GRE Math 1 Flashcards

Question

Ratios

Answer 1

*must be expressed in lowest, reduced terms* Relationships to each other ; values represent smallest possible quantity of each item... but you can have any multiple of each item Real vs. Relationship Quantities - Relationship quantities do not dictate the actual quantity they are just a representation of the smallest possible quantity Ration Formula: (ratio)(x/total) (ratio)(actual) 4/9 = x/36; x = 16 *if given only an actual number, the total of the ratio (a:b...a+b) must be a factor of the actual number ex) pg. 13 lab has 55 rabbits what could be the ratio of white rabbits to brown rabbits? 3:8 (8+3) bc 11 is a factor of 55 - when dealing with ratios, the total number needs to be divisible by the sum of the pieces of the ratio when dealing with fractions, always try to clear the fractions: - if you are given one fraction and a whole number multiply by the denominator to clear the fraction - if you are given more than one fraction... do the division problem and then reduce ... put colon between final fraction: 16/3 becomes 16:3

Answer 2

*practice converting precent to fraction* type of ratio in which the total is always 100 ... always try to get denominator of fraction to 100 o figure out what to multiple the numerator by part/whole --> part/100 ex) 18 is what percent of 3? what percent of means (x/100) 18 = (x/100)(3) x=600 Percents to Fractions & Vice Versa: *to convert a percent to a fraction divide the percent by 100% ex) 1/4% = (1/4)/100 = (1/4)(1/100) = 1/400 * * to convert a fraction to a percent multiply it by 100% ex) (12/5)(100%)=(12)(20%)=240% *review photo in phone of common percent equivalents Decimals to Percents and Vice Versa: * to convert decimal to percent multiply it by 100% ex) (.002)(100%)= 0.2%* - use the rules for multiplying by a power of 10... move the decimal point two places to the right and insert a percent sign - to convert a percent to its decimal equivalent use the rules for dividing by powers of 10... move the decimal point two places to the left... what is the decimal equivalent of 32%? .32 *if the denominator is 25, when trying to convert a fraction to a percent, you only have the multiply both the numerator and denominator by 4 in order to get the fraction to be over 100, which will give you the percent ex) (7/25)(100%)= (7)(4%)= 28%* Percent Formula: - part/whole(100%) Percent Increase Formula: - increase(100%)/original Percent Decrease Formula: - decrease(100%)/original * the original is the base from which the change occurs...it may or may not be the first number mentioned in the problem* ex) Two years ago, 450 seniors graduated from Inman High School. Last year 600 seniors graduated. By what percentage did the number of graduating seniors increase? - the original is the figure from the earlier time: 450 - the increase is 600 - 450, or 150. So the percent increase is: 150(100%)/450 = 33 and 1/3 %

Answer 3

AVG = Sum of Values / Number of Values or... AVG as the balance: difference between each term and the avg to determine what the unknown value needs to be in order to balance the other side of the avg ex) avg of 3,4,5 and x is 5 ... what is x? diff bw 3&5 is 2; diff bw 4 and 5 is 1; diff bw 5 and 5 is 0... thus x must be 5+2+1+0 to balance the avg... x = 8 *test question* the avg of six numbers is 6. If 3 is subtracted from each of four of the numbers, what is the new avg? x = sum of the 6 numbers x/6=6 (multiply both sides by 6 to solve for x) x = 36 but... now we know that 3 is subtracted from 4 of those numbers so we multiply 3(4) and subtract that from 36... 24 becomes the new sum but... now we have to find the new avg. so we do 24/6 and find that the new average is 4. Avg of consecutive, EVENLY spaced numbers: when consecutive numbers are evenly spaced, the avg will be the middle value: ex) 6,7,8...7 is the avg if there is an even number, there won't be on middle value. so to find the avg add the values and then divide by 2 Combining averages: iff there is an equal number of terms in each set, you can combine ex) suppose there are two bowlers and you must find their average score per game.. one has an avg of 100 and the other 200. Iff they bowled the same number of games, you can find their combined average...they both bowled 4 games so plug that into the formula and get: 4(100) + 4(200)/8 = 150

