Reference Flashcards
percent increase formula
Percent increase = amount of increase / original whole * 100%
Price goes from $80 to $100: 20/80*100% = 25%
percent formula
Part = Percent * Whole
What is 12% of 25? Part
45 is what percent of 9? Percent
15 is 3/5% of what number? Whole
percent decrease formula
Percent decrease = amount of decrease / original whole * 100%
Price goes from $100 to $80: 20/80*100% = 25%
How to predict whether a sum, difference or product will be ODD or EVEN
Pick numbers (2 for even, 3 for odd) ODD + EVEN = ODD ODD * ODD = ODD All others 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 6
Sum of digits is a multiple of 3, and the last digit is even
How to recognize multiples of 9
Sum of digits is a multiple of 9
How to recognize multiples of 12
Sum of digits is a multiple of 3, and last two digits is a multiple of 4.
How to find a COMMON FACTOR of two numbers
Break both numbers down to their prime factors to see which ones they have in common. Then multiply the shared prime factors to find all common factors.
135 = 3335
225 = 3355
Common factors are: 3,5,33=9, 35=15, 335=45
How to find a common multiple of two numbers
The product of two numbers is the easiest common multiple.
The LCM can be found by finding the prime factors of each number, then seeing the greatest number of times each factor is used. Multiply each prime factor the greatest number of times it appears.
28 = 227
42 = 237
LCM = 223*7 = 84
How to find the AVERAGE of CONSECUTIVE numbers
The average of evenly spaced consecutive numbers is simply the average of the smallest number and the largest number.
The average of all integers from 13 to 77 is:
(13+77)/2 = 90/2 = 45
How to COUNT CONSECUTIVE numbers
The number of integers from A to B inclusive is B - A + 1
How to find the SUM of CONSECUTIVE numbers
Sum = Average * Number of terms
What is the sum of the integers from 10 through 50, inclusive?
Average = (10 + 50) / 2 = 30
Number of terms = 50 - 10 + 1 = 41
Sum = 30 * 41 = 1,230
How to use a ratio to determine an ACTUAL number.
The ratio of boys to girls is 3 to 4. If there are 135 boys, how many girls are there?
Set up a proportion using the given ratio.
3/4 = 135/g 3g = 540 g = 180
How to use actual numbers to determine a RATE.
Anders typed 9,450 words in 3.5 hours. What was his rate in words per minute.
Identify the quantiles and the units to be compared. Keep the units straight.
9,450 words / 210 minutes = 45 words per minute.
How to solve an inequality
Same as equations, just remember to reverse the inequality sign if you multiply or divide by a negative number.
How to spot SPECIAL RIGHT triangles
3:4:5
5:12:13
30-60-90: 1:sqrt(3):2
45-45-90: 1:1:sqrt(2)
How to find the AREA of a PARALLELOGRAM
Area = (base)(height)
How to find the AREA of a TRAPEZOID
A trapezoid is a quadrilateral having only two parallel sides. You can always drop a perpendicular line or two to break the figure into a rectangle and a triangle or two triangles and combine the areas.
Alternatively:
Area = (average of parallel sides) * (height)
How to find the DISTANCE BETWEEN POINTS on the coordinate plane
If two points have the same x coordinates or the same y coordinates, just subtract the numbers that are different.
Otherwise, make a right triangle and use the Pythagorean theorem or apply special right triangle attributes if applicable.
How to find the SLOPE of a LINE
Slope = rise/run = (y2-y1)/(x2-x1)
How to solve a SIMPLE INTEREST problem
interest = principal * rate * time balance = P(1 + rt)
How to solve a COMPOUND INTEREST problem
balance = P * (1 + r/C)^(tC)
How to solve a REMAINDER problem
When n is divided by 7, the remainder is 5. What is the remainder when 2n is divided by 7?
Pick a number by taking a multiple of 7 and adding 5 to it.
n=12, 2n=24 which leaves a remainder of 3.
How to find the NEW AVERAGE when a number is added or deleted
Use the sum of the terms of the old average to help you find the new average.
How to use the ORIGINAL AVERAGE and NEW AVERAGE to figure out WHAT WAS ADDED or DELETED
Number added = (new sum) - (original sum)
Number deleted = (original sum) - (new sum)
How to find an AVERAGE RATE
Average A per B = Total A / Total B
Average speed = Total distance / Time
How to solve a COMBINED WORK problem
The inverse of the time it would take everyone working together equals the sum of the inverses of the times it would take each working individually.
