Math Reference
How to use a ratio to determine an ACTUAL NUMBER
Set up a proportion using the given ratio. The ratio of boys to girls is 3 to 4. If there are 135 boys, how many girls are there? 3/4 = 135/g 3 x g = 4 x 135 3g = 540 g = 180
How to find the SLOPE of a LINE
Slope = Rise/Run = Change in y/Change in x What is the slope of the line that contains the points (1, 2) and (4, -5)? Slope = -5 - 2/ 4-1 = -7/3 = -7/3
How to solve PROBABILITY problems where probabilities must be multiplied
(1) The case of the probability that two events occur: The probability that both event A and event B occur is the probability that even A occurs multiplied by the probability that event B occurs, given that event A occurred [conditional probability] Except when events A and B do not depend on one another, the probability that B occurs given that A occurs is not the same as the probability that B occurs. If 2 students are chosen at random to run an errand from a class with 5 girls and 5 boys, what is the probability that both students chosen will be girls? The probability that the first student chosen will be a girl is 5/10. The probability that the second will be a girl, given that the first is a girl, is 4/9. 1/2 x 4/9 = 2/9. (2) The probability that three events occur: The probability that events A, B, and C occur is the probability that A occurs multiplied by the conditional probability that B occurs given that A occurred multiplied by the conditional probability that C occurs given that both A and B have occurred.
How to find the AREA of a SECTOR
A sector is a fraction of the circle's area. Set up the interior angle measure as a fraction of 360, which is the degree measure of a circle around the central point. Area of sector = n / 360 x [pi]r^2
How to find the AREA of a TRIANGLE
Area = 1/2 (base) (height)
How to find the AREA of a CIRCLE
Area = [pi]r^2
How to recognize MULTIPLES of 2, 3, 4, 5, 6, 9, 10, and 12
2: Last digit is even 3: Sum of digits is a multiple of 3 4: Last two digits are a multiple of 4 5: Last digit is 5 or 0 6: Sum of digits is a multiple of 4, and last digit is even 9: Sum of digits is a multiple of 9 10: Last digit is 0 12: Sum of digits is a multiple of 3, and last two digits are a multiple of 4
How to spot SPECIAL RIGHT TRIANGLES
3:4:5 5:12:13 30-60-90 where the side lengths are multiples of 1, square root of 3, and 2, respectively 45-45-90 where the side lengths are multiples of 1, 1, and square root of 2, respectively
How to SIMPLIFY BINOMIALS
A binomial is a sum or difference of two terms. Use the FOIL method to simplify two binomials that are multiplied together. Multiply the First terms, the Outer terms, the Inner terms, and the Last terms; combine like terms. (3x + 5)(x - 1) = 3x^2 - 3x + 5x -5 = 3x^2 + 2x - 5
How to handle LINEAR EQUATIONS
A linear equation is expressed in the form y = mx + b, where m = the slope of the line (rise/run) b = the y-intercept (the point where the line crosses the y-axis)
How to handle NEGATIVE POWERS
A number raised to the exponent -x is the reciprocal of that number raised to the exponent x. n^-1 = 1/n n^-2 = 1/n^2 5^-3 = 1/5^3 = 1/ 5 x 5 x 5 = 1/125
How to FACTOR certain POLYNOMIALS
A polynomial is an expression consisting of the sum of two or more terms, where at least one of the terms is a variable. Classic polynomial equations: ab + ac = a (b + c) a^2 + 2ab + b^2 = (a + b^2) a^2 - 2ab + b^2 = (a - b^2) a^2 - b^2 = (a - b) (a + b)
How to solve an INEQUALITY
Add, subtract, multiply, and divide both sides by the same thing; reverse the inequality sign if you multiply or divide by a negative quantity. Rewrite 7 - 3x > 2 in its simplest form 7 - 3x - 7 > 2 - 7 -3x > -5 x < 5/3
How to MULTIPLY/DIVIDE VALUES WITH EXPONENTS
Add/subtract the exponents. n^a x n^b = n^a+b 2^3 x 2^4 = 2^(3+4) = 2^7 n^a / n^b = n^a-b 2^8 / 2^2 = 2^8-2 = 2^6
How to solve a FUNCTION problem
An algebraic expression of only one variable may be defined as a function, usually symbolized by f or g, of that variable. What is the minimum value of x is the function f(x) = x^2 - 1? If x is 1, then f(1) = 1^2 - 1 = 0. To find the minimum value, since x^2 cannot be negative, input 0: x = 0: f(0) = 0^2 - 1 = -1.
