GRE Math
Decimal of 1/8, 3/8, 5/8 and 7/8
0.123, 0.375, 0.625, 0.875
Divisibility Rule for 2
All even numbers (even in one's place) are divisible by 2
Definition of integer
All positive or negative whole numbers (including zero which is neither positive nor negative)
Definition of number
Any number; positive, negative, zero, whole number, fraction or decimal
Dividing with decimals
For division, move the decimals to the right until the denominator is a whole number ex: (0.56/.0007) * 10,000 = 5600 / 7 = 800 ex: (0.00013/0.025) * 4 = 0.0052
Equation for a percent decrease
"Y is decreased by 30%" or "x is 30% less than Y" (multiplier for a P% decrease) = 1- (P% as a decimal) ex: After an item was discounted 80% the new price is %150. What was the original price? 1-.8 = .2 x*.2 = 150 x = $750
How to find the greatest common factor (GCF)
1) Find the prime factorization ex: GCF of 360 and 800 360 = 36x100 = 6x6x10=3x2x3x2x5x2 = 3^2x2^3x5 800 = 80x10 = 40x2x5x2 = 2x2x5x2x2x5x2 = 2^5x5^2 2) What is the highest power of 2 that they have in common? Each have 3 factors of 2 Highest power of 3? Only one has 3 so it is zero Highest power of 5? 1 Thus GCF = 2^3 x 5 = 8x5 = 40
Decimals of 1/20, 1/40, 1/600
1/20 = 1/2 * 1/10 = .5*.1 = 0.05 1/40 = 1/4 * 1/10 = .25*.1 = 0.025 1/600 = 1/6 * 1/100 = 0.166666 * 0.01 = 0.0016666
240 is 30% of what number?
240=.3x 240/.3=x 2400/3=x 800=x
Divisibility Rule for 6
A number must be divisible by 2 AND 3 So it must be even and the sum of the digits must be divisible by 3 ex: 267,914,296 = 2+6+7= 15+9+1+4= 29+2+9+6=46 (not divisible by 3 or 6)
Memorize prime numbers 1-60
A prime number is only a factor of itself and 1 *1 is not a prime number *2 is the only even prime # -If a number less than 100 is not divisible by any prime divisor left than 10, then the number has to be prime (2, 3, 5, 7) Prime numbers : 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59
Associative Property
Ability to group numbers in different groupings; addition and multiplication are associative, subtraction and division are generally not a + (b+c) = (a+b) + c a * (b*c) = (a*b) * c
Hoe to change fractions to percents
Change to a decimal and move 2 places over for % ex: 3/8 = 0.375 = 37.5% ex: 2/3 = .666667 = 66.67%
Consecutive Integers Properties
Consecutive - in a row, following one another; could be positive, negative or both 1) A set of n consecutive integers will always one number divisible by n 2) If n is odd, then the sum of a set of n consecutive integers will always be divisible by n 3) In a set of 4 consecutive, must have 2 even 2 odd In a set of 3, could have either 2 even, 1 odd or 2 odd, 1 even
Combining Ratios Strategy 2 - Solve for the absolute quantity in each term
Ex: One cup of butter is enough for 12 of Kathy's cookies and 1 cup of sugar is enough for 8 of her cookies. If she used 5 more cups of sugar than butter, how many cookies can she make? B:C = 1:12 Multiply by 2 = 2:24 S:C = 1:8 Multiply by 3 3:24 B:S:C = 2:3:24 or B=2n S=3n and C=24n n = 5 -> C=24(5) = 12*10 = 120 cookies
Combining Ratios Strategy 1- Find the common element
Find the least common multiplier Ex: On a high school team the ratios of sophmores to juniors is 2:3 and the ratio of juniors to seniors is 5:6. Sophomores are what fraction of the whole team? S:J = 2:3 (multiply by 5) J:Sr = 5:6 (multiply by 3) S:J = 10:15 J:Sr = 15:18 Soph:Junior:Senior = 10:15:18 Whole: 10+15+18=43 Soph:Whole = 10/43
Compound interest
Interest on interest; the more interest that's accrued, the larger the amount of the next interest payment; no two successive interest payments are ever the same -With large amounts of money and/or long periods of time, the difference between successive interest payments becomes substantial A = P(r^y) P= principal y=years principal will be multiplied by the % increase multiplier A=total amount in account after y years r= multipler (1 + I/100n) -Can change compounding period quarterly n=4, monthly n=12, daily n=365 ex: Suppose the bank pays 5% annual interest, compounding quarterly 5/4 = 1.25% ex: If Susan invests $1000 in an account that yields 5% annual, compounding quarterly, how much does she have in 6 years? Quarterly percent = 5/4 = 1.25% Multiplier = (1+ 1.25/100) = 1.0125 A = 1000*1.0125^6*4
Divisibility Rule for 4
Look at the last 2 digits, if they form a 2 digit number divisible by 4, then the whole number is divisible by 4
What is 75% of 280?