Answer 4

count the number of times something is multiplied by itself Raising a positive fraction less than 1 to a positive exponent greater than 1 results in a smaller value... The higher the exponent the smaller the result. When a negative number is raised to an even = the result is positive When a negative number is raised to an odd exponent, the result is negative Any number raised to the 0 power = 1 * An important rule to keep in mind is that we can only multiply and divide powers if the base numbers are the same. If our base numbers are different, then we cannot add, subtract, multiply, or divide exponents.* * You cannot combine exponents when adding or subtracting the bases!* To add or subtract with powers, both the variables and the exponents of the variables must be the same. You perform the required operations on the coefficients, leaving the variable and exponent as they are: 2x^2 + 3x^2 = 5x^2 8y^3 - 5y^3 = 3y^3 In the testmaker's favorite exponent trap you'll see a variable squared. The square of a number is positive, so the student assumes that the number is positive, when it could be positive or negative. Keep in mind the rules from the previous slide. Don't forget: If x^2 = 4, that means that x could be 2 or - 2.

Answer 5

follow the same rules as exponents Note: The square root symbol refers to the positive root only. In other words: If x2 = 25, then x equals 5 or - 5. But root 25 equals 5 only. In order to multiply or divide numbers with exponents, all exponents needed to have the same base. The same is true for operations involving roots, as evidenced by the following rule and examples. Roots Tip #3: Square roots with the same base can be added or subtracted: 4 root 3 + 7 root 3 = 11 root 3 Roots Tip #6: When taking the square root of a fraction between 0 and 1, the result will always be larger than the fraction itself: the square root of 1/4 is 1/2 which is greater than 1/4... just as a fraction is always larger than its square. square root of 169 is 13 when dealing with radicals in QC, use both the positive & negative: ex) radical (x^sqaured+39) = 8 so we need to figure out what plus 39 will give 64. 25 will. so x = pos 5 & neg 5...test both when deciding the QC answer bc they could result in D being the answer *refer to picture in phone

Answer 6

use PEMDAS to isolate variables 1. Eliminate any fractions by multiplying both sides: (multiply by the LCM to eliminate fractions) (you can also cross multiply if the fractions are equal ex - 12+b/3 = b+10/3... 6(12+b) = 3(b+10) 2. Put all terms with the variable you are solving for on one side by adding or subtracting on both sides 3. Combine like terms 4. Factor out the desired variable 5. Divide to leave the desired variable by itself

Answer 7

To solve for all of the variables in a system of equations, we must have at least as many distinct linear equations n as we have distinct variables n There are two ways to solve systems: 1. Substitution - solve one equation for one of the variables, and substitute that variable into the other equation (easy when one of the equations is solved for) 2. Combination - add or subtract one equation from another equation to cancel out one of the variables (create equivalent fractions with LCM so that one of the variables cancels) (use when one of the variables will easily drop out)

Answer 8

*refer to picture in phone* polynomials & quadratics: - the FOIL method - difference of two squares: a^squared - b^sqaured can be factored into (a+b)(a-b) 9x^squared - 1 = (3x+1)(3x-1) - factoring polynomials of the form a^sqaured+2ab+b^squared: *a & b are perfect squares and the middle is the product of 2 times the perfect squares* the sign in front of the 2ab term will determine whether the sign is positive or negative... (a+b)^squared or (a-b)^squared: ex) x^squared + 6x + 9 = (x+3)^squared Solving quadratic equations: - factor then set each binomial equal to zero & solve for each variable : (x-1)(x-2) = 0 x-1=0 or x-2=0... x = 1 or 2 ... plug both values back into the unfactored equation and see if the equation works

Answer 9

Greater than: > Less than: < *dash under implies or equal to* Should be treated like equations with two exceptions: 1. when we multiply or divide an inequality by a negative number, we must reverse the direction of the inequality sign 2. Single-variable equations are usually solved for a specific value, whereas inequalities can only be solved for a range of values

Answer 10

3: 4:5 5: 12:13 7: 24:25 8: 15:17 9: 40:41 *also appear in multiples* Isosceles Right: (45 degrees, 45 degrees, 90 degrees) x:x:x(root 2) 30, 60, 90 right triangle: x:x(root 3):2x a^2+b^2=c^2 ex) what is the length of the hypotenuse of an isosceles right triangle of area 32? area of an isosceles right triangle is v^2/2 v^2/2 = 32 v^2= 64 v=8...the hypotenuse is 8(root)2

