GRE Math Foundations Flashcards
How to divide a decimal
- Move the decimal point in the divisor to the right to form a whole number
- Move the decimal point in the dividend the same number of places.
- Divide as though there were no decimals, then place the decimal point in the quotient
Example: 6.25 / 2.5 = 62.5 / 25 = 2.5
Percent Formula
Part = % x Whole
Example: What is 12% of 25?
* (12/100) x 25 = 300/100 = 3
Percent Increase / Decrease Formulas
% increase = (Amount of increase / Original) whole x 100%
Example: The price goes up from $80 to $100. What’s the % increase?
* 20/80 x 100% = 0.25 x 100% = 25%
% decrease = (Amount of decrease / Original) whole x 100%
How to recognize multiples of 2
Last digit is even
How to recognize multiples of 3
Sum of digits is a multiple of 3
How to recognize multiples of 4
Last two digits are a multiple of 4
How to recognize multiples of 5
Last digit is 5 or 0
How to recognize multiples of 6
Sum of digits is a multiple of 3, and last digit is even.
How to recognize multiples of 9
Sum of digits is a multiple of 9
How to recognize multiples of 10
Last digit is 0
How to recognize multiples of 12
Sum of digits is a multiple of 3, and last two digits are a multiple of 4
How to find a COMMON FACTOR of two numbers
Break both numbers down to their prime factors to see which they have in common. Then multiply the shared prime factors to find all common factors.
Example: What factors greater that 1 do 135 and 225 in common?
1.135 = 3 x 3 x 3 x 5
2. 225 = 3 x 3 x 3 x 5
3. The numbers share 3 x 3 x 5
4. Aside from 3 and 5, the remaing commong factors can be found by multplying 3,3, and 5 in every possible combination. 3x3=9, 3x5=15, and 3x3x5=45
5. Therefore, the common factors are 3, 5, 9, 15, & 45
How to find LCM (Least Common Multiple)
Finding the prime factorization of each number, the seeing the greatest number of times each factor is used. Multiply each factor the greatest number of times it appears.
Example: What is the LCM of 28 and 42?
* 28 = 2 x 2 x 7
* 42 = 2 x 3 x 7
* LCM = 2 x 2 x 3 x 7 = 84
Average of consecutive numbers
The average of evenly spaced numbers is the average of the smallest and largest number.
Example: The average of all the integers from 13 to 77
* (13+77)/2 = 90/2 = 45
How to count Consecutive Numbers
The number of integers from A to B inclusive is B - A + 1
Example: How many integers are there from 73 through 419, inclusive?
* 419 - 73 + 1 = 347
Sum of Consecutive Numbers
Sum = (Average) x (Number of terms)
Example: What is the sum of the integers from 10 throught 50, inclusive?
* Average: (10 + 50) / 2 = 30
* Number of terms: 50 - 10 + 1 = 41
* Sum: 30 x 41 = 1,230
How to count the Number of Possibilities
You can use multiplication to find the number of possibilities when items can be arranged in various ways.
Example: How many 3 digit numbers can be formed with the digits 1, 3, and 5 each used only once?
* *_ _ _ > List the quantitiy of number possibilities for each digit. For the 1st digit, there are 3 options, for the 2nd, there are 2, and for the 3rd, there is 1. *
* Multiply the possibilities: 3 x 2 x 1 = 6
How to calculate a simple Probability
Probability = Number of desired outcomes / Number of total possible outcomes
Example: What is the probability of throwing a 5 on a fair six-sided die?
* There is one desired outcome, 5. There are 6 possibile outcomes
* Probability = 1/6
How to FACTOR certain POLYNOMIALS:
ab + ac
a(b + c)
How to FACTOR certain POLYNOMIALS:
a^2 + 2ab + b^2
(a + b)^2
How to FACTOR certain POLYNOMIALS:
a^2 - 2ab + b^2
(a - b)^2
How to FACTOR certain POLYNOMIALS:
a^2 - b^2
(a - b)(a + b)
When dividing an inequality equation with a “-“ number
Divide both side by the “-“ number and reverse the inequality sign
How to handle Absolute Values
The absolute value of a number n, denoted by |n|, is defined as n, if n >/= 0 and -n, if n < 0. The absolute value of a number is the distance from zero the number on the number line, and is always positive. i.e.
Example1: |-5| = 5
Example 2: If |x - 3| < 2, what is the range of possible values for x?