Percents and Fractions - only use if percent is an easy fraction 3/4 * 280/1 = 3/1 * 70/1 = 210
What is 80% of 200?
Percents as multipliers "is" means equal "of" means multiply x= .8*200 = 160
Rules of proportions
Proportion is an equation in the form of fraction=fraction a/b=c/d -> ad=bc ex: 5/7=3/x -> 5x=21 -> x=21/5 *CAN do vertical cancellation in same fraction *CAN do horizontal cancellation *CAN'T do diagonal cancellation -when we multiply two fractions, cross-cancellation is legal (a/b * c/d) but when we are in proportion it is illegal
How to change percents to fractions
Put percent over 100 and simplify ex: 20% = 20/100 = 1/5 ex: 0.02 = 2/10,000 = 1/5,000
Equation to find the multipler
Some problems give us the starting and ending values and ask us to find the percent of increase or decrease multiplier = new price/old price ex: Price decreased from $250 to $200, what is the percent decrease? 200/250 = 4/5 = 0.8 1-0.8=.2 -> 20% decrease ex: Price of an item increased from $200 to $800, what's the percent increase? 800/200 = 4; 4-1=3 so there is a 300% increase ex: Price of an item increased from $60 to $102, what's the percent increase? 102/60 = 17/10 = 1.7; 1-1.7 = .7 so there is a 70% increase
Absolute Value
The distance of the number from the origin |6| = 6 |-14| = 14 |0| = 0 |x - 1| > 4 x > 5 OR x < -3
Multiplying/Dividing with Tens
When we divide any number by 10 or multiply by 0.1, we move the decimal point one place to the left ex: 0.02 / 10 = 0.002 39.85 * 0.1 = 3.985 64,000 / 0.0001 = 6.4 When we multiply any number by 10 or divide by 0.1, we move the decimal point on place to the right ex: 24 * 10 = 240 2.35/0.01 = 235 4.7/10^-4 = 4.7/0.0001 = 47,000
Distributive Property
a*(b+c) = a*b + a*c a*(b-c) = a*b - a*c
Factor, divisor and divisible definitions
-Factor and divisor mean the same thing -Every integer is a factor of itself and 1 -If C/A=B then A is a divisor of C because it divides evenly into C Three ways to say the same thing: 1) 8 is a factor of 24 2) 8 is a divisor of 24 3) 24 is divisible by 8 ex: 12 is not divisible by 8 because there is no positive integer
Divisibility Rule for 3
Add up all digits of the number, if the sum of the digits is divisible by 3 then the number is divisible by 3 ex: 135 = 1+3+5=9 (divisible by 3)
Prime Factorization
Any integer greater than 1 that's not prime can be expressed as a product of primes ex: 9= 3x3 12 = 2x2x3 96=2x48 = 2x6x8 = 2x2x3x2x2x2 = 2^5x3 1) r is a factor of Q 2) r is a divisor of Q 3) Q is divisible by r 4) Q is a multiple of r 5) every prime factor of r is included the the prime factorization of Q ex: 4680 = 2^3 x 3^2 x 5 x 13 Are the following factors? 25 = 5x5 no 120 = 12x10 = 2x2x3x5x2 yes 45=5x3x3 yes 180=18x10=3x3x2x5x2 yes 65=13x5 yes 85=5x17 no
Ideas of Multiples
Multiple: inverse relationship to factor ex: 91 is a multiple of 7 1) Just as every positive integer is a factor of itself, every positive integer is a multiple of itself 2) If we need the first five multiples of a number, we simply multiple the original number by the numbers {1,2,3,4,5} 3) If P is a multiple of r, then it must be true the (P-r) and (P+r) are also multiples of r ex: 2401 is a multiple of 7 2401+7 = 2408 and 2401 - 7 = 2394 are multiples of 7 4) If P & Q are multiples of r, then (P+Q) and (P-Q) are also multiples of r ex: 700 and 49 are both multiples of 7 700+49 = 749 and 700-49=651 are both multiples of 7 5) If P is a multiple of r, then any multiple of P is a multiple of r ex: any multiple of 52 is a multiple of 13 6) If P and Q are multiples of r, then the product P*Q is a multiple of r ex: 24 and 80 are multiples of 8 24 + 80 = 104 80-24 = 56 24 * 80 = 1920 these are all multiples of 8 CAN'T divide - if we divide 80 by 24 the quotient is not an integer
How to square a number ending in 5; e.g 75^2
Remove the five, add one to the remaining digit, find the product of these two #'s, put the number in front of 25; e.g 7x8 = 56 75^2=5625
Different ways to present ratios
