypotenuse?
10
Hector invested $6000. Part was invested in account with 9% simple annual interest, and the rest in account with 7% simple annual interest. If he earned $490 in the first year of these investments, how much did he invest in each account?
$3,500 in the 9% and $2,500 in the 7%.
There are 10 finalists for the school spelling bee. A first, second, and third place trophy will be awarded. In how many ways can the judges award the 3 prizes?
10! / (10-3)! = 720
B and C are both multiples of A. A is a factor of both B and C. What can you conclude?
(B + C) and (B - C) are also multiples of A (B * C) is also a multiple of A
A cylinder has a surface area of 22pi. If the cylinder has a height of 10, what is the radius?
1
What is the only number that is neither prime, nor the product of prime numbers?
1
a^0
1
a^0 =
1
10<all primes<20
11, 13, 17, 19
(12sqrt15) / (2sqrt5) =
(12/2) x (sqrt15 / sqrt5) = 6sqrt3
2³×7³
(2x7)³
(6sqrt3) x (2sqrt5) =
(6 x 2)(sqrt3 x sqrt5) = 12sqrt15
B is a multiple of A. A is a factor of B. What can you conclude?
(B + A) and (B - A) are also multiples of B Any multiple of B is also a multiple of A
If (a > b) and (c > d), what can you conclude?
(a + c) > (b + d)
If (a > b) and (d < c), what can you conclude?
(a - d) > (b - c) larger - smaller > smaller - larger
How to determine percent decrease?
(amount of decrease/original price) x 100%
Simplify the expression [(b^2 - c^2) / (b - c)]
(b + c)
Time
(distance)/(rate) d/r
sum of interior angles
(n-2) (180)
The sum of the measures of the n angles in a polygon with n sides
(n-2) x 180
Simplify the expression (p^2 - q^2)/ -5(q - p)
(p + q)/5
Distance
(rate)(time) d=rt
circle
(x-a)^2 + (y-b)^2 =r^2
Given the expression (x^A - B) where A is even and B is a perfect square, how would you factor it out?
(x^[A/2] + B)(x^[A/2] - B)
x^a y^a
(xy)^a
What is the maximum value for the function g(x) = (-2x^2) -1?
-1
There are 10 finalists for the school spelling bee. A first, second, and third place trophy will be awarded. How many different people can get the three prizes?
10! / 3!(10-3)! = 120
-3³
-27
6w^2 - w - 15 = 0
-3/2 , 5/3
For what values should the domain be restricted for the function f(x) = sqrt(x + 8)
-8
Describe the line y = -x.
-Has a slope of -1 -Passes through the origin -Reflected points above and below, change coordinates AND signs
Describe the line y = x.
-Has a slope of 1 -Passes through the origin -Any point above the line: y > x -Any point below the line: x > y -Reflected points above and below simply switch coordinates
X is the opposite of
-X
If a>b then
-a<-b
0 divided by 7
0
What is the only number that's neither positive nor negative?
0
What is the slope of a horizontal line?
0
The only number that is equal to its opposite
0 0=0
Probability of Event all cases
0≤P(E)≤1
1ⁿ
1
List the squares of every integer up to 15.
1 * 1 = 1 2 * 2 = 4 3 * 3 = 9 4 * 4 = 16 5 * 5 = 25 6 * 6 = 36 7 * 7 = 49 8 * 8 = 64 9 * 9 = 81 10 * 10 = 100 11 * 11 = 121 12 * 12 = 144 13 * 13 = 169 14 * 14 = 196 15 * 15 = 225
Probability of E not occurring:
1 - P(E)
Perfect Squares 1-15
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225
What is the general process for calculating the standard deviation of a list?
1. List all distances from the mean 2. Square the distances 3. Average the squared distances (variance) 4. Find the square root of the variance
What is the decimal form of 1/2, 1/3, 1/3....1/10?
1/2 ---> 0.5 1/3 ---> 0.333 1/4 ---> 0.25 1/5 ---> 0.2 1/6 ---> 0.167 1/7 ---> 0.143 1/8 ---> 0.125 1/9 ---> 0.111 1/10 ---> 0.10
1/2 divided by 3/7 is the same as
1/2 times 7/3
1/8 in percent?
12.5%
Legs 5, 12. Hypotenuse?
13
A triangle is inscribed in a semi circle with legs 5 and 12. What is the circumfermence of the semicircle?
13pi / 2
1/6 in percent?
16.6666%
Which is greater? 64^5 or 16^8
16^8 64^5 = (4^3)^5 = 4^15 16^8=(4^2)^8 = 4^16
If the 80th percentile of the measurements is 72degrees, about how many measurments are between 69 degrees and 72 degrees? Round your answer to the nearest tenth
18
The consecutive angles in a parallelogram equal
180°
What are the cubes of the numbers from 1 to 10?
1^3 = 1 2^3 = 8 4^3 = 64 5^3 = 125 6^3 = 215 7^3 = 343 8^3 = 512 9^3 = 729 10^3 = 1000
What is the coefficient of the x^2 term in the product of (x + 1)(x + 2)(x -1)?
2
What is the smallest prime number?
2
First 10 prime #s
2, 3, 5, 7, 11, 13, 17, 19, 23, 29 A prime number (or a prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself
List the prime numbers up to 60.
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59
In a triangle where the two legs are 4 and 3, what is the value of a line directly intersecting the middle coming from the meeting point of the two legs?
2.4. We calculate the area (6) and then turn the triangle on its side and use x as the height to calculate again. (5x)/2=6
A brick with dimensions 10. 15 and 25 weighs 1.5 kg. A second brick (same density) has dimensions 12, 18, and 30. What is the weight of the second brick?
2.592 kg
20<all primes<30
23, 29
If 8 schools are in a conference, how many games are played if each team plays each other exactly once?
28. n = 8, k = 2. n! / k!(n-k)!
How many 3-digit positive integers are even and do not contain the digit 4?
288 (8 * 9 * 4)
List all the powers of 2 up to 2^9
2^2 = 2 2^3 = 8 2^4 = 16 2^5 = 32 2^6 = 64 2^7 = 128 2^8 = 256 2^9 = 512
Nine coins are tossed simultaneously. In how many of the outcomes will the fourth coin tossed show heads?
