GRE Quantitative
0
0 is an integer 0 is an even integer 0 is a multiple of every number
Steps to solving inequality equations
1. Get rid of fractions and decimals 2. Get rid of all parentheses 3. Combine like terms on each side 4. Get all variables on one side 5. Get all plan numbers on other side 6. Divide both sides by coefficient of variable If dividing by negative, remember to flip sign!
n! factorial in fraction form
6!/8! = (6x5x4x3x2x1)/(8x7x6x5x4x3x2x1) = 1/(8x7) = 1/56. Cross out same numbers from numerator and denominator.
Factor exponential terms
6⁵-6³ = 6³(6²-1)
Lowest terms
A fraction is in lowest terms, or simplest form, if it cannot be simplified anymore.
Ratio example
A hospital has enough pills on hand to treat 10 patients for 14 days. How long will the pills last if there are 35 patients? 10x14=140 patient-days 140pd/35 patients = 4 DAYS
Negative root
A negative root without the parentheses results in a negative number. e.g., (-3)² = 9, BUT -3² = -9
Geometric sequence
A sequence in which the ratio between any two consecutive terms is the same If A1, A2, A3 is a geometric sequence whose common ratio is r, then An = A1r^(n-1) e.g., what is the 12th term of the sequence 3, -6, 12, -24, 48, -96...? A12=(3)(-2)^(12-1) = 3(-2,048) = -6,144
Reduce fractions before multiplying
Before multiplying any fractions, reduce by dividing any numerator and any denominator by a common factor.
Denominator
Bottom number in a fraction
Using extreme values in inequalities
Helping when solving: 1. Problems with multiple inequalities where the question involves the potential range of values for variables in the problem 2. Problems involving both equations and inequalities e.g., 0≤x≤3 and y<8, which of the following could NOT be the value of xy? Extreme values for x: Lowest value for x=0 Highest value for x=3 Lowest value for y=-∞ Highest value for y=LT8 Lowest value for xy= no lower limit Highest value for xy = (3)(LT8) = LT24. xy cannot be 24.
Inequalities for numbers between 0 and 1
If 0<x<1, and a is positive, then xa<a. If 0<x<1, and m and n are positive integers with m>n, then x^m < x^n <x. If 0<x<1, then √x > x. If 0<x<1, then (1/x)>x. In fact, 1/x>1.
Ratio example #2
If 15 workers can pave a certain number of driveways in 24 days, how many days will 40 workers take working at the same rate to do the same job? 15x24= 360 worker-days 360wd/40 workers = 9 DAYS If 15 workers can pave 18 driveways in 24 days, how many days would it take 40 workers to pave 22 driveways? 15wk x 24ys = 40wkdays 360wkdays/40wk = 9 days to do same amount of work 18 driveways/9 days = 2 driveways/day 22driveways x 1day/2driveways = 11 DAYS
Sequence problem
If each number in a sequence is 3 more than the previous number, and the 6th term is 32, what is the 100th number? From 6 to 100 there are 94 jumps. 94x3 = 282 32+282 = 314, the 100th number
Consecutive
In order from least to greatest
Rewriting bases
Lookout for perfect squares and cubes; you can change them into bases with exponents. e.g., 5³×25²→5³×(5²)²→5³×5⁴= 5⁷
Number
Made up of digit or collection of digits
Optimization problems
Minimization/maximization problems You need to focus on the largest and smallest possible values for each of the variables e.g., If -7≤a≤6 and -7≤b≤8, what is the maximum possible value for ab? Mina (-7)x Minb (-7) =49 Mina (-7)xMaxb (8) =-56 Maxa(6)xMinb(-7)=-42 Maxa(6)xMaxb(8)=48 Maximum value is 49.
5 Rules of working with exponents (MADSPM)
Multiply/Add, Divide/Subtract, Power/Multiply 1. When you see exponents with equal bases which are being Multiplied, Add the powers. 2. When equal bases are Divided, you Subtract the exponents. 3. When an exponent is raised to a Power, you Multiply the powers. 4. Negative exponent. 5. Zero Exponent. Other rules: 1. Rewrite terms using common bases. 2. Look for a way to factor the expression.