How to find the LENGTH of an ARC
An arc is a fraction of the circle's circumference. Use the measure of an interior angle of a circle, which has 360 degrees around the central point, to determine the length of an arc. Length of arc = n / 360 x 2[pi]r
How to find the AREA of a TRAPEZOID
Area = (Average of parallel sides) x (Height)
How to find the AREA of a PARALLELOGRAM
Area = (Base)(Height)
How to find the AREA of a SQUARE
Area = (Side)^2
How to find a COMMON FACTOR of two numbers
Break both numbers down to their prime factors to see which they have in common; multiply the shared prime factors to find all common factors. What factors greater than 1 do 135 and 225 have in common? Prime factors of 135 = 3 x 3 x 3 x 5 Prime factors of 225 = 3 x 3 x 5 x 5 The numbers share 3x3x5 in common The remaining common factors can be found by multiplying 3, 3, and 5 in every possible combination: 3 x 3 = 9, 3 x 5 = 15, and 3 x 3 x 5 = 45 The common factors of 135 and 225 are 3, 5, 9, 15, and 45
How to find the CIRCUMFERENCE of a CIRCLE
Circumference = 2[pi]r Circumference = [pi]d
How to use PEMDAS (Order of Operations)
Clean up Parentheses first; nested sets of parentheses are worked from the innermost set to the outermost set. Deal with Exponents or Radicals. Do the Multiplication and Division together, going from left to right. Do the Addition and Subtraction together, going from left to right.
How to find an AVERAGE RATE
Convert to totals. Average A per B = Total A/Total B If the first 500 pages have an average of 150 words per page, and the remaining 100 pages have an average of 450 words per page, what is the average number of words per page for the entire 600 pages? Total pages = 500 + 100 = 600 Total words = (500 x 150) + (100 x 450) = 120,000 Average words per page = 120,000/600 = 200 To find an average speed, you also convert to totals. Average speed = Total distance/Time Rosa drove 120 miles one way at an average speed of 40 miles per hour and returned by the same 120-mile route at an average speed of 60 miles per hour. What was Rosa's average speed for the entire 240-mile round trip? To drive 120 miles at 40 mph takes 3 hours; to return at 60 mph takes 2 hours; the total time is 5 hours: Average speed = 240 miles/5 hours = 48 mph
How to work with EQUILATERAL TRIANGLES
Equilateral triangles have three equal sides and three 60 degree angles.
How to solve a PERMUTATION problem
Factorials are useful for solving questions about permutations (the number of ways to arrange elements sequentially). How many ways are there to arrange 7 items along a shelf? 7 x 6 x 5 x 4 x 3 x 2 x 1 = 7! = 5040 If you're asked to find the number of ways to arrange a smaller group that's being drawn from a larger group, use the permutation formula: nPk = n! / (n - k)! where n = (the number in the larger group) and k = (the number you're arranging). Five runners run in a race. The runners who come in first, second, and third place will win gold, silver, and bronze medals, respectively. How many possible outcomes for gold, silver, and bronze medal winners are there? 5P3 = 5! / (5 - 3)! = 5!/2! = 5 x 4 x 3 = 60
How to add, subtract, multiply, and divide FRACTIONS
Find a common denominator before adding or subtracting fractions. 4/5 + 3/10 = 8/10 + 3/10 = 11/10 or 1 1/10 2 - 3/8 = 16/8 -3/8 = 13/8 or 1 5/8 To multiply fractions, multiply the numerators first and then multiply the denominators; simplify. 3/4 x 1/6 = 3/24 = 1/8 To divide by a fraction, multiply by its reciprocal (flip the numerator and the denominator). 5 / (1/3) = 5/1 x 3/1 = 15 (1/3) / (4/5) = 1/3 x 5/4 = 5/12
How to find the AVERAGE of CONSECUTIVE NUMBERS
Find the average of the smallest number and the largest number.
