GRE - Math
If there is a root in the denominator:
- Multiply the denominator and numerator by the root itself - adds the root to the numerator and gets rid of the root sign in the base WITH ADDITION/SUBTRACTION - Multiply the numerator and denominator by the expression but with an opposite sign - It squares the integer and gets rid of the root sign in the base in the denominator
How to set up an equation for arithmetic sequences:
1. Find the relationship between the numbers ex. +9 to each term 2. d = common difference An = nth term in sequence A1 = first term in sequence 3. use the format: An = A1 +/- (n-1)*d so in this case: An = 14 + (n-1)*9 4. If you're not given the specific numbers, find A1 first. Make the equation, and then solve for An
Sum of sequences
1. Find the terms included in the list 2. Use inclusive counting to find the number of variables in the list = N 3. Sum = N(A1 + An) / 2 A1 = first term to be included An = last term to be included
Mental Math: Squaring a number ending in 5
1. Take the first digit (n) and multiply it by n+1 2. Put that number infront of 25 to get answer Ex. 35^2 = 3 * 4 = 12 > 1225
To find the number of ODD factors for a positive integer:
1. prime factorization 2. list the exponents of the odd prime factors 3. add 1 to each number 4. multiply the new list together (even factors = total factors - odd factors)
1 foot equals
12 inches
5^3 5^4
125 625
2^4 2^5 2^6 2^7 2^8 2^9
16 32 64 128 256 512
Isosoles triangle
2 sides are the same length - an equilateral triangle can be isosceles because two or more angles are the same - every special fact about isosceles triangles is applicable to equilateral
Prime Numbers from 0 to 60
2, 3, 5, 7, 11, 13, 17, 19 23, 29, 31, 37, 41, 43, 47, 53, 59 - if a number is not divisible by a prime number less than 10, its a prime - Prime numbers MUST be positive
Work Equation
A = RT A = amount of work done
Divisibility Rules - 4
If the last two digits form a number divisible by 4
Divisibility Rules - 6
If the number is divisible by both 3 and 2
If x/y of a task can be completed in 1 min/hour/day, then:
It takes y/x min/hour/day to complete
For numbers less than 1, their squared value is:
Less than their original value so b > b^2 And their root value is larger than their original value: (square root) b < b
Divisibility Rule - 3 and 9
The sum of the digits is divisible by 3 The sum of the digits is divisible by 9
Why is finding the LCM important on the GRE?
The test may ask you directly or have you find the least common denominator for a fraction
Solve: | x - 1 | = 4
This means that x is 4 units away from 1 on the number line > So x = -3 or 5
If a score is in the pth percentile that means:
This score is larger than p% of the scores Ex. 95th percentile is better than 95% of the scores - Theres no such thing as 100th percentile
Number of Sides/Diagonals in Pentagons
Triangle = 3/0 Quadrilateral = 4 Pentagon = 5/2 Hexagon = 9
If it is asking you to compare two values:
You can: 1. add the same quantity to both 2. subtract the same quantity to both 3. multiply the same quantity to both > These will preserve the underlying relationships between the two quantities
When you need to find the square root of an exponent:
You divide it by 2 Ex. Square root of y^16 is y^8
Square root of Exponents
You divide the exponent by 2 Ex. Square root of y^16 is y^8
ab/cd =
a/c * b/d or a/d * b/c
a+b/c
a/c + b/c (same with subtraction) CAN ONLY BE DONE WITH SAME DENOMINATOR
a+b/c+d
a/c+d + b/c+d
(a + b)^n does NOT equal
a^n + b^n
Equation for concentration
amount of solute / total amount of solvent (x100%)
Difference of Two Squares
a² - b² = (a + b)(a - b) Ex. (25,002)^2 - (24,998)^2 They are actually (25,000 + 2) and (25,000 - 2) So we have (50,000)(4) *2a times 2b*
If a task can be completed in x amount of time, then:
convert the value to if it was in 1 min/hour/days so 1/x
Scale factors for Areas of similar figures
increases by k^2 - then multiply the smaller area by that scale factor to get larger area
Scale factors for Volumes of similar figures
increases by k^3 - then multiply that scale factor by the volume of smaller shape to get volume of larger shape
fraction * denominator =
numerator
Ratio of circumference to diameter
pi
The combined work rate of people/machines is:
the SUM of the individual work rates
Define absolute value
the distance a number is away from zero on a number line
Exponents: If a positive base is between 0 and 1,
the larger the exponent, the smaller the value will be
Outliers
High value outliers cause the mean to be higher than the median
Rules for Even and Odd Integers:
- Add/subract "likes" = EVEN Add/subtract "unlikes" = ODD - If there is an even in the multiplication, answer will always be even - Only if there are no evens in the multiplication will the answer be odd - Division cannot be predicted
How to tell if a question is inclusive counting
General subtraction excludes the lower endpoint so if your process is supposed to include both endpoints, you have to add 1
Unit digits questions (ex. What is the units digit of 57^123?)