1/r + 1/s = 1/t
How to determine a COMBINED ratio
a:b = 7:3, b:c = 2:5, what is a:c?
Multiply one or both ratios by whatever you need to get the terms they have in common to match.
a: b = 7:3 = 14:6
b: c = 2:5 = 6:15
a: c = 14:15
How to solve a DILUTION or MIXTURE problem
Straightforward setup: If 5 pounds of raising that cost $1 per pound are mixed with 2 pounds of almonds that cost $2.40 per pound, what is the cost per pound of the resulting mixture?
$1*5 + $2.40 * 2 = $9.80 per 7 pounds
$9.80/7 = $1.40 per pound
How to solve a DILUTION or MIXTURE problem
Balancing method: How many liters of a solution that is 10% alcohol must be added to 2 liters of a solution that is 50% alcohol to create a solution that is 15% alcohol?
Make the weaker and stronger substances balance.
(percent difference between WEAKER and DESIRED) * (amount of WEAKER) = (percent difference between STRONGER and DESIRED) * (amount STRONGER)
n(15 - 10) = 2(50 - 15)
n = 70/5 = 14
Add 14 liters of the 10% solution.
How to solve a GROUP problem involving BOTH/NEITHER
Total = Group 1 + Group 2 + Neither - Both
How to solve a GROUP problem involving EITHER/OR categories
The key to solving this type of problem is to organize the information in a grid.
E.g.
Doctors Dentists Total
Male
Female
Total
How to solve a PERMUTATION problem
How many ways are there to arrange n items?
n!
How to solve a PERMUTATION problem
Find the number of ways to arrange k items that are being drawn from a larger group n.
P(n,k) = n! / (n-k)!
How to solve a COMBINATION problem
C(n,k) = n! / k!(n-k)!
Picking numbers
Problems that seem difficult are good candidates for Picking Numbers. They include problems where either the question or the answer choices have variables, the problem tests a number property you don’t recall, or the problem and the answer choices deal with percents or fractions without using actual values.
Backsolving
Numerical choices are always in ascending or descending order. Start with either (B) or (D) first.
If you chose (B) but it was too large, the correct answer was A. If it was too small, try (D). If that was too large, the correct answer is C, and if it was too small then (E) is the answer.
Strategic guessing
This is a good strategy if you can eliminate choices by applying number-property rules or by estimating because gaps between answer choices are wide.
How to deal with STANDARD DEVIATION
You can often handle the questions using estimation, but formally:
- Find the average of the set
- Find the difference between the mean and each value in the set
- Square each of the differences
- Find the average of the squared differences
- Take the positive square root of the average
x^a * x^b
x^(a + b)
x^a / x^b
x^(a - b)
(x^a)^b
x^(ab)
0^x, for any x
0, also for 0^1
x^1, x != 0
1
sqrt(x^(-2))
(x^(-2))^(1/2) = x^(-1) = 1/x
How to add, subtract, multiply and divide ROOTS
You can add/subtract roots only when the parts inside the root are identical.
To multiply/divide roots, deal with what’s inside the root and outside the root separately.
How to solve MULTIPLE equations
When you see two equations of the GRE, they’re probably easy to combine in such a way that you get something closer to what you’re looking for.
How to handle LINEAR equations
y = mx + b
m is the slope, b is the y-intercept
the x intercept is when mx + b = 0
How to find one angle or the sum of all angles of a REGULAR POLYGON
Regular means that all angles in the polygon are of equal measure.
Sum of interior angles = (n-2)*180
Degree measure = sum/n
How to find the LENGTH of an ARC
n/360 * 2piradius
How to find the AREA of a SECTOR
n/360 * pi*radius^2
VOLUME of a RECTANGULAR SOLID
Volume = length * width * height
SURFACE AREA of a RECTANGULAR SOLID
Surface area = 2lw + 2lh + 2wh
How to find the DIAGONAL of a RECTANGULAR solid
Use Pythagoras or special triangle rules.
VOLUME of a CYLINDER
Volume = area of base * heigh = pir^2h
SURFACE AREA of a CYLINDER
Surface area = 2pir^2 + 2pir*h
SAMPLE standard deviation
Used when a sample of data is taken from a large population. Exactly like regular standard deviation, except that after summing up the squares, divide by n-1 instead of n (and then take square root like normally).
Given the mean and standard deviation, how many standard deviations is a number n from the mean?
mean - n / (standard deviation) = number of deviations above or below the mean.
Doing this for each value in the set is called standardization and in ANY group, most of the data will be within 3 standard deviations from the mean (above or below).