How to handle FRACTIONAL POWERS
Fractional exponents relate to roots. x^1/2 = square root of x x^1/3 = cube root of x x^2/3 = cube root of x^2 square root of x^-2 = (x^-2)^1/2 = x^(-2)(1/2) = x^-1 = 1/x
How to find the WEIGHTED AVERAGE
Give each term the appropriate "weight" The girls' average score is 30. The boys' average score is 24. If there are twice as many boys as girls, what is the overall average? Weighted average = (1 x 30) + (2 x 24)/3 = 78/3 = 26 *Do not just average the averages
How to solve an OVERLAPPING SETS problem involving BOTH/NEITHER
Group 1 + Group 2 + Neither - Both = Total 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 students are studying both French and Spanish? 65 + 51 + 53 - Both = 120 169 - Both = 120 Both = 49
How to use the PERCENT INCREASE/DECREASE FORMULAS
Identify the original whole and the amount of increase/decrease. Percent increase or decrease = (amount of increase or decrease/original whole) x 100% The price goes up from $80 to $100. What is the percent increase? Percent increase = 20/80 x 100% = 0.25 x 100% = 25%
How to use the PERCENT FORMULA
Identify the part, the percent, and the whole. Part = Percent x Whole Find the part: What is 12 percent of 25? Part = 12/100 x 25 = 300/100 = 3 Find the percent: 45 is what percent of 9? 45 = percent/100 x 9 4,500 = percent x 9 500 = percent Find the whole: 15 is 3/5 percent of what number? 15 = (3/5)(1/100) x whole 15 = (3/500) x whole whole = 15(500/3) = (7,500/3) = 2,500
How to use actual numbers to determine a RATE
Identify the quantities and the units to be compared. Keep the units straight. Anders typed 9,450 words in 3 1/2 hours. What was his rate in words per minute? 3 1/2 hours = 210 minutes (9,450 words / 210 minutes) = 45 words per minute
How to solve a COMPOUND INTEREST PROBLEM
If interest is compounded, the interest is computed on the principal as well as on any interest earned. To compute compound interest: (Final balance) = (Principal) x (1 + interest rate ^ ((time)(c))/c) where c is the number of times the interest is compounded annually. If $10,000 is invested at 8 percent annual interest, compounded semiannually, what is the balance after 1 year? Final balance = (10,000) x (1 + 0.08^((1)(2))/2) = (10,000) x (1.04)^2 =$10,816
How to work with FACTORIALS
If n is an integer greater than 1, then n! is the product of all the integers from 1 to n. 2! = 2 x 1 = 2 3! = 3 x 2 x 1 = 6 4! = 4 x 3 x 2 x 1 = 24 5! = 120 6! = 720 7! = 5040 8! = 40320 9! = 362880 10! = 3628800 By definition, 0! = 1
How to calculate a simple PROBABILITY
Probability = Number of desired outcomes / Number of total possible outcomes What is the probability of throwing a 5 on a fair six-sided die? There is one desired outcome: throwing a 5 There are six possible outcomes: throwing a 1, 2, 3, 4, 5, or 6 Probability = 1/6
How to solve a COMBINATION problem
If the order or arrangement does not matter, you are looking for the numbers of combinations, where nCk = n! / k! (n - k)! and n = (the number in the larger group) and k = (the number you're choosing) How many different ways are there to choose 3 delegates from 8 possible candidates? 8C3 = 8! / 3! (8-3)! = 8! / 3! x 5! = 8 x 7 = 56
How to find the DISTANCE BETWEEN POINTS on a COORDINATE PLANE
If two points have the same x or y coordinates, they make a line segment that is parallel to an axis. Subtract the numbers that are different; distance is always positive. What is the distance from (2, 3) to (-7, 3)? The y's are the same, so subtract the x's: 2 - (-7) = 9 If the points have different x and y coordinates, make a right triangle and use the Pythagorean theorem.
How to find the MAXIMUM and MINIMUM lengths for a SIDE of a TRIANGLE
If you know the lengths of two sides of a triangle, you know that the third side is somewhere between the positive difference and the sum of the other two sides. The length of one side of a triangle is 7. The length of another side is 3. What is the range of possible lengths for the third side? The third side is greater than the positive difference (7 - 3 = 4) and less than the sum (7 + 3 = 10) of the other two sides.