Test will always give ratios in the simplest form, absolute number of participants will be larger than the number in the given ratio (use scale factor) 1) p to q form: ratio of boys to girls is 3 to 4 2) fraction form: ratio of boys to girls is 3/4 3) colon form: ratio of boys to girls is 3:4 4) idiom form: for every 3 boys, there are 4 girls -To solve ratio problems, set 2 equivalent fractions equal (known as a proportion) ex: In a class, the ratio of boys to girls is 3:7. If there are 32 more girls than boys, how many boys are there? boys=3n girls=7n 7n-3n = 4n 4n = 32 n = 8 Number of boys = 3n = 3(8) = 24 *Each part is part of a whole factor Ex: Purpose concrete is creased using a 1:2:3 ratio of cement to sand to gravel. If we have 150 kgs of sand, how many kgs of concrete can we make? 1+2+3 = 6 total parts sand: concrete = 2:6 or 1:3 1/3 = 150/x -> 1/3x = 150 -> x=150*3 = 450 kgs of concrete
Decimal of 1/9 and 7/9
0.11111... and 0.7777...
Equation for a percent increase
"Y is increased by 30%" or "x is 30% greater than Y" (multiplier for a P% increase) = 1 + (P% as a decimal) ex: After a 30% increase, the price of something is $78. What was the original price? x*1.3=78 x=60
Rules of fractions
-Can multiply any expression by n/n because multiplying by 1 never changes the value -the product of any fraction with its reciprocal equals one 4/17*17/4 = 1 6*1/6 = 1 -one divided by any fraction equals the reciprocal of that fraction 1/(3/7) = 7/3 -If a number is bigger than 1, then its reciprocal is between 0 and 1 and vice versa
Operations with Fractions
-Can only add or subtract 2 fractions if they have a common denominators ex 1/5 + 3/5 = 4/5 Find a common denominator by multiplying both parts of a fraction by a number to make both denominators equal ex: 3/5 - 1/3 = 3/5(3/3) - 1/3(5/5) = 9/15 - 5/15 = 4/15 -Can multiply fractions by multiplying across the numerators and denominators (can cancel across to make smaller before you multiply) ex: 5/14 * 7/15 = 1/2*1/3 = 1/6 ex: 21/35*30/42=1/7*6/2=6/14=3/7 -Can divide fractions by multiplying by the reciprocal ex: (3/5)/2 = 3/5*1/2=3/10 ex: 6/(3/4) = 6*4/3 = 24/3 = 8
Comparing Fractions rules
-Changing numerators, same denominator: If a>b then (a/c) > (b/c) -Changing denominators, same numerator: If p>q, then (s/p) < (s/q) if p,q,s > 0 *Bigger denominators make smaller fractions 7/24>7/36 -If both the numerator gets bigger and the denominator gets smaller then the fraction gets bigger 3/8<4/7 -If we multiply both the numerator and the denominator by the same # then we get an equivalent fraction 3/7 * 12/12 = 36/84 -Can compare fractions by cross multiply ex: 7/11 and 5/8 7*8 and 5*11 56>55 --> 7/11>5/8
Divisibility Rule for 9
Add up all digits, if the sum is divisible by 9 ex: 1296 = 1+ 2+ 9+ 6 = 18 (divisible by 9)
Integer Property Strategies
-Make sure your dealing with integers; must use words "integer", "even", "odd", "prime" -Don't forget about zeros and negative -Question only talks about remainders if all the numbers involved are positive integers -Factors: 13 is a factor/divisor of 78 78 is divisible by/a multiple of 13 13 is part of the prime factorization of 78 -1 is not a prime -2 is the lowest and only even prime number -a negative squared is a positive -LCM = PxQ / GCF -> always cancel with P/GCF or Q/GCF -If test gives variables can always use substitution 1=odd 2=even -For all questions involving a remainder, remember these strategies: 1) Listing possible dividends (when divided by 6 have remainder of 3) 2) Using the rebuilding the dividend formula dividend = divisor * quotient + remainder -For consecutive numbers, often appear in variable form (t^2 - 2t)*(t-1) = t(t-2)*(t-1) = (t-2)*(t-1)*t This expression is the product of 3 consecutive integers only if we know that t is an integer
Factors of a perfect square
-The exponents of the prime factors of a square all must be even k = 2^6 x 3^4 x 5^2 just by looking we know its a perfect square k = 360^2 -A perfect square always has an odd number of factors ex: 36 = 6^2
Decimal of 1/7
0.143
Decimal of 1/6 and 5/6
0.1666666... and 0.833333...