Answer 11

In order to minimize the chances that you'll overlook something, you should follow these three steps every time you encounter a word problem: Determine the math concept being tested Determine what specific piece of information the question is asking you to find & label the variables or unknowns in a way that is easy to remember what they stand for Look for equations and/or other relationships in the language of the text...translate the problem into one or more equations sentence by sentence *refer to screenshot on desktop*

Answer 12

Addition & multiplication are associative: regrouping the numbers does not affect the result: ``` (3+5)+8 = 3+(5+8) 8+8 = 3+13 16 = 16 ``` (a+b)+c = a(b+c) (ab)c = a(bc)

Answer 13

The sum of angles around a point is 360 degrees; straight line is equivalent to an angle of 180 degrees Parallel lines never intersect, when they are cut by a transversal, corresponding angles are equal complementary angles: their angles sum to 90 acute angles: measure less than 90 obtuse angles: measure bw 90 and 180 supplementary: their measure sums to 180. adjacent angles are supplementary bc they lie along a straight line angle bisectors: a line or line segment bisects an angle if it splits the angle into two smaller, equal angles

Answer 14

Sum of interior angles is 180 degrees p = sum of all sides; a = 1/2(b)(h), where b & h must form a right angle Triangle inequality theorem: third side must be greater than the difference but smaller than the sum of the other two sides... *gives the relationship that sides of a triangle have to each other isosceles: two sides and two angles are equal equilateral: three sides are equal and all angles equal 60 degrees right: have one 90 degree angle interior and exterior angles: the sum of interior angles is 180 an exterior angle is equal to the sum of the remote interior angles...three exterior angles of any triangle add up to 360 QC and triangles: - if you know two angles, you know the third - to find the area, you need the base and the height - in a right triangle, if you have two sides, you can find the third. and if you have two sides you can find the area - in isosceles right triangles and 30/60/90 triangles, if you know one side you can find everything when a triangle is inscribed in a circle in such a way that one side of the triangle coincides with the diameter of the semicircle, the triangle is a right triangle ...may be the hypotenuse of the right triangle

Answer 15

area = pie(radius squared) diameter = 2r circumference = 2(pie)r or pie(d) pie = the ratio of the circumference of any circle to its diameter...c/d area of a sector: as a *fraction of the circles area* is equal to the degree measure of the corresponding central angle as a fraction of 360... area of sector/area = x(is the central angle)/360 arc length: as a fraction of circle's circumference is equal to the degree measure of the corresponding central angle as a fraction of 360... length of arc/circumference = x(is the central angle)/360 A line that has exactly one point in common with a circle is tangent to the circle, and it is perpendicular to the radius If a triangle is inscribed in a circle so that one of its sides is a diameter of the circle, then the triangle is a right triangle central angle: an angle formed by two radii chord: a line segment that joins two points on a circle. the diameter is the longest chord of a circle; half of the area of the circle is behind the diameter tangent: a line that touches only one point on the circumference of a circle...a line drawn tangent to a circle is perpendicular to the radius at the point of tangency

Answer 16

closed plane figure formed by 3 or more line segments Sum of interior angles of polygon with n sides: (n-2)180 * you can also figure out the interior angles by dividing the polygon into triangles...draw a diagonal from any vertex to all nonadjacent vertices...then multiply the number of triangles by 180 quadrilaterals: any 4 sided figure ... two triangles put together (special right triangles, pythagorean triplets are important) parallelogram: quadrilateral whose opposite sides are parallel - including rectangles, squares - opposite sides and angles are equal - area = (b)(h) - the diagonals bisect each other - rectangle: diagonals are equal, but are not perpendicular; perimeter = 2(l+w) - square: diagonals are equal & perpendicular so you can use the 45,45,90 rule to find the length of the diagonal QC w parallelograms: - in a parallelogram, if you know two adjacent sides, you know all of them; and if you know two adjacent angles you know all of them - in a rectangle if you know two adjacent sides, you know the area - in a square if you are given virtually any measurement (area, length of a side, length of a diagonal) you can figure out the other measurements a polygon is inscribed in a circle if all the vertices of the polygon lie on the circle. a polygon is circumscribed about (drawn around) a circle if all of the sides of the polygon are tangent to the circle

