Math GRE Review
What is a Prime Number?
an integer greater than 1 that has no factors other than 1 and itself (0 and 1 are not prime #s)
What is Standard Deviation?
measure of how spread out a group of numbers is. The more spread out the numbers, the larger the SD.
Important Binomial Products
(a+b)(a-b) = a²-b² (a+b)² = a²+2ab+b² (a-b)² = a²-2ab+b²
.029 x 10³
(move decimal 3 places to the right) = 29
416.03 x 10⁻⁴
(move decimal 4 places to the left) = .041603
6.25/2.5 =
(move decimals) 62.5/25 = 2.5
(x/y)√(y/x²) =
**Square the entire expression to get rid of the radical! = (x²/y²) (y/x²) = 1/y² Turn it back into a radical, √(1/y²) = *1/√y*
Find the fraction: 80%
4/5
What are multiples?
An integer that is divisible by another integer is a multiple of that integer (ex: 12 is a multiple of 3, since 12 is divisible by 3)
How do you find the average (mean) of a group of numbers?
Average = sum of values/number of values
Percent Problems Equation
Percent x Whole = Part
What are factors?
The factors (divisors) of a number are the positive integers that evenly divide into that number (ex: 36 has nine factors: 1,2,3,4,6,9,12,18,36)
What is the smallest prime #?
The number 2 is the smallest prime and the only even prime number
even# + odd# =
odd#
odd# x odd# =
odd#
If 2 students are chosen at random from a class with 5 girls and 5 boys, what's the probability that both students chosen will be girls?
*Probability that the 1st student chosen will be a girl = 5/10 *Probability that the 2nd student chosen will be a girl = 4/9 *Probability that both students chosen will be girls = 1/2 x 4/9 = 2/9
Probability Equation
# of desired outcomes/# of total possible outcomes
Percent Increase/Decrease Equations
% increase/decrease = (change/original amount) x 100%
If a fair coin is tossed 4 times, what's the probability that at least 3 of the 4 tosses will come up heads?
*2 possible outcomes/toss x 4 tosses = 16 possible outcomes *5 possible desired outcomes Probability = 5/16
Anything raised to the zero power = ?
1 Ex: 4⁰ = 1
The price of an antique is reduced by 20% and then this price is reduced by 10%. If the antique originally cost $200, what is its final price?
1) 20% = X/$200, X = $40 2) $200-$40=$160 3) 10% = X/$160, X=$16 4) $160-$16 = **$144**
Properties of 1
1) ax1 = a 2) a/1 = a 3) 1ⁿ = 1 4) is a factor of every integer 5) is the smallest positive odd integer 6) is NOT a prime #
Properties of 0
1) only # that is neither +/- 2) a multiple of every integer 3) is an even # 4) a/0 = undefined 5. 0/a = 0
Multiple Event Probability where each individual event must occur a certain way (2 independent events happening simultaneously)
1. Figure out the probability for each individual event 2. Multiply the individual probabilities together
Multiple Event Probability where individual events can have different types of outcomes
1. Find total number of possible outcomes (find total for each individual event and multiply those together) 2. Find number of desired outcomes
√2
1.4
√3
1.7
Find the fraction: 33¹/₃%
1/3
Find the fraction: 16²/₃%
1/6
Find the fraction: 12.5%
1/8
Powers of 10
10
2¹⁰
1024
Powers of 10
10⁶ = 1,000,000
Find the percent: 1/8
12.5%
11²
121
5³
125
13 is 33¹/₃% of what number?
13 = 1/3X X = 39
45.1 x 0.3 =
13.53
12²
144
2⁴
16
13²
169
Find the percent: 1/6
16²/₃%
14²
196
√5
2.2
Find the fraction: 66²/₃%
2/3
Find the fraction: 40%
2/5
Find the percent: 1/5
20%
15²
225
16²
256
3³
27
Find the fraction: 30%
3/10
Find the fraction: 60%
3/5
Find the fraction: 37.5%
3/8
2⁵
32
Find the percent: 1/3
33¹/₃%
20²
400
Find the percent: 1/20
5%
Find the fraction: 83¹/₃%
5/6
Find the fraction: 62.5%
5/8
25²
625
2⁶
64
4³
64
Find the fraction: 87.5%
7/8
2³
8
3⁴
81
What is 25% of 36?
= .25 x 36 = 9
John travels 30 miles in 2 hrs and then 60 miles in 3 hrs. What is his average speed in miles/hr?
Average A per B = Total A/Total B = (30+60)miles/(2+3)hrs = 18mph
If "a" is inversely proportional to "b" and b=5 when a=3, what is the value of "b" when a=10?