2^9 / 2 = 256
2⁵/2³
2²
(2²)³
2⁶
200 <_ x <_ 300. How many values of x are divisible by 5 & 8?
3
x^2 = 9. What is the value of x?
3, -3
30< all primes<40
31, 37
In a Regular Polygon, the measure of each exterior angle
360/n
In any polygon, all external angles equal up to
360°
3/8 in percent?
37.5%
List all the powers of 3 up to 3^4
3^2 = 9 3^3 = 27 3^4 = 81
If r, t, s & u are distinct, consecutive prime numbers, less than 31, which of the following could be an average of them (4, 4.25, 6, 9, 24, 22, 24)
4.25, 6, 22
What's another way to state the fraction 1/(3/4)?
4/3 A fraction is equal to its reciprocal when you flip the numerator and denominator
40 < all primes<50
41, 43, 47
A company places a 6-symbol code on each product. The code consists of the letter T, followed by 3 numerical digits, and then 2 consonants (Y is a conson). How many codes are possible?
441000 = 1 * 10 * 10 * 10 * 21 * 21
Employee X is paid 19.50 per hour no matter how many a week. Employee Y earns 18 for the first 40 and 1.5 the hourly wage for every hour after that. If both earned the same amount and worked the same in one week, how many did each work?
48
In similar hexagons, the ratio of the areas is 16:25. What is the ratio of their corresponding sides?
4:5
For similar triangles, the ratio of their corresponding sides is 2:3. What is the ratio of their areas?
4:9. The ratio of the areas of two similar triangles equals the square of the ratio of the corresponding sides.
List all the powers of 4 up to 4^4
4^2 = 16 4^3 = 64 4^4 = 256
Simplify (a^2 + b)^2 - (a^2 - b)^2
4a^2(b)
What is the side length of an equilateral triangle with altitude 6?
4sqrt3. The triangle can be divided into two equal 30-60-90 triangles with side 6 as the side in which 6 = xsqrt3. So x =2sqrt3...
25^(1/2) or sqrt. 25 =
5 OR -5
5 bakeries sell an average of 300 muffins per bakery per day. If 2 stop making muffins but the total muffins sold stays the same, what is the average of muffins per bakery sold among the remaining?
500
50 < all primes< 60
53, 59
What is the area of a regular hexagon with side 6?
54sqrt3. (divide the hexagon into 6 congruent equilateral triangles.
List all the powers of 5 up to 5^4
5^2 = 25 5^3 = 125 5^4 = 625
Reduce: 4.8 : 0.8 : 1.6
6 : 1 : 2
60 < all primes <70
61, 67
5/8 in percent?
62.5%
What are the smallest three prime numbers greater than 65?
67, 71, 73
In a normal distribution, what percent of the population falls within one standard deviation of the mean? Two standard deviations?
68% of the population falls within one SD (34% below the mean, 34% above the mean) 95% of the population falls within two SD Only 5% will be more than two SD from the mean (2.5% each on the high and low ends)
Convert 0.7% to a fraction.
7 / 1000
What is the third quartile of the following data set: 44, 58, 63, 63, 68, 70, 82
70
70 < all primes< 80
71, 73, 79
What number between 70 & 75, inclusive, has the greatest number of factors?
72
5/6 in percent?
83.333%
7/8 in percent?
87.5%
-3²
9
Ratio of ages of Anna and Emma is 3:5 and of Emma and Nicolas is 3:5. What is the ratio of Anna to Nicholas' ages?
9 : 25
The four angles around a point measure y, 2y, 35 and 55 respectively. What is the value of y?
90
The percent decrease of a quantity
= (actual decrease/Original amount) x 100%
If a lamp decreases to $80, from $100, what is the decrease in price?
= (actual decrease/Original amount) x100% = 20/100x100% = 20%
If a lamp increases from $80 to $100, what is the percent increase?
= 25%. = (actual increase/original amount) x 100% = 20/80 x 100% = 1/4 x 100% = 25%
What is a bisector? What is a perpendicular bisector?
A bisector is a line that cuts a shape or angle into two equal parts A perpendicular bisector crosses another line at a perpendicular angle In a triangle, a perpendicular bisector divides one side in half
What is a central angle?
A central angle is an angle formed by 2 radii.
What is a chord of a circle?
A chord is a line segment joining two points on a circle.
What is a weighted average?
A comparison between the total average and the individual averages of two or more parts Calculate with sum / N / mean formulas, or with proportions
What is a transversal?
A line that cuts across two parallel lines
What is an angle bisector?
A line that divides an angle in half
What is a median? (geometry)
A line that passes through a triangle from one vertex to the halfway point of the opposite side
0 is
A multiple of every integer
How do you get a root out of the denominator?
A number with a root in the denominator is not fully simplified a) If the root is isolated, just multiply the fraction by the root over itself (equaling 1) - this puts it in the numerator b) If the root is part of a binomial, you can use the difference of two squares to solve it - multiply the binomial by itself, but change from addition to subtraction (or vice versa) and solve
How are percentiles used to represent a population?
A percentile indicates the percent of scores, out of the total number, that were lower The distance between percentiles does not necessarily correspond to actual distance between scores! The highest possible percentile is the 99th, the lowest possible is the 0th
What is a trapezoid?
A quadrilateral with only one pair of parallel sides (the bases) The angles connected to either of the legs must be supplementary If the legs are equal, it is known as a symmetrical / isosceles trapezoid - the two angles on each base will be equal, as well as the diagonals
What is a parallelogram?
A quadrilateral with two pairs of parallel and equal sides "Big Four" properties of a parallelogram - if any one of these is true, they're all true! 1. Opposite sides are parallel 2. Opposite sides are equal 3. Opposite angles are equal 4. Diagonals bisect one another Types of parallelograms: rhombus, rectangle, square (the big four properties are true for these shapes as well)
What is it called when a point is reflected to the quadrant opposite it (i.e. I to III or II to IV)?
A reflection about the origin.
What is a 30-60-90 triangle?
A right triangle with angles of 30, 60, and 90 Sides are 1, 2, and square root of 3
What is a polygon?
A shape with at least three straight, closed sides 3 sides - triangle 4 sides - quadrilateral 5 sides - pentagon 6 sides - hexagon 8 sides - octagon In a regular pentagon, all sides are equal and all angles are equal
What is a chord on a circle?