Answer 17

Uniform solids: solids in which the measure of each dimension is constant through the entire object volume of uniform solid: (area of base)(height) volume of cylinder - (pie)(r squared)(height) volume of a rectangular solid - (l)(w)(h) volume of a cube - edge cubed ; SA of cube - 6(edge squared)...there are 12 edges in a cube SA - sum of the area of all of the faces of the solid (top,ends, sides) SA rectangular solid - 2lw+2lh+2hw Lateral SA of a cylinder - (circumference of height)(base) = (2)(pie)(r)(h) Total SA of a cylinder - areas of circular ends + lateral surface area = (2)(pie)(r-squared) + (2)(pie)(r)(h) lateral (pipe) surface of a cylinder: unrolls into a rectangle with a length equal to the circumference of the circular base and the height of equal to that of the cylinder

Answer 18

Slope of line - change in y / change in x equation of a straight line in the two dimensional coordinate plane is typically expressed as y = mx+b ; b = y-intercept ; m is the slop to find the x-int, set y=0 & solve for x to find the y-int, set x=0 & solve for y if line is parallel to y-axis slope is undefined ... the change in x = 0... therefore has the equation x=a, where a is the x-intercept lines that are parallel to the x-axis have a slope of 0 and therefore have the equation y=b (y-intercept) distance in the coordinate plane - pythagorean theorem : in order to find the distance between two points on the coordinate plane you have to draw right triangle...if the line segment is not parallel to either of the axes, it becomes the hypotenuse of a right triangle to find the length of a line segment parallel to one of the axes, all you have to do is find the difference bw the end point coordinates that do change: - parallel to x use the x coordinates - parallel to y use the y coordinates

Answer 19

If neither the numerator nor the denominator are the same, you have three options: 1. turn both fractions into their decimal equivalents 2. express both fractions in terms of a common denominator and then see which new equivalent fraction has the largest numerator 3. cross multiply the numerator of each fraction by the denominator of the other... the greater result will wind up next to the greater fraction - ex) which is greater 5/6 or 7/9? 5*9= 45 ; 6*7=42... since 45 is bigger 5/6 is the larger of the two

Answer 20

- the reciprocal of a fraction bw 0 and 1 is greater than both the original fraction and 1 : ex) the reciprocal of 2/3 is 3/2, which is greater than both 1 and 2/3 - the reciprocal of a fraction bw -1 and 0 is less than both the original fraction and -1 ex) the reciprocal of -2/3 is -3/2, which is less than both -1 and -2/3

Answer 21

relationship bw speed, distance & time: speed = quantity / time this can also be written as .... ``` time = distance/speed distance = (speed)(time) ``` in general for rate problems speed equations works with rate & quantity as substitutes: rate = quantity/time ex) The shortest distance between the starting point and the ending point of a trip if a vehicle is driven 60 miles north and 80 miles east ... this means draw a right triangle with 60 as the height since it is mentioned first and 80 as the base... the hypotenuse is the actual answer

Answer 22

question type: how long it takes a number of people working together to complete a task two people or two machines: T= ab/a+b a = amount of time it takes person a to complete the task b = amount of time it takes person b to complete the task more than two people: 1/a + 1/b + 1/c = 1/T

Answer 23

in order to solve for the NUMERICAL value of an unknown, you need as many distinct equations as you have variables ... if there are two variables you need two distinct equations Implications for QC: to have enough information to compare the two quantities, you must have at least as many equations as you have variables * make sure you REALLY have two distinct equations: ex) 2x+3y = 8 and 4x+6y=16 are the same equations in different forms* *when asked to solve for one variable in terms of the other, isolate the desired variable on one side of the equation and move all of the constants and other variables to the other side*

Answer 24

list of numbers. the value of a number in a sequence is related to its position in the list... the subscript part of each number will give the position of the each element in the series... most questions give a formula that defines each element

Answer 25

question asks how many unordered subgroups can be formed from a larger group n!/k!(n-k)! n = number of items in the group as a whole k = the number of items in each subgroup formed ! = factorial ... the product of an integer & all of the integers below it... descending order down to 1... 4! = 24...(4)(3)(2)(1) when the same numbers are in the numerator and denominator you can cancel them in order to reduce the terms when asked to find the potential combinations from multiple groups: 1. find the # of possible combinations in each group and then multiply them to get the actual answer (pg.213, premier) ex) how many groups can be formed consisting of 2 people from room A and 3 people from room B if there are 5 people in room A and 6 people in room B - use the formula for room A: the number of combinations of 2 in a set of 5 - use the formula for room B: the number of combinations of 3 in a set of 6 - multiply the results when asked to find the number of possible subgroups when choosing from one item from a set ... the number of possible subgroups will always equal the number of items on the set: ex) Restaurant A has 5 appetizers, 20 main courses, and 4 desserts. If a meal consists of 1 appetizer, 1 main course, and 1 dessert, how many different meals can be ordered at Restaurant A? - the number of possible outcomes from each set is the number of items in the set: there are 5 appetizers, 20 main courses, 4 possible desserts ...(5)(4)(20)= 400 different meals