- All that matters is the unit digit - Look for patterns of unit digits in squares Ex. 7^1 = 7 7^2 = __9 7^3 = __3 (from 9x7 = 6*3*) 7^4 = __1 (from 3x7 = 2*1*) 7^5 = __7 (from 1x7 = *7*) and so on the pattern continues every FOURTH number - So whenever we have a multiple of 4, the unit digit will be 1
Dividend vs Divisor
- Dividend is the number that is being divided - Divisor is the number we are dividing by
Simplifying square roots multiplied/divided by square roots:
- Factor the large numbers out and try to find perfect squares - Roots DO NOT distribute over addition + subtraction so (square root) (P+Q) does not equal (square root) P + (square root) Q
Percentiles
- Halfway between percentiles is not the same as halfway between scores > think about the normal distribution curve and if it's decreasing or increasing between those values
What numbers can you cancel in a proportion?
- Horizontally (numerators and denominators) - Vertically (corresponding numerator and denominator) NO CROSS CANCELLING
To find distance of x from origin numbers (using absolute value)
- If the origin number is positive, it will be the absolute value of x-n > and if the origin number is negative, it will be x+n
How to solve for average speed
Average velocity = total distance / total time
In most word problems, "of" means.... and "is" means...
"of" = multiply "is" = equals
Mental Math: Dividing by 5
(either one) 1. Double, then divide by 10 2. Divide by 10, then double
b^n/m is:
(m root) b^n - if its a high number, find the root first and then raise that product to the n value
Sum of angles in an n-sided polygon
(n-2)180 - divide the sum by n to get degrees of each angle
| x - 37 | = 25. This equation has two solutions, x1 and x2. If D = | x1 - x2 | what is the value of D?
*Don't need to find x2 and x1 The values of x2 and x1 are 25 units in opposite directions of 37. Either way, the distance between x2 and x1 is 50
Mental Math: Finding the next square
*If you know n2 but not (n+1)2* 1. Add n to n^2 2. Then add (n+1) to that value > You will get (n+1)^2
Rules of Exponents
*MUST HAVE SAME BASE* - If the exponents are powers = multiply them - If they are multiplied = add them - If they are divided = subtract them
Ex. How to find the sum of all to integers from 45 to 155 inclusive?
- Add them going forward and in reverse 45 + 46....154 + 155 155 + 154..46 + 45 (add those numbers then divide by 2) 200 + 200 + 200 + 200 - Next, find the number of integers from 45 to 155 inclusive (155-45) + 1 = 111 - So... (111)(200) = 11,100
Standard deviation
- Numbers below the mean are negative, ones above are positive - However, it is a distance so technically the numbers can only be positive or zero - Sets with numbers close to the mean will have a smaller standard deviation than sets with numbers spread out - If you multiply every number in the set by X, the standard deviation also gets multiplied by X - If all numbers on list are identical, SD = 0 - Large outliers creates a larger SD - If numbers added are farther from the mean than the SD, the SD gets larger (ex. 40 and 60 are added, mean is 50, and SD = 5. Mean doesn't change but SD increases bc they are 10 away from mean, not 5) > same goes with smaller (ex. if 47 and 53 are added, they would decrease the SD bc they are only 3 from the mean, not 5)
Raising racial expressions to higher powers
- Raise the exponent to both the integer in front (if there is one) and the radical
Mental Math: Squaring a number ending in 0
- Square the first number then add two 0's
When writing ages in past or future:
- Substitute the subtraction or addition of years with variable in parenthesis to equation Ex. In 8 years, Tom will be twice as old as Sam (T + 8) = 2 (S + 8)
Similar Figures
- have equal angles - can set up proportions with the side lengths
Quadrilateral (and Diagonal properties)
- sum of angles is 360 (bc contains two triangles) - has two diagonals - diagonals of parallelograms bisect each other - diagonals of rhombuses are perpendicular - diagonals of rectangles are congruent
Different denominators with same numerator
- the number with a larger denominator is SMALLER
Cubed roots
- x^3 = positive, has one positive solution x^3 = negative, has one negative solution - You can find the cube root of a negative number (can't do that for square roots) -
If b < 1 and n > m then:
0 < b < m (root) b < n (root) b < 1
Fraction to Decimal memorizations: 1/10 1/8 1/5 1/3 2/5
0.1 0.125 0.2 0.33 0.4
a^0 a^1
1 a
a^-n (a/b)^-n
1 / a^n (b/a)^n
Two Travelers Same Direction
1. (Faster traveler in front) Subtract the speeds to determine the speed the gap is expanding 2. (Slower traveler in front) Subtract the speeds to determine the speed the gap is shrinking
Two Travelers Opposite Direction (2 tips)
1. (towards each other) Add the speeds to determine the speed the gap is shrinking 2. (away from each other) Add the speeds to determine the speed the gap is expanding
Consecutive Integers Rules
1. A set of n consecutive integers will always have one number divisible by n 2. If n is odd, the sum of set of n consecutive integers will always be divisible by n.