A straight line between any two points on the circumference The longest chord is the diameter
What is an isosceles right triangle?
A triangle with one right angle and two 45 degree angles The sides must proportionally equal 1 and 1 (for the legs) and square root of 2 (for the hypotenuse)
What is an isosceles 30-60-90 triangle?
A triangle with one right angle, one 30 degree angle and one 60 degree angle The sides must proportionally equal 1 (short leg), square root of 3 (long leg), and 2 (hypotenuse)
What is a boxplot?
A type of graph that visually represents the minimum value, first quartile, median, third quartile, and maximum value These five values are not considered part of any group First part: min - Q1 Second part: Q1 - median Third part: median - Q3 Fourth part: Q3 - max Note that the "box" shows the middle 50% of the group (the interquartile range)
Area of a Parallelogram:
A=(base)(height)
area of a trapezoid
A=½ (b1+b2)(h)
area of a circle
A=πr2
In a QC problem, what are some things you could do to both sides without changing the nature of the comparison?
Add or subtract the same number to / from both sides Multiply or divide by the same POSITIVE number on both sides Cross multiply proportions, if you have a fraction on both sides Square or find the square root of both sides (ONLY if both sides are positive) NOTE: You may need to perform multiple operations in order to get comparable items!
When multiplying exponential #s with the same base, you do this to the exponents...
Add them. i.e. (5^7) * (5^3) = 5^10
How do you multiply terms with different rooted numbers and solutions?
All numbers outside the root can be multiplied together All numbers inside the root can be multiplied together Simplify the result NOTE: This does not apply to addition and subtraction (must have the same number under the root sign in order to perform calculations)
What do you know about the angles created by a transversal crossing two parallel lines?
All the "big" angles are equal All the "little" angles are equal The two angles that lay along one straight line must add up to 180 degrees
How do you know if a number is divisible by 3?
All the digits added up together must be divisible by 3
How do you know if a number is divisible by 9?
All the digits added up together must be divisible by 9
What are the real numbers?
All the numbers on the number line (negative, rational, irrational, decimal, integer). All the numbers on the GRE are real. (-2, 1, .25, 1/2, pi)
What do you know about the angles in a triangle?
All three angles must add up to 180 degrees Therefore, each triangle must have at least two acute angles (the third angle may be obtuse or right) The largest angle is opposite the largest side (and vice versa)
What is an exterior angle?
An angle which is supplementary to an interior angle.
What is an inscribed angle?
An angle whose vertex is located on the circumference of the circle The degrees of the angle are half that of the arc If both legs of the inscribed angle meet at the end of the diameter, the inscribed angle is 90 degrees If the legs of two different inscribed angles meet at the end of the same chord (from the same side), the angles are equal
What is an arc of a circle?
An arc is a portion of a circumference of a circle.
What type of number is zero?
An even integer Remember that zero can be a numerator but never a denominator
Define a "monomial"
An expression with just one term (-6x, 2a^2)
If point B shares an x-coordinate with point A and a y-coordinate with point C, what can you conclude?
Angle ABC is a right angle
What can you conclude from the prime factorization of N?
Any multiple of N must be divisible ONLY by the prime numbers in that factorization If N = 2 times 7 times 11, then a multiple of N will also be divisible by 2, 7, and 11 as well as any multiple of those numbers
What is an arc on a circle?
Any section of the circumference You can measure the length of the arc, the degrees of its angle (beginning at the center), and its area (the total slice)
What is a quadrilateral?
Any shape with four straight sides The four angles must add up to 360 degrees (think of it as two triangles) The line from one vertex to another is called a diagonal Types of quadrilaterals: parallelogram, rhombus, rectangle, square, trapezoid - there are also quadrilaterals that have no parallel sides and don't fall in any of these categories
What is a right triangle?
Any triangle that contains a right angle The sides that connect at the right angle are the legs - the opposite side is the hypotenuse The Pythagorean Theorem applies only to right triangles: a^2 + b^2 = c^2 (where c is the hypotenuse)
How do you calculate the area of an arc?
Arc area = (arc degrees / 360) * circle area Remember that circle area = (pi * radius)^2
How do you calculate the length of an arc?
Arc length = (arc degrees / 360) * circle circumference Remember that circumference = 2 * (pi)(radius)
How do you calculate the area of a triangle?
Area = (base * height) / 2 NOTE: The base can be any side, while the height is the altitude of the triangle relative to that side (as a perpendicular line)
How do you calculate the area of a circle?
Area = (pi * radius)^2
How do you calculate the area of a quadrilateral?
Area = base * height If the bases aren't equal, use the average: (base 1 + base 2) / 2 If you don't know the height, divide the shape into triangles / rectangle if possible - if there's a right angle, you can use the Pythagorean Theorem
What are the two different kinds of sequences?
Arithmetic - adding the same amount each time to get the next term Recursive - each term is determined by one or more previous terms
What can you assume from diagrams drawn on the test? What can you not assume?
Assume that a line that appears straight is actually straight DO NOT ASSUME (unless stated explicitly): a) That lines are perpendicular, parallel, or of the same length even if they appear to be b) That any shape is perfectly square (even if it appears so) or is drawn to scale c) That angles are acute / right / obtuse even if they appear to be
If you have N number of consecutive integers, what can you conclude?
At least one of the integers is divisible by N If N is odd, then the sum of all the integers must be divisible by N If N is even, half of the integers are even and half are odd
circumference of a circle
C=2πr
How do you find the least common multiple of two integers?
Calculate the GCF by multiplying common prime factors What do you have to multiply the GCF by to get each number? Multiply these two numbers and the GCF to get the LCM
How do you calculate the number of multiples of x between y and z?
Calculate the number of multiples of x in y Calculate the number of multiples of x in z Subtract the smaller from the larger
How do you calculate the number of INCLUSIVE multiples of x between y and z?
Calculate the number of multiples of x in y Calculate the number of multiples of x in z Subtract the smaller from the larger and add 1
How do you find the greatest common factor of two integers?
Calculate the prime factorization of each number List the factors they have in common (including common exponents) and multiply to get the GCF
How do you calculate the average speed of two different legs of travel?
Calculate the rate of each leg individually (R = D / T) Average speed of the trip = total distance / total time
How do you solve a work / rate problem where two people or machines are working at different rates?