Remainder formula
Remainder must be smaller than divisor; possible remainder range from 0 to N-1. When a positive integer a is divided by a positive integer b, you first find the greatest multiple of b that is less than or equal to a. That multiple of b can be expressed as the product qb, where q is the quotient. The remainder is equal to a minus that multiple of b, or r=a-qb. If m and n are positive integers and if r is the remainder when n is divided by m, then n is r more than a multiple of m. That is, n=mq+r where q is an integer and 0≤r<m.
Absolute value
The distance between the number x and 0 on the number line is called the absolute value of x, written as |x|.
Unique prime factors
The individual prime factors of a number, regardless of how many times they appear. e.g., 12→ 2×6 →2×2×3 The unique prime factors are 2 and 3.
Successive percent changes (example)
The monthly enrollment at a preschool decreased by 8% and increased 6% during the next month. What was the cumulative percent change for the two months? 100-8 = 0.92 100+6 = 1.06 (1.06)(0.92) = 0.9752 100-97.52 = 2.48, the cumulative percent change in the enrollment for 2 months, a 2.48% decrease
Divisor
The number you divide by
Least Common Mulitple (LCM)
The product of all the primes that appear in any of the factorizations, using each prime the largest number of times it appears in any of the factorizations. e.g., 108×240 108→2×2×3×3×3 240→2×2×2×2×3×5 LCM = 2×2×2×2×3×3×3×5 = 2,160.
Greatest Common Divisor/Factor (GCF)
The product of all the primes that appear in each factorization, using each prime the smallest number of times it appears in any of the factorizations. e.g., 108×240 108→2×2×3×3×3 240→2×2×2×2×3×5 GCF = 2×2×3 = 12.
Sum
The result of addition
Quotient
The result of division
Multiplying/dividing fractions
To multiply two fractions, multiply the two numerators and multiply the two denominators To divide one fraction by another, first invert the second fraction - find its reciprocal - then multiply the first fraction by the inverted fraction
Numerator
Top number in a fraction
System of equations
Two basic methods to solving systems of linear equations: Substitution (set one equation equal to variable and substitute in that variable into first equation) Elimination (Cancel out variables that are the same with opposite signs)
Compound inequalities
Two inequalities in one statement You can perform operations on a compound inequality as long as you remember to perform those operations on every term in the inequality, not just the outside terms.
Equivalent inequalities
Two inequalities that have the same solution set
Quadratic equation
When a quadratic equation has solutions, it can be found using the quadratic formula. Can also be solved using factoring. Have TWO solutions.
Percent increase
amount of increase/base (change/original) - base smaller number amount of decrease/base - base larger number change/base
Rules for adding/subtracting square roots
a√r + b√r = (a+b)√r a√r - b√r = (a-b)√r
Multiplying/Dividing square roots
a√r × b√s = (a×b)√rs √(a/b) = (√a)/(√b)
Another sequence problem
if Sn=3^n, what is the units digit of S65? 3^1=3 3^2=9 3^3=27 3^4=81 3^5=243 3^6=729 3^7=2187 3^8=6561 Repeating of 3, 9, 7, 1, 3, 9, 7, 1... Pattern repeats even 4 terms, so 64th term will be 4th term... 65th term = 3.
Rules of square roots
(√a)² = a √a² = a √a√b = √ab (√a)/(√b) = √(a/b) For odd order roots, there is exactly one root for every number n, even when n is negative. For every even order roots, there are exactly two roots for every positive number n and no roots for any negative number n.
Properties of evenly spaced sequences
1. Mean = Median 2. Mean and Median = Average of the 1st and last term. 3. Sum = Avg × N (# items in sequence) Counting integers Consecutive → (Last # - First #) + 1 Multiples → (Last # - First #) ÷ Increment + 1 Inclusive: include extremes Exclusive: exclude extremes
Standard deviation
1.Find the mean of n numbers 2. Calculate difference between each of n and mean 3. Square each of the differences 4. Take average of n squared differences 5. Take square root of avg calculated Normal distribution is a bell shaped curve
Benchmark values
1/10, 1/5, 1/4, 1/3, 1/2, 2/3, 3/4 Estimate computations using benchmark values; try to make your rounding errors partially cancel each other out by rounding some numbers up and some numbers down. e.g., Approximately what is 10/22 or 5/18 of 2,000? →10/22≈ little under half, so round up to ½ (11/20) →5/18≈ little over ¼, so round down to ¼ (5/20) (½)(¼)(2,000) = 250.