5 Steps in calculating standard deviation: (don't need to know exactly how to do this but might ask about these steps)
1. Calculate the mean 2. Subtract the mean from every number on the list to get another list of deviations 3. Square every deviation to make a third list 4. Find the average of the third list (the squared deviations) to get the average deviation aka the variance 5. Take the square root of the variance to get the standard deviation
To find GCF for large numbers:
1. Do prime factorization 2. Pull out the common numbers with their powers 3. Multiply them together to get GCF ex. for GCF of 360 and 800 - 360 > [2^3, 3^2, 5] - 800 > [2^5, 5^2] They both have 2^3 and 5 in common so 8*5 = 40
To find LCM for large numbers
1. Do prime factorization and find the GCF 2. Multiply the GCF by another factor that would result in the number you're trying to solve for 3. Multiply the GCF by those factors to get LCM ex. 24 and 32 - 24 > [2^3, 3] - 32 > [2^5] so GCF = 8 - 24 = 8*3 and 32 = 8*4 so the LCM = 8*3*4 = 96
To find the number of factors a positive integer has:
1. Find the prime factorization and write it in terms of exponents ex. 2*2*5*3*5*7 > 2^2 * 3 * 5^2 * 7 2. Create a list of the exponents = [2,1,2,1] 3. Add 1 to every number = [3,2,3,2] 4. Multiply all those numbers together to get # of factors 3*2*3*2 = 36
6^3 7^3 8^3 9^3 10^3
216 343 512 729 1000
3^3 3^4
27 81
Diameter of a circle
2r
Circumference of a circle
2πr
Pythagorean Theorem triplets
3, 4, 5 5, 12, 13 8, 15, 17 7, 24, 25
4^3 4^4
64 256
Area of a trapezoid
A=1/2(b1+b2)h - find the average of the bases and multiply by height
Area of a quadrilateral
A=bh
Area of a circle
A=πr²
Define Integer
All positive and negative WHOLE numbers including zero.
Quick way to compare fractions
Cross multiply then insert a "greater than" or "less than" sign ex. 7/11 ?? 5/8 7*8 ?? 5*11 56 > 55 so 7/11 > 5/8
Equation for motion
D = RT
When looking for the median or another value of n in percentages:
Ex. what is the median number of x (with percentages listed) - add up the percentages till you get to the "range" where your value is (ex. the median or 50%) - the value you're looking for is in that range but you might have to find the average of two numbers in that range - ex. the median number of tvs a house has falls in the "3" category by adding up the percentages. If its out of 100 tvs for example, you have to do 3 + 3 / 2 = 3
Fraction Busting Technique
Multiply the fractions by the least common multiplier to get a whole number ex. 5/6 + 1/4 > multiply by 12 5/6 * 12/1 = 10 and 1/4 * 12/1 = 3 so its 10 + 3
Is it always true that the root of k^2 = k
No, it is true for positive numbers and zero. NOT negatives ex. k = -4 4 does not equal -4
Ex. The price of an item increased from $200 to $800. What was the percent increase? (trick question!)
Normally you would think 800/200 = 4 so a 400% increase BUT NO that is the multiplier. So 4 - 1 = 3 and it is a 300% increase
If the square root sign is printed on the test you should consider:
Positive roots ONLY - However, if your calculation leads you to finding a square root yourself, consider both positive and negative roots
Quartiles
Q1 = median of the lower list aka numbers below the mean Q2 = median of the upper list aka numbers above the mean - Median of the WHOLE set is excluded from the upper and lower sets
Exponents: Negative bases between -1 and 0, ex. -1/2
The denominator gets larger and the sign is based on if the exponent is even or odd (even = positive, odd = negative) (-1/2)^2 = 1/4 (-1/2)^3 = -1/8
If a number is half a standard deviation from the mean:
The distance of it from the mean is half the number of the SD. Ex. mean = 50, SD = 20 60 would be half a standard deviation from the mean bc it's 10 measurements away which is 20/2
For ANY arithmetic sequence:
The mean equals the median
For non arithmetic sequences:
The mean only equals the median when the sequence is evenly spaced and symmetrical
When dividing a number by a divisor...
The remainder is always less than the divisor