* Represent the range, so (x - 3) < 2 and (x - 3) > -2
* x - 3 < 2 = x < 5
* x - 3 > -2 = x > 1
* So, 1 < x < 5
Geometry Pages
105, 110-113, 133-140
Similar Triangles
In similar triangles, corresponding angles are the equal and corresponding sides are proportional.
Special Right Triangles
3:4:5
5:12:13
30°:60°:90° ; sides are multiples of 1, √3, and 2 respectively
45°:45°:90° ; sides are multiples of 1, 1, and √2 respectively
Simple Interest Equation
Interest = Principle x rt
Where r is defined as the interest rate per payment period and t is defined as the number of payment periods
Example: If 12,000 is invested at 6% simple annual interest, how much interest is earned after 9 months?
* Since the interest rate is annual and we are caultating how much interest accrues after 9 months, we will expess the payment period as 9/12
* (12,000) x (0.06) x 9/12 = 540
Compound Interest Equation
(Final Balance) = (Principal) x (1 + interest rate/c)^(time)(c)
Where c = the number of times the interest is compounded annually
Example: If 10,000 is invested at 8% annual interest, compounded semiannually, what is the balance after 1 year?
* Final Balance = (10,000) x (1+ 0.08/2)^(1)(2)
* (10,000) x 1.04^2
* = 10,816
Ones I skipped
60 - 64
Average Rate (Speed)
Rate = Distance / Time
Combined Work Problem
The work formula states: The inverse of time it would take everyone working together equals the sum of the inverses of the times it would take each working individually. In other words:
1/r + 1/s = 1/t
Where “r” and “s” are the separate times and “t” is the time it would take the individuals to work together
Example:If it takes Joe 4 hrs to paint a room and Pete twice as long to paint the same room, how long would it take the two of them, working together, to paint the same room, if each of them works at his respective individual rate?
* 1/4 + 1/8 = 1/t
* 2/8 + 1/8 = 1/t
* 3/8 = 1/t
* t = 1/(3/8) = 8/3
* It would take 8/3 hrs, or 2 hours and 40 min
Combined Ratio Problem
Multiply one or both ratios by whatever you need in order to get the terms they have in common to match.
Example: The ratio of a to b is 7:3 The ratio of b is 2:5. What is the ratio of a to c?
* Multiply each value of a:b by 2 and each value of b:c by 3
* You get 14:6 and 6:15
* Thus a:c = 14:15
Dilution / Mixture Problem
There are 2 methods: straightforward set-up, and the balancing method.
- Straightforward set-up example: If 5 lbs of raisins that cost $1 per pound are mixed with 2 lbs of almonds that cost $2.40 per pound, what is the cost per pound of the resulting mixture?
* ($1)(5) + ($2.4)(2) = $9.8 = total cost for 7 lbs of the mixture
* Cost per lb = $9.8/7 = $1.40 - Balancing method example: How many liters of a solution that is 10% alcohol by volume must be added to 2 L of a solution that is 50% alchol by volume to create a solution that is 15% alcohol by volume?
* The balancing method example: Make the weaker and stronger (or cheaper and more expensive, etc.) substances balance. That is, (% diff. between the weaker solution and the desired solution) x (amount of weaker solution) = (%difference between stronger and the desired solution) x (amount of stronger solution). Make n the amount, in liters of the weaker solution.
* n(15-10) = 2(50-15)
* 5n = 2(35)
* n = 70/5 = 14
Overlapping Sets
Group 1 + Group 2 + Neither - Both = Total
Example: Of the 120 students at a certain language school, 65 are studying French, 51 are studying Spanish, and 53 are studying neither language. How many are studying both French and Spanish?
* 65 + 51 + 53 - Both = 120
* 169 - Both = 120
* Both = 49
Permutation Formula
nPk = n! / (n-k)!
“the number of ways to arrange elements sequentially”
n = # in the larger group
k = # you’re arranging
Combination Formula
nCk = n! / k!(n-k)!
Use when the arrangement order doesn’t matter.
Probability
# of desired outcomes / # of total possible outcomes
Standard Deviation
- Find the ave of the set
- Find diff between mean and each value in set
- Square each difference
- Find the average of squared differences
- Take the positive square root of the average
Note: the greater the spread, the higher the standard deviation
Exponent Properties
n^a x n^b
n^(a + b)
Exponent Properties
(n^a)^b
n^ab
Exponent Properties
n^-a
a/n
Exponent Properties
n^(a/b) or n^(1/2)
√n