How to solve a COMBINED WORK PROBLEM
In a combined work problem, you are given the rate at which people or machines perform work individually and you are asked to compute the rate at which they work together (or vice versa). The work formula states: The inverse of the time it should 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) where r and s are the times it would take each party to complete a job working by themselves, and t is the time it would take the two of them working together. *All these variables must stand for units of TIME and must all refer to the amount of time it takes to do the same task. If it takes Joe 4 hours 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? Joe takes 4 hours; Pete takes 8 hours: (1/4) + (1/8) = (1/t) 2/8 + 1/8 = 1/t 3/8 = 1/t t = 8/3 It would take them 2 hours 4 minutes to paint the room together
How to solve a SEQUENCE problem
In a sequence problem, the nth term in the sequence is generated by performing an operation, which will be defined for you, on either n or on the previous term in the sequence. The term itself is expressed as a-sub-n. What is the positive difference between the fifth and fourth terms in the sequence 0, 4, 18, . . . whose nth term is n^2(n-1)? The fifth term: a-sub-5 = 5^2(5 - 1) = 25(4) = 100 The fourth term: a-sub-4 = 4^2(4 - 1) = 16(3) = 48 So the positive difference between the fifth and fourth terms is 100 - 48 = 52.
How to use actual numbers to determine a RATIO
Put the number associated with "of" on the op and the number associated with "to" on the bottom. Ratio = of / to Ratios should always be reduced to lowest terms. The ratio of 20 oranges to 12 apples is 20/12 or 5/3
How to solve a DILUTION or MIXTURE problem
In dilution or mixture problems, you have to determine the characteristics of a resulting mixture when different substances are combined; or you have to determine how to combine different substances to produce a desired mixture. Straightforward setup: If 5 pounds of raisins 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 = total cost for 7 pounds of the mixture The cost per pound is $9.80/7 = $1.40 Balancing method: How many liters of a solution that is 10 percent alcohol by volume must be added to 2 liters of a solution that is 50 percent alcohol by volume to create a solution that is 15 percent alcohol by volume? Make the weaker and stronger substances balance (Percent difference between the weaker solution and the desired solution) x (amount of weaker solution) = (percent difference between the stronger solution and the desired solution) x (amount of stronger solution) n is the amount, in liters, of the weaker solution n (15 - 10) = 2 (50 - 15) 5n = 2 (35) n = 70 / 5 = 14 So 14 liters of the 10 percent solution must be added to the original, stronger solution
How to work with SIMILAR TRIANGLES
In similar triangles, corresponding angles are equal, and corresponding sides are proportional.
How to work with ISOSCELES TRIANGLES
Isosceles triangles have at least two equal sides and two equal angles. In a GRE question, you'll need to use that information to find the LENGTH of a SIDE or a MEASURE of an ANGLE.
How to SIMPLIFY A RADICAL
Look for factors of the number under the radical sign that are perfect squares; find the square root of those perfect squares. Keep simplifying until the term with the square root sign is as simplified as possible (when there are no other perfect square factors - 4, 9, 16, 25, 36... - inside the root-sign). square root of 48 = (square root of 16)(square root of 3) = 4(square root of 3) square root of 180 = (square root of 36)(square root of 5) = six(square root of 5)
How to find the dimensions or area of an INSCRIBED or CIRCUMSCRIBED FIGURE
Look for the connection! Is the diameter the same as a side or a diagonal? If the area of a square is 36, what is the circumference of a circle circumscribing the square? To get the circumference, you need the diameter or radius. The circle's diameter is also the square's diagonal: the diagonal of the square is 6 square root of 2 (the diagonal transforms the square into two separate 45-45-90 degree triangles). Circumference = [pi](Diameter) = 6[pi]square root of 2.