Decimal of 1/5, 2/5, 3/5 and 4/5
0.2, 0.4, 0.6, 0.8
Decimal of 1/3 and 2/3
0.33333333... and 0.6666666...
Operations with Fractions continued
1) Can cancel any part of the numerator with any part of the denominator ab/cd = a/c*b/d ex: 27(y+5)(2y-2)/6(y-1) = 9(y+5)*2(y-1)/2(y-1) = 9(y+5) 2) Can separate a fraction into 2 fractions by adding or subtracting in the numerator a+b/c = a/c + b/c d-e/f = d/f - e/f *We CAN'T separate a fraction into 2 fractions by adding or subtraction in the denominator a/(b+c) =/= a/b+a/c d/(e-f) =/= d/e - d/f *CAN split up the numerator but the denominator must stay unchanged a+b/c+d = a/c+d + b/c+d 3) Multiplying a fraction by its denominator = numerator ex: 4/7 * 7 = 4 ex: x/5 = 3 -> 5(x/5) = 3(5) -> x=15 4) Simplifying complex fractions ex: (x+1/6)/(x+6/15) Multiply total fraction by 15/15 =(x+1/6)*15/(x+6) = 15x + 5/2/(x+6) Multiply total fraction by 2/2 =30x+5/2x+12
How to find the least common multiple (LCM)
1) Find prime factorization and the GCF ex: 24 and 32 24 = 6x4 = 3x2x2x2 = 3 x 2^3 32 = 8x4 = 2x2x2x2x2 = 2^5 GCF = 2^3 = 8 2) Write each number in the form of GCF times another factor 24 = 8x3 32=8x4 The LCM is the product of these 3 factors LCM = 8x3x4 = 8x12 = 96 *LCM helps in adding and subtracting fractions because the LCM is the LCD ex: 1/10 - 1/35 GCF 10=2x5 35=5x7 GCF=5 LCM = 2x5x7= 70 1/10 (7/7) - 1/35 (2/2) = 7/70 - 2/70 = 5/70 = 1/14 *If A is a factor of R, then the LCM of A and R must be R ex: The LCM of 8 and 24 is 24 *If A and B have no factors in common greater than 1 then their LCM would have to be their product AxB ex: 7 and 15 -> LCM = 7x15
How to count factors of large numbers
1) Find prime factorization of number ex: 8400 = 84x100= 7x12x10x10 = 7x3x2x2x5x2x5x2= 2^4x3x5^2x7 2) Make a list of the exponents of the prime factors {4,1,2,1} 3) Add 1 to every number on the list {5,2,3,2} 4) Multiply the numbers together 5x2x3x2=60 The number 8400 has 60 factors (including 1 and 8400) *To find the number of odd factors, repeat this procedure but ignore the factors of 2 ex: 21600 = 2^5 x 3^3 x 5^2 {3 x 2} +1 = {4 x 3} = 12 odd factors *To find the even factors, find total factors and subtract odd factors 72 total factors - 12 = 60 even factors
Even and Odd Integer properties
1) Zero is an even number *test loves this* 2) Evens and odds include both positive and negative numbers 3) Evens and odds pertain only to integers, any non-integer is neither even nor odd 4) All even numbers as divisible by 2 - the prime factorization of an even integer always contains 2 5) No odd number is divisible by 3 - prime factorization of a positive odd number will never contain a factor of 2 *Add or subtract "likes" we get EVEN ex: E+E = E E - E = E O+O = E O-O=E *Add or subtract "unlikes" we get ODD ex: E+O = O E - O = O *E*E=E 2X4=8 E*O=E 4X3=12 O*O=O 5X3=15 *At least one even factor in a product = even *Only way to equal odd in a product is if every factor is odd
First 15 perfect squares
1^2 = 1 6^2 = 36 11^2=121 2^2 = 4 7^2=49 12^2=144 3^2 = 9 8^2=64 13^2=169 4^2 = 16 9^2= 81 14^2=196 5^2 = 25 10^2= 100 15^2=225
Sequential Percent changes