Divisibility terminology
12 is divisible by 3. 12 is a multiple of 3. 12/3 is an integer. 12=3n, where n is an integer 12 items can be shared among 3 people so that each person has the same number of items. 3 is a divisor of 12. 3 is a factor of 12. 12/3 yields a remainder of 0. 3 "goes into" 12 evenly.
Prime numbers under 30
2, 3, 5, 7, 11, 13, 17, 19, 23, 29 0 is not a prime number 1 is not a prime number 2 is the only even prime number Prime numbers are always positive integers All prime numbers are odd but 2; prime+prime=even, unless one prime is 2. So if sum of two primes is odd, one of the primes=2. If you know 2 cannot be one of the primes, then the sum=even.
Complex fractions
Division in disguise Multiply top and bottom fractions by common denominator, OR multiply numerator by reciprocal of denominator.
Integer rules
Even + Even = Even Odd + Odd = Even Even + Odd = Odd Even x Even = Even Even x Odd = Even Odd x Odd = Odd Even÷ Even = Even, Odd, Non-integer Even ÷ Odd = Even, Non-integer Odd ÷ Even = Non-integer Odd ÷ Odd = Odd, Non-integer *Plug in numbers if you forget rules*
Factors and Multiples
Fewer Factors, More Multiples
FOIL Method (Distribution)
First Outer Inner Last
Arithmetic with remainders
For a number to result in a certain remainder, it has to be equal to a multiple of x plus that remainder. e.g., Multiples of 5 with remainder of 4: 5x0+4=4, 5x1+4=9, 5x2+4=14...
Combining exponential terms
Multiplication: When multiplying exponential terms that share a common base, add the exponents. e.g., 2² × 2³ = 2⁵ Division: When dividing exponential terms with a common base, subtract the exponents. e.g., 2⁴/2² = 2² When something with an exponent is raised to another power, multiply the two exponents together. e.g., (2²)³ = 2⁶
Fraction problems with unspecified amounts
Pick real numbers to stand in for the variables; choose smart numbers equal to common multiples of the denominators of the fractions in the problem... BUT if you are given an actual number somewhere in the problem (no variable), then do not use smart numbers. Use numbers given.
Important values to remember
Powers of 2 = 2,4,8,16,32,64,128 Powers of 3 = 3,9,27,81 Squared and cubed values 4²=16 5²=25 6²=36 7²=49 8²=64 9²=81 10²=100 11²=121 12²=144 13²=169 14²=196 15²=225 2³=8 3³=27 4³=64 5³=125
Negative exponents
Put the term containing the exponent in the denominator of a fraction and make the exponent positive. When you see a negative exponent, think reciprocal!
Interquartile range
Q3-Q1 Shows where the middle half of the data lies
LCM and GCF
The product of the GCF and LCM of two numbers is equal to the product of the two numbers. [GCF(x & y)]×[LCM(x &y)] = xy e.g., 180 & 240 GCF= 12, LCM = 2160 12×2160 = 180×240 25,920 = 25,920
Product
The result of multiplication
Difference
The result of subtraction
Real numbers
The set of real numbers consists of all rational and irrational numbers. Includes all integers, fractions, and decimals. Can be represented by the real number line.
Even exponents
When asked about problems with even exponents, be cautious, as base could be either positive or negative. Positive number is called (principal) square root. e.g., If x²=16, does x=4? Not necessarily, because x can also equal -4.
Special products
x²-y²=(x+y)(x-y) x²+2xy+y²=(x+y)(x+y) x²-2xy+y²=(x-y)(x-y)
Triangle inequality
|a+b|≤|a|+|b|
Simplify roots through prime factorization
√12 = √(4×3) = √4×√3 =2√3
Rules involving inequalities
1. Adding/subtracting a number to an inequality preserves it. 2. Adding inequalities in the same direction preserves them. (if a<b and c<d, then a+c<b+d) 3. Multiplying/dividing an inequality by a positive number preserves it. (if a<b, and c is positive, then ac<bc and (a/c)<(b/c)) 4. Multiplying/dividing an inequality by a negative number reverses it. (if a<b, and c is negative, then ac>bc and (a/c)>(b/c) 5. Taking negatives reverses and inequality. (if a<b, then -a>-b and if a>b, then -a<-b) 6. If two numbers are each positive or negative, then taking reciprocals reverses an inequality. (if a and b are both (+) or (-) and a<b, then (1/a)>(1/b)).