How to solve certain QUADRATIC EQUATIONS
Manipulate the equation so that it is equal to 0. Factor the left side (reverse FOIL by finding two numbers whose product is the constant and whose sum is the coefficient of the term without the exponent). Break the quadratic into two simple expressions. x^2 + 6 = 5x x^2 - 5x + 6 = 0 (x - 2) (x - 3) = 0 x -2 = 0 OR x - 3 = 0 x = 2 OR 3 x^2 = 9 x = 3 OR -3
How to determine a COMBINED RATIO
Multiply one or both ratios by whatever you need in order to get the terms they have in common to match. The ratio of a to b is 7:3. The ratio of b to c is 2:5. What is the ratio of a to c? Multiply each member of a:b by 2 and multiply each member of b:c by 3: a:b = 14:6 and b:c = 6:15 a:b:c = 14:6:15 a:c = 14:15
How to handle a value with an EXPONENT RAISED TO AN EXPONENT
Multiply the exponents. (n^a)^b) = n^ab (3^4)^5 = 3^20
How to solve an OVERLAPPING SETS problem involving EITHER/OR CATEGORIES
Organize the information in a grid. At a certain professional conference with 130 attendees, 94 of the attendees are doctors, and the rest are dentists. If 48 of the attendees are women and 1/4 of the dentists in attendance are women, how many of the attendees are male doctors? Doctors Dentists Total Male 55 27 82 Female 39 9 48 Total 94 36 120 Thus, 55 of the attendees are male doctors
How to find the PERIMETER and AREA of a RECTANGLE
Perimeter = 2(Length + Width) Area = (Length)(Width)
How to solve a REMAINDERS problem
Pick a number that fits the given conditions and see what happens. When n is divided by 7, the remainder is 5. What is the remainder when 2n is divided by 7? Find a number that leaves a remainder of 5 when divided by 7, by taking any multiple of 7 and adding 5 to it: n = 12 2n = 24, which when divided by 7 leaves a remainder of 3
How to deal with STANDARD DEVIATION
Standard deviation is a measure of how spread out a set of numbers is - how much the numbers deviate from the mean. The greater the spread, the higher the standard deviation. Find the average of the stet Find the differences 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
How to determine COMBINED PERCENT INCREASE/DECREASE when no original value is specified
Start with 100 as a starting value. A price rises by 10 percent one year and by 20 percent the next. What's the combined percent increase? Say the original price is $100: Year one = $100 + (10% of 100) = 100 + 10 = 110 Year two = $110 + (20% of 110) = 110 + 22 = 132 From 100 to 132 is a 32 percent increase
How to use the AVERAGE to find the SUM
Sum = (Average) x (Number of terms) 17.5 is the average of 24 numbers. What is the sum of the 24 numbers? Sum = 17.5 x 24 = 420
How to find the SUM of CONSECUTIVE NUMBERS
Sum = (Average) x (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 x 41 = 1,230
How to find the sum of all the ANGLES of a POLYGON and one angle measure of a REGULAR POLYGON
Sum of the interior angles in a polygon with n sides: (n - 2) x 180 The term "regular" means all angles in the polygon are of equal measure. Degree measure of one angle in a regular polygon with n sides: ((n - 2) x 180) / n What is the measure of one angle of a regular pentagon? ((5 - 2) x 180) / 5 = 540 / 5 = 108
Basics of PROBABILITY
Suppose that a random process is performed. There is then a set of possible outcomes that can occur. An event is a set of possible outcomes. We are concerned with the probability of events. When all the outcomes are all equally likely, the basic probability formula is: Probability = Number of desired outcomes/ Number of total possible outcomes
How to find the SURFACE AREA of a CYLINDER
Surface area = 2[pi]r^2 + 2[pi]r x height
How to predict whether a sum, difference, or product will be ODD or even
Take simple numbers like 2 for even numbers and 3 for odd numbers and see what happens. If m is even and n is odd, is the product mn odd or even? m = 2 and n = 3 2 x 3 = 6, which is even, so mn is even
How to handle CUBE ROOTS
The cube root of x is just the number that when used as a factor three times gives you x. Both positive and negative numbers have only one cube root. The cube root of a number is always the same sign as the number itself. (-5) x (-5) x (-5) = -125, so the cube root of -125 is -5. (1/2) x (1/2) x (1/2) = 1/8, so the cube root of 1/8 is 1/2.
How to find a COMMON MULTIPLE of two numbers
The least common multiple (LCM) can be found by finding the prime factorization 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. 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
How to COUNT CONSECUTIVE NUMBERS
The number of integers from A to B inclusive is B - A + 1 How many integers are there from 73 through 419, inclusive? 419 - 73 + 1 = 347
How to find the x- and y-INTERCEPTS of a line
The x-intercept of a line is the value of x where the line crosses the x-axis; it is the value of x when y = 0. The y-intercept is the value of y where the line crosses the y-axis; it is the value of y when x = 0. The y-intercept is also the value b of the equation y = mx + b.