2 or more percent changes that follow in a sequence *When you have 2 or more changes in a row, NEVER add or subtract the percents. Must take percent of the first and then the percent of the second ex: Anne wants to buy a shirt she saw last week that was $100. When she goes to buy it she notices the price has increased 30%. If she uses her 30% employee discount to buy the shirt, how much will she pay? 100*1.3 = 130 1 - .3 = .7 130* .7 = $91
Properties of Remainders
20 / 6 yields 3 with a remainder of 2 20 = the dividend 6=divisor 3=quotient 2= remainder 0 < or = remainder < divisor D/S = Q + r/s S=divisor *Trick Question that the test loves: What is the smallest positive integer that when divided by 12 has a remainder of 5? It's 5 -> if the divisor is larger than the dividend the integer quotient = 0 and the remainder equals the dividend 5/12 = 0 with r=5 Rebuilding the dividend: D = S x Q + r ex: When positive integer N is divided by positive integer P, the quotient is 18 with a remainder of 7. When is divided by (P +2) the quotient is 15 with a remainder of 1. What is N? N = 18*P + 7 N = 15(P + 2) + 1 = 15P +31 18P + 7 = 15P + 31 3P = 24 P = 8 N = 15(8+2) + 1 = 120 + 30 + 1 = 151
Definition of rate
Any ratio with different units in the numerator and denominator. Rates are ratios -Must set up an equation of form: ratio=ratio is called a proportion *MUST know there are 360 degrees in a revolution ex: A bumblebee wing flaps 1440 times in 8 seconds. How many times does it flap in a minute? 1440/8 = 720/4 = 360/2 = 180 180 flaps/ second 1 minute = 60 seconds 180*60= 10,800 ex: If gold has a density of 20 grams/cm^3 and the price of gold is $50/gram, then what's the price of the gold in a 2cm*2cm*2cm cube of gold? 8cm^3 * 20 gram/cm^3 * $50/gram = 8*20*50= $8000
How to change from percents to decimals
Divide by 100 and move decimal point 2 places to the left ex: 4% = 0.04 ex: 0.25% = 0.0025
How to square a number not ending in zero or five; e.g 41^2, 69^2, 84^2
Find the square of the closest number ending in five or zero, add the number and add the original number 41^2 = 40^2 + 40 + 41 = 1600 + 81 = 1681 69^2 = 70^2 - 70 - 69 = 4900 - 140 + 1 = 4761 84^2 = 85^2 - 85 - 84 = 7225 - 160 - 9 = 7056
56 is what percent of 800?
Finding the percent 56=800x 56/800=x 7/100=x 0.07=x 7%=x
Multiplying with decimals
For multiplication count the number of digits to the right of the decimal point ex: 6.25 * 0.048 -> 2+3 = 5 decimal points Now ignore the decimal points 625*48 = 1250*24 = 2500*12 = 5000*6 = 30,000 = .3 ex: (.03)^3 = (.03)*(.03)*(.03) -> 2+2+2=6 3^3 = 27 -> 0.000027
GEMDAS
Grouping symbols (paraenthesis, brackets, square root sign, long fraction bar, exponent slot 3^x-7) Exponents, Multiplication and Division, Additional and Subtraction
Divisibility Rule for 5
If last digit is 5 or 0, then divisible by 5
GCD - LCM formula
LCM = PXQ / GCF ex: the LCM of 48x75 48 = 12x4 = 2x2x2x2x3 = 3 x 2^4 75 = 15x5 = 3x5x5 GCF = 3 LCM = 48x75 / 3 = 48x25 = 24x50 = 12x100 =1200
How to change decimals to percents
Multiply by 100 and move decimal point 2 places to the right ex: 0.68 = 68% ex: 0.075 = 7.5% ex: 2.3 = 230%
In word problems, is means.... of means....
equals multiply ex: What is 3/5 of $400 3/5 * 400/1 = 3/1 * 80 = 340= 240