Calculate the rate of each person (R = A / t) Total rate = sum of the two individual rates
What are the rules for cancelling in proportions?
Cancel between numerators, or cancel between denominators Do NOT cancel diagonally!
How else can you express a term with a negative exponent?
Change the sign and put it under a 1 to create the reciprocal: x^-3 = 1 / (x^3) If the term is already in the denominator, change the sign and flip the numerator and denominator: 1 / x^-4 = x^4
How do you calculate the circumference of a circle?
Circumference = 2 * (pi)(radius)
When solving the quadratic equation (x + a)(x + b), what would become the coefficient? What would become the constant?
Coefficient = sum of a + b Constant = product of a * b
How do you calculate a weighted average using proportions?
Compare size of each group to total size Divide the group into parts and draw a diagram - remember that the smaller group is farther from the average! Divide the value difference into parts Calculate the distance from one part to the average, or whatever is asked for
How do you solve a problem involving mixtures with different amounts and concentration levels?
Concentration = (amount of solute / total amount of solution) * 100 Must solve for every variable in the different solutions before calculating the final amount / concentration 8L of 60% solution -> 4.8L solute + 4L of ___% solution -> ___ solute = 12L of 50% solution -> 6L solute Amount of solute in the second solution must be 1.2L Solve for concentration: 1.2 / 4 = 0.3 or 30%
How do you calculate multiple percent increases and decreases?
Convert percent changes into multipliers Multiply the individual multipliers to get the multiplier of the total difference Convert back to percent change, or multiply the multiplier by the old total to get the new total
When would you use a counting formula to solve a probability problem? When would you use a probability formula?
Counting - items being selected from a set with certain restrictions, all outcomes are EQUALLY likely Probability - simple events such as coin flip / cards / dice, use of the words "independent" or "mutually exclusive" in the problem, different outcomes are NOT equally likely
How can you turn a fraction into a decimal?
Create a proportion - one side has the fraction, the other side is x/100 Cross multiply, then simplify if possible
How do you calculate the distance between two points on a line?
Create a right triangle The legs are the horizontal distance and the vertical distance a^2 + b^2 = c^2
How do you compare fractions where both the numerator and denominator of one fraction are larger than those of the other fraction?
Cross-multiply, keeping the fractions in the same order Fraction #1 num. * fraction #2 den. = fraction #1 den. * fraction #2 num. Whichever side is larger corresponds to the original order of the fractions
How do you calculate all possible dividends, given the divisor and remainder?
Divisor + remainder = possible dividend Divisor - remainder = possible dividend Keep adding the divisor to build the list - remember that the list begins with the remainder itself! Remainder, Remainder divisor + remainder, divisor + divisor + remainder, etc.
How do you rebuild the dividend from a mixed numeral, given the divisor?
Dividend = (divisor * quotient) + remainder Given that the divisor is 3, the quotient is 5, and the remainder is 2: 5 and 2/3 Dividend = (3 * 5) + 2 = 17 Improper fraction: 17/3
What are the different parts of a mixed numeral called?
Dividend, divisor, quotient, remainder If you divide 11 by 5 you get 2 and 1/11 11 is the divisor 5 is the dividend 2 is the quotient 1 is the remainder
How do you solve a motion / travel problem involving two objects?
Draw a diagram of the problem for clarity Are the objects going in the same direction, or different directions? Calculate the rate of each object individually (may end up with unknown variables) Same direction: total rate = faster rate - slower rate Opposite direction: total rate = sum of both rates NOTE: If you're calculating the different between the rates, it may be easier to apply D = RT to the gap itself
Which quadrant is the upper right hand?
I
Is 0 even or odd?
Even
What are the rules for multiplication with odd or even numbers?
Even * even = even Odd * odd = odd Even * odd = even If there are any even numbers in the equation, the result will be even! Therefore if the result of an equation is odd, ALL the numbers in the equation are also odd!
What are are the rules for addition and subtraction with odd or even numbers?
Even + even = even Even - even = even Odd + odd = even Odd - odd = even Odd + even = odd Odd - even = odd The ONLY time you get an odd result is when you mix and match!
0 is a multiple of
Every number
How do exponents apply to multiple variables?
Exponents distribute across both multiplication and division: (xy)^4 = x^4 * y^4 (x/y)^5 = x^5 / y^5 NOTE: This does not apply to addition and subtraction!
What are the two types of diagonals in a rectangular solid?
Face diagonal - stays on the surface, can be calculated with the Pythagorean Theorem if you know the length of the sides Space diagonal - passes through the interior, can be calculated with the Pythagorean Theorem applied in 3D AG^2 = AE^2 + EH^2 + HG^2 (height, width, depth) In a cube, it's just AG^2 = (length of one side * 3)^2
Whats the difference between factors and multiples?
Factors are few, multiples are many.
How do you calculate the total number of degrees of all angles in a polygon?
Figure out how many triangles it contains The number of triangles is 2 less than the number of sides (one triangle has 3 sides) Multiply - each triangle has 180 degrees
How can you test the Pythagorean Theorem on a triangle with very long sides?
Find the greatest common factor between the sides you know, and scale it down proportionally You may recognize a Pythagorean triplet in the scaled down equation Otherwise - don't forget the multiply the solution by the scale factor!
How do you calculate the sum of an arithmetic sequence?
First term + last term Multiply by the number of terms in the sequence Divide by 2 [(x1 + xn)(n)] / 2 NOTE: Think of this as adding all the high-low pairs, then dividing to account for duplicates
Which quadrant is the upper left hand?
II
What is the best strategy for problems where there are variables in the answer choices?
First try to solve it algebraically - this would give you the only correct answer If that doesn't work, pick random values that would work in the parameters of the question: a) Start with numbers that are easy to calculate, like 0 and 1 b) Try these values with every answer choice to eliminate some possibilities c) If there are still multiple possibilities, try "real" values - prime numbers are a good strategy
How is a quadratic equation graphed?
Forms a parabola The vertex is either the high or lowest point, and also forms the line of symmetry The parabola will open upwards for a quadratic coefficient greater than 1 (and vice versa) The parabola will be thin if the absolute value of the quadratic coefficient is greater than 1 (and vice versa) The solutions (if there are any) are the x-intercepts
Which quadrant is the lower left hand?