How to find the ORIGINAL WHOLE before percent increase/decrease
Think of a __ percent increase over x as 1.__x and set up an equation. After decreasing by 5 percent, the population is now 57,000. What was the original population? 0.95 x (Original population) = 57,000 Divide both sides by 0.95 Original population = 57,000 / 0.95 = 60,000
How to find the SURFACE AREA of a RECTANGULAR SOLID
To find the surface area of a rectangular solid, you have to find the area of each face and add the areas together. Surface area = 2(lw) + 2(wh) + 2(lh)
How to solve a simple LINEAR EQUATION
Use algebra to isolate the variable. Do the same steps to both sides of the equation. 28 = -3x - 5 28 + 5 = -3x, 33 = -3x 33/-3 = -3x/3 -11 = x
How to count the NUMBER OF POSSIBILITIES
Use multiplication to find the number of possibilities when items can be arranged in various ways. How many three-digit numbers can be formed with the digits 1, 3, and 5 each used only once? The first digit has three possibilities: 1, 3, or 5 The second digit has two possibilities... The third digit has one possibility... Multiply the possibilities together: 3 x 2 x 1 = 6
How to find the DIAGONAL of a RECTANGULAR SOLID
Use the pythagorean theorem twice, unless you spot special triangles.
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. Michael's average score after four tests is 80. If he scores 100 on the fifth test, what's his new average? Find the original sum from the original average: Original sum = 4 x 80 = 320 Add the fifth score to make the new sum: New sum = 320 + 100 = 420 Find the new average from the new sum: New average = 420/5 = 84
How to use the ORIGINAL AVERAGE and NEW AVERAGE to figure out WHAT WAS ADDED OR DELETED
Use the sums. Number added = (New sum) - (Original sum) Number deleted = (Original sum) - (New sum) The average of five numbers is 2. After one number is deleted, the new average is -3. What number was deleted? Find the original sum from the original average: Original sum = 5 x 2 = 10 Find the new sum from the new average: New sum = 4 x (-3) = -12 The difference between the original sum and the new sum is the answer. Number deleted = 10 - (-12) = 22
How to find an ANGLE formed by INTERSECTING LINES
Vertical angles are equal. Angles along a line add up to 180 degrees.
How to find the VOLUME of a CYLINDER
Volume = Area of the base x Height = [pi]r^2 x height
How to find the VOLUME of a RECTANGULAR SOLID
Volume = Length x Width x Height
How to find an angle formed by a TRANSVERSAL across PARALLEL LINES
When a transversal crosses parallel lines, all the acute angles formed are equal, and all the obtuse angles formed are equal. Any acute angle plus any obtuse angle equals 180 degrees.
How to add, subtract, multiply, and divide POSITIVE AND NEGATIVE NUMBERS
When addends have the same sign, add their absolute values and give the sum the same sign as the addends. -3 + -9 = -12 When addends have different signs, subtract the absolute values and give the sum the sign of the addend with the greater absolute value. 3 + -9 = -6, -3 + 9 = 6 In multiplication and division, when the signs are the same, the product or quotient is POSITIVE. 6 x 7 = 42, 96 / 8 = 12 When the signs are different, the product or quotient is NEGATIVE. -6 x 7 = -42, -96 / 8 = -12
How to handle GRAPHS of FUNCTIONS
When graphing a function, the output - f(x) - becomes the y-coordinate. In the function f(x) = x^2 - 1, the output is a parabola. *In the event of a parabola, pick out obvious points on the graph, such as (1,0) and (0,-1), plug these values into the answer choices, and eliminate until one answer choice is left.
How to solve MULTIPLE EQUATIONS
When you see two equations with two variables, combine. If the question doesn't ask for the variables separately, don't solve for them separately. If 5x - 2y = -9 and 3y - 4x = 6, what is the value of x + y? 5x - 2y = -9 +[-4x + 3y = 6] ... x + y = -3
How to solve a SIMPLE INTEREST problem
With simple interest, the interest is computed on the principal only and is given by Interest = Principle x rt where r is the interest rate per payment period and t is defined as the number of payment periods. If 12,000 is invested at 6 percent simple annual interest, how much interest is earned after 9 months? Since the interest rate is annual and we are calculating how much interest accrues after 9 months, we will express the payment period as 9/12: (12,000) x (0.06) x 9/12 = $540
How to ADD, SUBTRACT, MULTIPLY, and DIVIDE ROOTS
You can add/subtract roots only when the parts inside the root-sign are identical. square root of 2 + 3(square root of 2) = 4(square root of 2) square root of 2 + square root of 3 cannot be combined. To multiply/divide roots, deal with what's inside the root-sign and outside the root-sign separately. (2(square root of 3)) x (7(square root of 5)) = (2x7) (square root of 3x5) = 14 square root of 15.
How to handle EXPONENTS with a base of ZERO and bases with an EXPONENT of ZERO
Zero raised to any nonzero exponent equals zero. 0 raised to the 0 power is undefined.