Switching between improper fraction and mixed fraction
1. You can always split the numerator of a fraction into different parts and thus split a fraction into multiple fractions e.g., 5/4 → 4/4 + 1/4 2. Figure out the largest multiple of the denominator that is less than or equal to the numerator. In example above, 4 is the largest multiple of 4 that is less than 5; split up the fraction.
When to simplify exponential expressions
1. You can only simplify exponential expressions that are linked by multiplication or division. You cannot simplify expressions linked by adding or subtracting but in some cases you can factor them to manipulate them. 2. You can simplify exponential expressions linked by multiplication or division if they have either a base or an exponent in common. e.g., (7⁴)(7⁵) = 7⁴+⁵ = 7⁹ (3⁴)(12⁴) = (3×12)⁴ = 36⁴
Quartile
3 quartiles (Q1, Q2, Q3) which divide the data into 4 groups Same as P25, P50, P75 Median = Q2 = P50
Equation clean-up
Always get variables out of denominators Simplify grouped terms within the equation Combine like terms
Exponent of 1 or 0
An exponent of 1 → Any number that does not have an exponent implicitly has an exponent of 1. An exponent of 0 → Any nonzero base raised to the power of 0 is equal to 1.
Fibonacci sequence
An= An-1 + An-2... A1=1 A2=1 A3= 1+1 =2
Improper fraction
Any fraction in which the numerator is larger than the denominator (e.g., 5/4) or in which the numerator and denominator are the same (e.g., 8/8).
Sequences
Consecutive integers: Increments of 1. Consecutive multiples: multiples of the increment. Evenly spaced sequences: constant increments. All sequences of consecutive integers are sequences of consecutive multiples. All sequences of consecutive multiples are evenly spaced sequences. All evenly spaced sequences are fully defined if the following three parameters are known: 1. The smallest (first) or largest (last) number in the sequence. 2. The increment (always 1 for consecutive integers). 3. The number of items in the sequence.
Mixed Numbers
Consists of an integer and a fraction Convert to improper fraction or decimal when doing arithmetic with mixed fractions. To convert, multiply the integer by the denominator and add the numerator.
Factoring
Reversing the process of distribution FOIL ax²+bx+c=0 F=ax² O+I = bx L = multiply O and I = c
Rules
For any positive numbers a&b, a% increase of b = b% increase of a If a<b, the percent increase in going from a to b is always greater than the percent decrease in going from b to a. If a # is the result of increasing another # by k%, to find the original #, divide by (1+k%) If a # is the result of decreasing another # by k%, to find the original #, divide by (1-k%)
Rate of increase
For positive bases bigger than 1, the greater the exponent, the faster the rate of increase.
Intervals
Four types: y<x<z y≤x<z y<x≤z y≤x≤z Four types with one endpoint: x<4 x≤4 x>4 x≥4 The entire real number line is also considered to be an interval.
Quant Comp FROZEN
Fractions Repeats (in question stem) One Zero Extremes Negatives
Prime factorization: Factor foundation rule
If a is divisible by b, and b is divisible by c, then a is divisible by c. e.g., 12÷6 & 12÷3 → 12÷6 If d is divisible by two primes, e and f, then d is also divisible by e×f e.g., 20÷2 & 20÷5 → 20÷(2×5)
Increasing by x percent
If a quantity is increased by x percent, then the new quantity is (100+x)% of the original. Thus, a 15% increase produces a quantity that is 115% of the original.
Comparing fractions
If same denominator, compare numerators. If same numerator, fraction with smaller denominator is greater. To determine if two fractions are equivalent, cross multiply. The fractions are equivalent if the two products are equal.
Example of finding original number after percent increase
If the population of a town in 1990 was 3,000 and this represents an increase of 20% since 1980, to find the population in 1980, divide 3,000 by (1+20%) = 3,000/120 = 2,500.
Multiples/divisor rule
If you add or subtract multiples of N, the result is a multiple of N. -or- If N is a divisor of x and y, then N is a divisor of x+y. e.g., 21 and 35 are multiples of 7 (7 is a divisor of 21 and 35) → 21+35=56, also divisible by 7.