III
Which quandrant is the lower right hand?
IV
If a variable has an even exponent, what can you conclude?
If must have a positive solution x^2 cannot be equal to -4
What's the difference between a bar graph and a histogram?
In a bar graph, each bar represents a distinct group (such as different countries) In a histogram, each bar represents a range of values (such as SAT scores)
How do you calculate the number of combinations of a set (size N) where the items can be in any order?
In combinations, order does not matter - ABC = BAC = CAB Create a factorial for the total number of items (N!) Set the group size as R - divide by the different between all permutations and the permutations of the group, multiplied together Total combinations = N! / [(N - R)! * (R!)]
How do you use *substitution* to solve for the intersection point between two linear binomial equations?
Isolate one of the variables in one of the equations Plug that solution into the other equation and solve for both variables
How do you solve an equation that contains a square root?
Isolate the root on one side Square both sides to get rid of the root If a quadratic equation is produced, simplify by factoring it out Test the possible solutions using the original equation NOTE: With a square root, no solution can be negative - you can rule these out without testing them!
Given a formula such as x = (a * 2b)/y, how would you predict the effect of changing one variable on the other variables?
Isolate the variable being manipulated Solve the formula using 1 for the other variables Solve the formula again using a different number (as specified by the question) Note how the solution changes
If two of a triangle's sides are also radii of a circle, what can you conclude?
It must be an isosceles triangle - the radii are equal lengths Therefore, the two angles on the base are equal A line down the middle of an isosceles triangle is simultaneously its median, angle bisector, perpendicular bisector, and altitude (height)
How do you calculate the number of permutations of a set (size N) where some the items are identical?
In permutations, order matters - ABC is not the same as BCA Create a factorial for the total number of items (N!) Set the number of unique items as K - divide by the frequency of each unique item, multiplied together Total permutations = N! / (n1!)(n2!)...(nK!)
How do you calculate the number of permutations of a set (size N) where you use ALL of the items?
In permutations, order matters - ABC is not the same as BCA Total permutations = N! = (number of options at stage 1) * (number of options at stage 2), etc... How many ways to seat 5 people? 5 times 4 times 3 times 2 times 1 = 5! NOTE: If there are restrictions, make sure to start with the most limiting rule
How do you calculate the number of permutations of a set (size N) where you use SOME of the items?
In permutations, order matters - ABC is not the same as BCA Total permutations = N! = (number of options at stage 1) * (number of options at stage 2), etc... If there are fewer stages you reduce the length of the factorial 5 people and 3 possible seats = 5 times 4 times 3
P and r are factors of 100. What is greater, pr or 100?
Indeterminable.
What is the standard deviation?
Indicates the average distance from the total The farther the values are from the mean (on average), the larger the standard deviation will be Therefore, adding values that are farther from the mean will increase the SD (and vice versa) Adding or subtracting the same number to / from every number on the list has NO effect on standard deviation Multiplying every number on the list by N also multiplies the standard deviation by N
How many multiples does a given number have?
Infinite.
If the sum of the squares of two sides of a triangle equals the square of the third side, what can you conclude?
It's a right triangle The angle where the shorter sides meet is 90 degrees
If a line down a triangle divides one angle in half and the opposite side in half as well, what can you conclude?
It's an isosceles triangle (and possibly equilateral) THEREFORE: -Two of the sides and two of the angles are equal -The line hits the opposite side at a right angle -The length of the line is also the altitude of the triangle
How do you know if a number less than 100 is a prime number?
It's only a prime number if it's not divisible by 2, 3, 5, or 7
Given two integers (x and y), if you have either the LCM or GCF, how do you calculate the other?
LCM = (x * y) / GCF GCF = (x * y) / LCM
How does adding an extreme outlier to a list affect the mean, median, and mode?
MEAN - significant change MEDIAN - little to no change MODE - probably no change
One is (a prime or not?)
NOT A PRIME
Given a variable with a positive exponent, what can you conclude?
NOTHING unless you know that it's a positive integer! The product may be higher or lower, positive or negative, integer or fraction
The product of odd number of negative numbers
Negative
What is the formula for calculating compound interest?
New amount = original amount * (multiplier ^ units of time) Units of time should be consistent with what the question is asking for (or be converted in the formula)
How would you measure the radius / diameter / circumference / area of a circle shown on the coordinate plane?
Note the (x, y) coordinates at the center of the circle and a point on the circumference Calculate the slope using rise over run: m = (y2 - y1) / (x2 - x1) If the center of the circle is at the origin, then x^2 + y^2 = radius^2
7 divided by 0
Null
a/0
Null
What do you know about the angles created by two intersecting lines?
Opposite angles are congruent (equal) The two angles that lay along one straight line must add up to 180 degrees
How would you calculate the probability of both A and B happening, if those events are NOT independent?
P(A and B) = P(A) * P(B given A) One event occurring would affect the probability of the other event (such as drawing cards without replacement), so the probability of B must account for this change
How would you calculate the probability of both A and B happening, if those events are independent?
P(A and B) = P(A) * P(B) One event does not affect the probability of the other (as in flipping a coin), so the total probability of both happening is the product of the probability of each event
How would you calculate the probability of either A or B happening, if those events are mutually exclusive?
P(A or B) = P(A) + P(B) The events cannot happen at the same time (such as choosing a red shirt or a blue shirt), so the total probability is the sum of the probability of each event
How would you calculate the probability of either A or B happening, if those events are NOT mutually exclusive?
P(A or B) = P(A) + P(B) - P(A and B) Must account for the overlap, given that the events could potentially occur at the same time (such as drawing a spade and a king) and you're only looking at one trial
If Event is impossible
P(E) = 0
If E is certain
P(E) = 1/1 = 1
Perimeter of a rectangle
P= 2L + 2w
The Perimeter of a rectangle
P=2(l+w)
The Perimeter of a Square
P=4s (s=side)
A quadrilateral where two diagonals bisect each other
Parallelogram
Pi is a ratio of what to what?
Pi is the ratio of a circle's circumference to its diameter.
What strategies could you use for a QC problem involving variables?
Pick numbers to use as the variables (consistent across both sides) - start with 0, then a negative number, then a fraction, then a positive number Solve with algebra - remember square of a sum, square of a difference, and difference of two squares - if you have one side, you can calculate the other side
0 Is neither
Positive or Negative
How do you find the prime factorization of a number with an exponent that's too large to calculate?