Answer 26

permutations differ from combinations in that permutations are ordered... by definition each combination larger than 1 has multiple permutations. "how many ways/arrangements/orders/schedules are possible? indicates a permutation question

Answer 27

the numerical representation of the likelihood of an event or combination of events...expressed as a ratio of the number of desired outcomes to the total number of possible outcomes probability of any event occurring cannot exceed 1 (probability of 1 represents a 100% chance of an event occurring), and it cannot be less than 0 (a probability of 0 represents a 0% chance of an event occurring) *see examples on phone of pg.215* probability of dependent events: when the probability of a later event occurring varies according to the results of an earlier event...see example on phone of page 216

Answer 28

8 16 32 64

Answer 29

Sum of digits is a multiple of 3 and last two digits is a multiple of 4

Answer 30

pos, neg, zero, one , fraction

Answer 31

1. add/subtract same value 2. multiply/divide same POS value 3. square both columns to remove radical 4. cross multiply to compare fractions

Answer 32

x^2 - 9 = (x+3)(x-3) x^2+6x+9 = (x+3)(x+3) = (x+3)^2 x^2-6x+9= (x-3)(x-3) = (x-3)^2 x^2+2x(radical 7)+7 = (x+radical 7)(x+radical 7) x^2+x+1/4 = (x+1/2)(x+1/2)

Answer 33

parallelogram: quadrilateral whose opposite sides are parallel - including rectangles, squares, rhombus - opposite sides and angles are equal - area = (b)(h) - the diagonals divide into two identical triangles QC w parallelograms: - in a parallelogram, if you know two adjacent sides, you know all of them; and if you know two adjacent angles you know all of them

Answer 34

- all angles are 90 degrees - opposite sides are equal - diagonals are equal in length and bisect each other , but are not perpendicular; - perimeter = 2(l+w) - type of parallelogram QC - in a rectangle if you know two adjacent sides, you know the area

Answer 35

- all angles are 90 degrees - all sides are equal - diagonals are equal in length, bisect each other and meet at right angles; diagonals divide square into two right angles of equal size (they are perpendicular so you can use the 45,45,90 rule to find the length of the diagonal) - area = side squared or diagonal squared/2 - type of rectangle QC - in a square if you are given virtually any measurement (area, length of a side, length of a diagonal) you can figure out the other measurements

Answer 36

- One pair of opposite sides is parallel. - Isosceles trapezoid has equal leg lengths - Area = base 1 + base 2 / 2 * height (average of the bases * height)

Answer 37

- opposite sides are parallel - opposite angles are equal - all sides are equal - diagonals bisect one another and meet at right angles - Area= diagonal 1 * diagonal 2 / 2

Answer 38

equals 1 a^0 = 1

Answer 39

two equal sides and two equal angles | A square can turn into two isosceles right triangles 45-45-90

Answer 40

can be split into 30-60-90 triangle Area = root 3 / 4 * side squared Height = 1/2 (side) * root 3

Answer 41

positive slope line equation is y = x and the slope is 1 so a point on the line would be (1,1) negative slope line equation is y = -x and the slope is -1 so a point on the line would be (-1,1)

Answer 42

means all of the angles are equal and all of the sides are equal

Answer 43

Thousandth

Answer 44

3 Feet/Yard

Answer 45

Can also be called divisors

Answer 46

A positive integer that is not prime

Answer 47

include zero because it means positive or zero

Answer 48

measure of how far from zero

Answer 49

(Last - First) + 1 Ex) How many integers are there from 14 to 765, inclusive? 765 - 14 + 1 = 752

Answer 50

(Last - First) / Increment + 1 "How many multiples of 4?" "How many even numbers?" Ex) How many multiples of 7 are there between 100 and 150? 105 is the first number and 147 is the last number (not 100 and 150 because those are not multiples of 7) 147 - 105 = 42 42 / 7 = 6 6 + 1 = 7

GRE Math 1 Flashcards

(85 cards)