By definition, "a" is inversely proportional to "b" if the product "ab" is a constant. 1) 3x5 = 15, the constant is 15 2) 10 x b =10b 3) 10b=15 **b=³/₅**
What does it mean if 7 is directly proportional to x?
By definition, y is directly proportional to x if the quotient y/x is a constant.
If a class of 10 boys & 15 girls took a test & the average grade of the boys was 80, and for the girls was 90, what was the class average?
Calculate the weighted averages! (10x80) + (15x90) / (10+15) = 86 *b/c the class is 40% boys & 60% girls, calculate the average as: (.4x80)+(.6x90)= 86
Simplify a Radical
Look for perfect squares (4,9,16,25,36...) inside the √. Factor them out and "unsquare" them. ex. √48= √16 x √3 = 4√3 ex2. √72= √(36x2) = √36 x √2 = 6√2
How do you change a decimal to a percent?
Multiply by 100 (move decimal 2 places to the right) ex: 0.17 = 17%
Multiplying Decimals
Multiply the #s and then move the decimal point the total # of places in the answer
In a classroom of 30 students, the ratio of the boys in the class to students in the class is 2:5. How many boys?
Multiply the ratio by the total = 2 boys/5 students x 30 students = 12 boys
Prime Numbers
Numbers that can only divide evenly by 1 or themselves
Dividing Fractions
Invert the 2nd fraction and multiply
What is the Absolute Value of a number?
It is the number of units a number is away from 0 on a number line.
A building has 2/5 of its floors below ground. What is the ratio of the # of floors above ground to the # below ground?
Pick Numbers! Pick a value for the total # of floors, one that is divisible by 2 and 5. Let's pick 10. 2/5 x 10 = 4 floors below ground & 6 above. Therefore, the ratio is 6:4 or 3:2
IF you have 12 shirts in a drawer and 9 of them are white, what is the probability of picking a white shirt at random?
Probability = 9 white shirts/12 shirts total = 3/4
How do you convert a percent to a fraction?
Put the given percent over 100, remove the percent symbol (%) and simplify if needed.
Compare 5/7 and 9/11
Quick Tip: Cross-multiply! 5x11>9x7
How do you change a percent to a decimal?
Remove the percent symbol and divide by 100 (move decimal 2 places to the left) ex: 0.8% = .008
How to find the sum of a group of numbers
Sum of values = average value x number of values
How to find the average of consecutive #s
Take the average of the largest and smallest # in the set. (i.e. for the set of consecutive #s 5....172 would be (172+5)/(2)= 88.5)
What is the Greatest Common Factor (GCF) of a pair of numbers?
The largest number that will divide evenly into both (factor) 1,2,4,8 = factors of 8. 1,2,3,6 = factors of 6. GCF = 2
How do you count consecutive integers? ex: How many integers are there from 73 through 419, inclusive?
The number of integers from A to B inclusive is B-A+1.
Multiply & Divide Roots
To multiply/divide roots deal with what's inside the √ and outside the √ separately. (2√3) (7√5) = (2x7) (√3x√5) = 14√15 10√21 ÷ 5√3 = (10/5) (√21/√3) = 2√7
Integer
Whole numbers and their opposits. {...-3,-2,-1,0,1,2,3...} These are rational numbers. Negative, positive, include 0 (neither negative or positive)
Add & Subtract Roots
You can add/subtract roots only when the parts inside the radical are identical. √2 + 3√2 = 4√2 √2+√3 = cannot be combined! √2-3√2 = 2√2
A negative # raised to an odd power
a negative number Ex: (-3)³ = -27
A negative # raised to an even power
a positive number Ex: (-3)² = 9
Rules for Powers
a³x a⁴ = a³⁺⁴ a³/a⁴ = a³⁻⁴ (a³)⁴= a³°⁴ (a/b)³ = a³/b³ (ab)³ = a³b³ a⁻³ = 1/a³ 0ⁿ = 0
even# + even# =
even#
even# x even# =
even#
even# x odd# =
even#
odd# + odd# =
even#
Dividing Decimals
move decimals point in the divisor (outside #) until the # is a whole number, move the decimals point the same # of places in the dividend (inside #)
For 0<x<1, Which is larger; ³√x OR √x?
³√x , X will always be smaller than it's square root, and the cube root will be less than the square root ex: √¼ = ½
(¹/₁₆)² =
¹/₂₅₆, A decimal or fraction less than 1, raised to a power will yield a # smaller than the original
Rules for Roots
√(a/b) = (√a)/(√b) √(ab) = √a x √b (a)^(¹/n) = ⁿ√a a^(m/n) = ⁿ√a^m