Prime factorize the number independently of the exponent, then distribute the exponent accordingly: 24^5 = (2 times 2 times 2 times 3)^5 = (2^3 times 3)^5 = 2^8 * 3^5 NOTE: This means that if two terms have the same base number and different exponents, you can factor out the larger power to reduce the equation
Positive integers that have exactly 2 positive divisors are
Prime numbers (2, 3, 5, 7, 11, 13, 17, 19, 23)
How are the four quadrants of a coordinate plane arranged?
Quadrant 1: x,y Quadrant 2: -x, y Quadrant 3: -x, -y Quadrant 4: x, -y
What is a rhombus?
Quadrilateral -> parallelogram -> rhombus Has four equal sides - may be a square or diamond shape The diagonals are perpendicular Subject to the big four properties: equal / parallel sides, equal opposite angles, bisecting diagonals
How are quartiles used to represent a population?
Quartiles are individual values (not groups) that are evenly spaced throughout a population The population is divided into four equal parts (by quantity alone) There are five relevant values: minimum value, first quartile, median, third quartile, maximum value These values are NOT counted as part of any group - they stand alone! First part: min - Q1 Second part: Q1 - median Third part: median - Q3 Fourth part: Q3 - max
When presented with a graph for data interpretation, what are the important strategies to keep in mind?
Read ALL labels, captions, etc. Try to get a general understanding of the information ALWAYS double check whether units shown are consistent with units used in the problem! NOTE: graphs are generally drawn to scale (unlike geometry diagrams)
Given the prime factorization of a large number (that is also a perfect square), how do you find the square root of that number?
Reduce each exponent of the factors by half and multiply
How do you calculate a weighted average using the sum / N / mean formula?
Reduce figures by orders of 10 if necessary Calculate mean for each group and total
What are the Pythagorean triplets?
Sets of numbers that work within the Pythagorean Theorem Any of these triplets would show that it's a right triangle Also, if you know you have a right triangle and you know two of the sides, you automatically know the third *3, 4, 5* 6, 8, 10 9, 12, 15 12, 16, 20 *5, 12, 13* 10, 24, 26 15, 36, 39 *8, 15, 17* 16, 30, 34 *7, 24, 25* 14, 48, 50
Which is greater? 200x^295 or 10x^294?
Relationship cannot be determined (what if x is negative?)
How do you solve a term with a fractional exponent?
Restate the term as a root - the denominator of the exponent fraction is the root number x^(1/4) = root4(x) If the numerator of the fraction is not 1, separate the numerator and denominator to create a fraction with a numerator of 1 Then you can root it and keep the other exponent - MUST APPLY IT TO THE WHOLE TERM! x^(3/4) = x^(3 * 1/4) = (root4(x))^3
How is the slope of a line calculated?
Rise over run m = (y2 - y2) / (x2 - x1) For any two points on the line, vertical distance / horizontal distance
How would you use rounding to make a comparison in a QC problem?
Round up or down to numbers that are easier to work with Remember to make a note of whether you rounded up or down!
circular cylinder surface area
SA=2(πr^2) + 2πrh
How can you simplify an equation by substituting a function?
Specify a variable for some part of the equation Replace every instance of that part with the variable Once you have a value for the variable, solve the original part of the equation with that result
What are the approximate square roots of 2, 3, and 5?
Square root of 2 = 1.4 Square root of 3 = 1.7 Square root of 5 = 2.2
How do you use *elimination* to solve for the intersection point between two linear binomial equations?
Stack the equations so the same variables line up If one of the variables can be cancelled, just add the variables and solutions Otherwise, multiply both parts of one of the equations to make it possible to cancel out a variable Solve for the remaining variable NOTE: If the variables disappear, there is not a single point of intersection a) If the result is a constant (like 10 = 10), then the lines are the same (infinite solutions) b) If the result is never true (like 2 = 5), then the lines are parallel (no solutions)
What is the best strategy for backsolving a word problem?
Start with answer choice C) and plug it into the problem to see if it's right If it's not right, you know whether to go higher or lower
When dividing exponential #s with the same base, you do this to the exponents...
Subtract them. i.e (5^7)/(5^3)= 5^4
What is the fundamental counting principle and when is it used?
The FCP is a way to calculate the number of possible combinations in any ordered list Total combinations = (number of options at stage 1) * (number of options at stage 2), etc... The number of options changes depending on whether the options are independent or dependent If there are different types of restrictions applied, start with the most limiting restriction at stage 1 - consider how the number of options changes at each stage
How do you calculate unknown values in ratio problems? (3 shirts for every 5 jackets, etc.)
Use the scale factor (N) to create equations: 3 shirts = 3N 5 jackets = 5N ...where N is the value to multiply by in order to get the total number of items 5N - 3N = 2N (the difference between the number of items in each group) 5N + 3N = 8N (the total number of items)
How to recognize a multiple of 6
Sum of digits is a multiple of 3 and the last digit is even.
How do you calculate one-time percent increases and decreases?
TO FIND THE MULTIPLIER Increase: add 1 (30% increase = 1.3) Decrease: subtract from 1 (25% decrease = 0.75) TO FIND THE PERCENT INCREASE / DECREASE Find the multiplier: new total / old total Remember that the percent change is relative to 1! (Multiplier of 1.4 indicates a 40% increase, multiplier of 0.7 represents a 30% decrease)
If a variable has an odd exponent, what can you conclude?
The solution must have the same sign as the variable x^3 = 8, x must be positive x^3 = -8, x must be negative
In a QC problem, if there is a definite quantity for each side that could potentially be calculated, what can you conclude?
The answer cannot be D), because the values must potentially be able to be compared
If the probabilities of two different events add up to more than 1, what can you conclude?
The events must be mutually exclusive
If the bases of two equal exponents are the same, what can you conclude?
The exponents must also be equal - they can be set up as an equation and solved NOTE: If the bases are not equal, try to fraction out the larger number to create an equal base
Describe the relationship between 3x^2 and 3(x - 1)^2
The graph of 3(x - 1)^2 is a translation (shift) of the graph one unit or space to the right.
What is the "range" of a series of numbers?