Integer divisibility rules
Integer is divisible by... 2 if integer is even. 3 if sum of integer's digits is divisible by 3. 4 if integer is divisible by 2 twice, or if two digits at end of integer are divisible by 4. 5 if integer ends in 0 or 5. 6 if integer is divisible by both 2 and 3. 8 if integer is divisible by 2 three times or if last 3 digits are divisible by 8. 9 if sum of integer's digits is divisible by 9. 10 if integer ends in 0.
Fraction operations (knowing what to expect)
Multiplying by a fraction between 0 and 1 creates a product SMALLER than the original number. Also true when the original form is a fraction. e.g., 8 × ¼ = 2, 2<8 ½ × ¼ = 1/8, 1/8<½ Dividing by a fraction between 0 and 1 yields a quotient, or result, LARGER than the original number. e.g., 6 ÷ ¾ = 8, 8>6 ½ ÷ ¼ = 2>½
Denominator rule
NEVER split the denominator!
Fractions
Number in the form a/b, where b≠0 Rational numbers
Terms
Numbers and expressions used in an equation
Integer
Numbers that have no fractional or decimal part Fractions are neither even, odd, nor an integer Each of the multiplied integers is a factor or divisor of the product
Digit
Numbers that make up other numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 (10 digits)
Order of operations (PEMDAS)
Parentheses Exponents Multiplication & Division (do all M & D together in same step, going from left to right) Addition & Subtraction (do all A & S together in same step, going from left to right)
Percentile
Percentiles divide a set of data into 100 roughly equal groups ex) 63rd percentile means that 63% if the data in the group is less than or equal to that number and the rest of the data is greater than that number.
Isolating a variable
Perform PEMDAS in reverse
PITA method
Plug In The Answer method Sometimes you can plug in values instead of setting up equations.
Arithmetic sequence
Sequence in which the difference between any two consecutive terms is the same. If A1, A2, A3 is an arithmetic sequence whose common difference is d, then An = A1 + (n-1)d e.g., If the 8th term of an arithmetic sequence is 10 and the 20th term is 58, what is the first term? A20= A1 + 19d = 58 A8 = A1 + 7d = 10 12d=48, d=4 10= A1 +7d = A1+28 = -18
Cross multiplication
Set up fractions next to each other, cross-multiply the fractions and put each answer by the corresponding NUMERATOR. (NOT the denominator) Cross multiplication can be used to compare fractions or solve equations. Can be used to compare multiple fractions.
Switching mixed numbers to improper fractions
Think of mixed numbers as whole number of 1, convert to common denominator, and add. e.g., 5¾ → 5/1 + ¾ → 20/4 + ¾ = 23/4.
Adding/subtracting fractions
To add or subtract two fractions with different denominators, first find a common denominator, which is a common multiple of the two denominators
Weighted average
To calculate the weighted average of a set of #s, multiply each number in the set by the # of times it appears, add all the products, and divide by the total number of #s in the set. e.g., On Thursday 20 of the 25 students took a test and their avg was 80. On Friday, the other 5 took the test and the avg was 90. What was the avg for the entire class? (20(80) + 5(90))/25 = 82
Solving algebraic equations involving exponential terms
Unknown base→x³=8; solve by taking cubed root of both sides Unknown exponent→ if you have the same base, you can set the exponents equal to each other to solve for x. When solving an equation with an even exponent, look for two solutions (positive and negative).
Sequences
When a sequence consists of a group of k terms that repeat in the same order indefinitely, to find the nth term, find the remainder (r), when (n) is divided by (k). The (r)th term and the nth term are equal. e.g.) what is the 1000th digit to the right of the decimal point in the expansion of 5/7? 5/7 = 0.714285714285... Since the sequence contains 6 digits, 1000/6 = 166.666...; the quotient is 166, 166x6=996, 1000-996=4. The remainder is 4. Therefore, the 1000th term is the same as the 4th term, which is 2.
Fractional base
When the base of an exponential expression is a positive proper fraction (between 0 and 1), as the exponent increases, the value of the expression decreases. Same goes for decimals between 0 and 1.
Compound base
When the base of an exponential expression is a product, you can multiply the base together and then raise it to the exponent or you can distribute the exponent to each number in the base. e.g., (2×5)³=(7)³=343 (5-2)³=3³=27