The greatest value minus the smallest.
What do you know about the cubed root of a number vs. the square root of the same number?
The solution of the cubed root will be smaller NOTE: The higher the root number, the closer the solution is to 1 - regardless of the number being rooted
How to recognize a # as a multiple of 4
The last 2 digits are a multiple of 4. (i.e 144 .... 44 is a multiple of 4, so 144 must also be a multiple of 4.)
How do you know if a number is divisible by 4?
The last two digits of the number must be divisible by 4
What is a major arc?
The longest arc between points A and B on a circle's diameter.
What are the rules for adding, subtracting, multiplying and dividing exponents?
To multiply two terms with the same base, add the exponents: x^3 * x^6 = x^9 To divide two terms with the same base, subtract the exponents: x^5 / x^2 = x^3 To raise a term to another power, multiply the exponents: (x^6)^2 = x^12 NOTE: You cannot add two terms with different exponents, even if they have the same base! x^2 + x^4 = ? HOWEVER: If two terms have the same base AND the same exponent, you can add their coefficients (2^x + 2^x = 4^x)
What is the units digit of a number? How do you calculate it for large exponents?
The number in the ones place Units digit of 14,542 = 2 To calculate the units digit of a large exponent, look at the units digit of the base number: a) List the first 5 multiples of that number b) Note the pattern c) Look at the exponent number to figure out where it would land in the pattern
How do you know if a number is divisible by 6?
The number must be divisible by both 2 and 3
What is the "solution" for a set of inequalities.
The overlapping sections.
What is the "solution" for a system of linear equations?
The point of intersection of the systems.
What can you conclude about the exponential growth of a positive number less than 1?
The product will decrease exponentially but always be positive
What can you conclude about the exponential growth of a negative number between 0 and -1?
The product will decrease in absolute value, getting increasingly close to zero, with the sign changing each time
What can you conclude about the exponential growth of a negative number less than -1?
The product will increase in absolute value, but the sign will change each time - wild swings up and down getting further from zero
If the root is even (square root, root4, root6, etc.) what can you conclude?
The result must be positive Square root(x) = -2 * -2 = 4 (negatives cancel out) An odd root could be positive or negative: cube root(x) = -3 * -3 * -3 = -9
What is the Difference of Two Squares?
The result of multiplying binomials that are identical except for addition / subtraction - this results in a perfect square (a + b)(a - b) = *a^2 - b^2*
What is the Square of a Sum?
The result of squaring a binomial with addition only (a + b)^2 = (a + b)(a + b) = *a^2 + 2ab + b^2*
What is the Square of a Difference?
The result of squaring a binomial with subtraction only (a - b)^2 = (a - b)(a - b) = *a^2 - 2ab + b^2*
Describe the relationship between the graphs of x^2 and (1/2)x^2
The second graph is less steep.
What is the "domain" of a function?
The set of input values for a function.
What is a minor arc?
The shortest arc between points A and B on a circle's diameter.
What do you know about the sides of a triangle?
The sum of any two sides must be greater than the third side
How to recognize if a # is a multiple of 12
The sum of the digits it a multiple of 3 and the last two digits is a multiple of 4. (i.e 144: 1+4+4=9 which is a multiple of 3, and 44 is a multiple of 4, so 144 is a multiple of 12.)
What are the roots of the quadrinomial x^2 + 2x + 1?
The two xes after factoring.
How do you calculate the surface area of a cube?
Think of the faces as 6 different squares Surface area = (length of one edge *6)^2
How do you calculate the surface area of a rectangular solid that is not a cube?
Think of the faces as pairs of rectangles Surface area = 2lw + 2wh + 2lh
How do you calculate the surface area of a cylinder?
Think of the surface area as the label on a can: circumference * height, plus 2x circle area Circle area = (pi * radius)^2 Circumference = 2 * (pi)(radius) Surface area of a cylinder = 2 * (pi)(radius)(radius + height)
How would you calculate the probability of an event happening at least once?
Use the complement rule - subtract the probability of that event NOT happening from 1 Probability of pulling the queen of hearts at least once in 3 attempts, with replacement: P(Q) = 1 - P(no Q) = 1 - (1/52)^3
How do you calculate X% of Y?
Turn the percent into a decimal Multiply the decimal and Y - make the calculation easier by multiplying by 10 or 100 to remove the decimal Either multiply both the numerator and denominator, or divide the result by 10 or 100 Remember to multiply by 100 if you're calculating the percent ---> OR: break X% into easy-to-calculate pieces (30% = 10% * 3)
What is the minimum number of factors for any positive integer?
Two - the integer itself, and 1
What are "supplementary angles?"
Two angles whose sum is 180.
What are complementary angles?
Two angles whose sum is 90.
What is an isosceles triangle?
Two sides and two angles are equal A line down the middle of an isosceles triangle is simultaneously its median, angle bisector, perpendicular bisector, and altitude (height) This is the ONLY triangle (besides equilateral) where one line can play more than one of these roles! Isosceles right triangle: angles 90-45-45, Pythagorean theorem
What is the slope of a vertical line?
Undefined, because we can't divide by 0.
circular cylinder volume
V=πr^2h
How do you calculate the volume of a cube?
Volume = (length of one edge)^3 ...Hence the term cubing
How do you calculate the volume of a rectangular solid that is not a cube?
Volume = lwh
How do you calculate the volume of a cylinder?
Volume of a cylinder = (pi)(radius^2)(height)
When is a point in the coordinate plane NOT in any quadrant?
When it is located on the x-axis, the y-axis, or the origin
When are shapes congruent?
When they have BOTH the same shape and the same size They may be oriented differently in space
When are shapes similar?
When they have the same shape, but DIFFERENT sizes Similar figures still have equal angles! Also, the corresponding sides between similar shapes are proportional Divide one leg of the larger triangle by the congruent leg of the smaller triangle to get the scale factor If two triangles have a pair of equal angles, they must be similar
When does a function automatically have a restricted domain (2)?
When we need to avoid having a zero in the denominator or avoid taking the square root of a number.
When can you use a matrix instead of a Venn diagram?
When you have only two categories (any number of groups per category) For example: a) Hair color - blonde, brunette, redhead b) Height - under 5', 5-6', over 6'
How do you solve an absolute value equation?
Write it out with both a positive and negative solution Solve both equations Check your solutions in the original absolute value equation - if it doesn't make sense, it's not a valid solution
How do you transform an absolute value inequality into a regular inequality?
Write out the inequality with positive and negative solutions Solve each equation and combine the solutions: x < a x > b a > x > b
How do you calculate the total number of factors in a large number?
Write out the prime factorization List the exponents of each prime factor Add 1 to each of the exponents Add these digits together to get the total number of factors
How do you calculate the number of EVEN factors in a large number?
Write out the prime factorization List the exponents of each prime factor Add 1 to each of the exponents Add these digits together to get the total number of factors Calculate the number of ODD factors by leaving the exponent of 2 out of this process Subtract the odd factor number from the total factor number
How do you calculate the number of ODD factors in a large number?
Write out the prime factorization List the exponents of each prime factor EXCEPT the exponent of 2 (if 2 is included in the factorization) Add 1 to each of the exponents Add these digits together to get the total number of factors
How would you simplify the square root of a number that isn't a perfect square?
Write out the prime factorization of the number within the root sign Any pairs of factors (2 times 2, 3 times 3) can be brought outside the root sign as a single number root2(28) = root2(7 times 2 times 2) = 2(root2(7))
What are the intercepts of a line, and how are they calculated?
X-intercept is the point at which the line crosses the x-axis; find this by plugging in a value of 0 for y Y-intercept is the point at which the line crosses the y-axis; find this by plugging in a value of 0 for x
Can you simplify sqrt72?
Yes, because you can factor out a perfect square (36). Sqrt(36 x 2) = sqrt36 X sqrt2 = 6sqrt2.
5x^2 - 35x -55 = 0
[(7+ sqrt93) /2], [(7 - sqrt93) / 2]
What are the five different types of word problems and how are they solved?
a) *Age* problems - assign variables, don't forget to account for time passing! b) Motion / *travel* problems - D = RT, remember to keep units consistent c) *Work* / rate problems - A = RT, create a proportion, remember to keep units consistent, simplify before cross-multiplying d) *Growth and decay* problems - compound interest formula [total = principle * (multiplier ^ time)] e) Mixture / *solution* problems - create a proportion, remember to keep units consistent, concentration = (amount of solute / total amount of solution) * 100
Given the prime factorization of a number, how can you tell that it's a perfect square?
a) All the exponents of the factors are even b) The number of factors is odd (ignoring the exponents)
How do you solve a quadratic equation?
a) Gather all terms on one side and set the expression equal to 0 b) Simplify if possible (any common factor between all terms?) c) Factor out the equation - solutions are the constants d) Check both solutions in the original expression - if the expression doesn't equal 0, the solution is not valid
What should you NOT assume about an unknown number unless it is specified?
a) Whether it's an integer b) Whether it's positive or negative It could also be zero!
How do you simplify a complex fraction?
a) Write it out as a division problem b) Flip one of the fractions and change it to a multiplication problem c) Simplify if possible BEFORE solving
a<b then a - b is positive or negative?
a-b is negative
a>b then a - b is positive or negative?
a-b is positive
pythagorean theorem
a^2 + b^2 = c^2 for right triangles
If a is inversely porportional to b, what does it equal?
ab=k (k is a constant)
If a is negative and n is even then aⁿ is (positive or negative?)
aⁿ is positive
area of a triangle
bh/2
c[h(c)]
c>1, stretched vertically 0<c<1 shrunk vertically
percent change
change/base
Rate
d/t (distance)/(time)
Find distance when given time and rate
d=rt so r= d/t and t=d/r
formula for distance problems
distance=rate×time or d=rt
What is the graph of f(x) shifted left c units or spaces?
f(x + c)
What is the graph of f(x) shifted upward c units or spaces?
f(x) + c
What is the graph of f(x) shifted downward c units or spaces?
f(x) - c
What is the graph of f(x) shifted right c units or spaces?
f(x-c)
h(x+c)
left c
How can you rearrange the formula for sum / mean / number of items (N) to calculate for each variable?
mean = sum / N sum = mean * N N = sum / mean
Suppose you have a set of n objects, and you want to select k of them, but the order doesn't matter. What formula do you use to determine the number of combinations of n objects taken k at a time?
n! / (k!)(n-k)!
If you have a set of n objects, but you only want to order k of them, what formula do you use to determine the number of permutations?
n! / (n-k)!
Inequalities sign switch
only when multiplying or dividing by a negative
p'''
reflection about the origin
p'
reflection about the x axis
p''
reflection about the y axis
How could root2(xy) be expressed in another way?
root2(x) * root2(y) Rooting distributes across multiplication and division
If an inequality is multiplied or divided by a negative number....
the direction of the inequality is reversed.
The larger the absolute value of the slope...
the steeper the slope.
0^0
undefined
h(x) +c
up c
#3 What is an important property of a 30-60-90 triangle?
• The ratio of the length of the three sides is x:x√3:2x
Factor x^2 - xy + x.
x(x - y + 1)
Consecutive integers
x, x+1, x+2
quadratic equation
x= -b±√(b^2-4ac) / 2a for ax^2+bx+c=0
quadratic formula
x=(-b±√(b^2-4ac))/2a
(x^2)^4
x^(2(4)) =x^8 = (x^4)^2
x^4 + x^7 =
x^(4+7) = x^11
x^6 / x^3
x^(6-3) = x^3
x^a x^b
x^a+b
x^a / x^b
x^a-b
(x^a)^b
x^ab
(x+y)²
x²+2xy+y²
(x-y)²
x²-2xy+y²
(x+y)(x-y)
x²-y²
Suppose that the graph of f(x) is the result of stretching y=x + 5 away from the x-axis by a factor of 2. What is the new equation for the graph f(x)?
y = (x + 5)/2
Suppose that the graph of f(x) is the result of sliding the graph of y=2x^2 down 3 units of spaces. What is the new equation?
y = 2x^2 - 3
How do you calculate the number of multiples of x in y?
y divided by x (round down)
If y is directly proportional to x, what does it equal?
y/x is a constant
slope
y2-y1 / x2-x1 rise over run
parabola
y=ax^2+bx+c
Slope
y₂-y₁/x₂-x₁
circle: c/d=
π
√a/√b
√(a/b)
√a√b
√ab
