GRE Math Reference (Kaplan 10th Edition pages 616 to 639)

How to add, subtract, multiply, and divide fractions:

- Find a common denominator before adding or subtracting fractions. - To multiply fractions, multiply the numerators first and then multiply the denominators; simplify if necessary. (ex: 3/4 x 1/6 = 3/24 = 1/8) - You can also reduce before multiplying numerators and denominators; simplify if necessary. - To divide by a fraction, multiply by its reciprocal. To write the reciprocal of a fraction, flip the numerator and the denominator. (ex: 5 / 1/3 = 5/1 x 3/1 = 15)

How to add, subtract, multiply, and divide decimals:

- To add or subtract, align the decimal points and then add or subtract normally. Place the decimal point in the answer directly below existing decimal points. (ex: 3.25 + 4.4 = 7.65) - To multiply with decimals, multiply the digits normally and count off decimal places (equal to the total number of places in the factors) from the right. (ex: 2.5 x 2.5 = 6.25) - To divide by 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. (ex: 6.25 / 2.5 is changed to 62.5 / 25 which equals 2.5)

How to add, subtract, multiply, and divide whole numbers:

- You can check addition with subtraction: (ex: 17 + 5 = 22; 22 - 5 = 17) - You can check multiplication with division: (ex: 5 x 28 = 140; 140 / 5 = 28)

How to find the circumference of a circle:

-2πr, where r is the radius -πd, where d is the diameter

How to simplify binomials:

-A binomial is a sum or difference of two terms. -To simplify two binomials that are multiplied together, use the FOIL method. Multiply the First Term, then the Outer terms, followed by the Inner terms, and then the Last terms. Lastly, combine like terms. -Ex: (3x + 5) (x - 1) = 3x² - 3x + 5x - 5 = 3x² + 2x - 5

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. -Different classic polynomial equations: ab + ac = a(b + c) a² + 2ab + b² = (a + b)² a² - 2ab + b² = (a - b)² a² - b² = (a - b) (a + b)

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. -Use the area formulas for those familiar shapes. -Alternatively, you could apply the general formula for the area of a trapezoid. -Formula: area = (average of parallel sides) x (height) -So, area of rectangle = 8 x 5 = 40, area of triangle = 1/2(4 x 5) = 10, area of trapezoid = 40 +10, area of trapezoid = (8 + 12 / 2) x 5 = 50

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. -Ex: What factors greater than 1 do 135 and 225 have in common -First find the prime factors of 135 and 225 -135 = 3 x 3 x 3 x 5 -225 = 3 x 3 x 5 x 5 -The numbers share 3 x 3 x 5 in common -Thus, aside from 3 and 5, 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 -Therefore, the common factors of 135 and 225 are 3, 5, 9, 15, and 45

How to predict whether a sum, difference, or product will be odd or even:

-Don't bother memorizing the rules -Just take simple numbers (such as 2 for even numbers and 3 for odd numbers) and see what happens -Ex: If m is even and n is odd, is the product mn odd or even? 2 x 3 = 6, so mn is even

How to work with equilateral triangles:

-Equilateral triangles have three equal sides and three 60 ̊ angles. -If a GRE question tells you that a triangle is equilateral, you can bet that you'll need to use that information to find the length of a side or the measure of an angle.

How to solve a permutation problem:

-Factorials are useful for solving questions about permutations (i.e., the number of ways to arrange elements sequentially). For instance, to figure out how many ways there are to arrange 7 items along a shelf, you would multiply the number of possibilities for the first position by the number of possibilities remaining for the second position, and so on. (So, 7 x 6 x 5 x 3 x 2 x 1 or 7!) -If you're asked to find the number of ways to arrange a smaller group that's being drawn from a larger group, you can either apply logic or you can use the permutation formula: nPk = n! / (n - k)! where n = the number in the larger group and k = the number you're arranging

How to find the hypotenuse or leg of a right triangle:

-For all right triangles, the Pythagorean theorem is a² + b² = c², where a and b are the legs and c is the hypotenuse.

How to find the area of a rectangle:

-Formula: a = (length)(width)

How to find the area of a parallelogram:

-Formula: area = (base)(height) -So, area = 8 x 4 = 32.

How to find the area of a square:

-Formula: area = (side)²

How to find the area of a triangle:

-Formula: area = 1/2(base)(height). -Base and height must be perpendicular to each other. -Height is measured by drawing a perpendicular line segment from the base--which can be any side of the triangle--to the angle opposite the base. -In the example: area = 1/2(8)(3) = 12

How to find an average rate:

-Formula: average A per B = Total A / Total B -Ex: 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) = 75,000 + 45,000 = 120, 000 -Average words per page = 120,000 / 600 = 200

How to find an average speed:

-Formula: average speed = total distance / total time -Ex: 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 at 40 mph takes 3 hours. To return at 60 mph takes 2 hours. The total time, then, is 5 hours. -average speed = 240 miles / 5 hours = 48 mph

How to find the perimeter of a rectangle:

-Formula: perimeter = 2(length + width)

How to find the slope of a line:

-Formula: slope = rise/run = change in y/change in x -Ex: What is the slope of the line that contains the points (1,2) and (4, -5)? slope = -5 - 2 / 4 - 1 = -7/3

How to find a weighted average:

-Give each term the appropriate "weight." -Ex: 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 -Hint: Don't just average the averages.

How to use actual numbers to determine a rate:

-Identify the quantities and the units to be compared. Keep the units straight. -Ex: Anders typed 9450 words in 3.5 hours. What was his rate in words per minute? First, convert 3.5 hours to 210 minutes. Then set up the rate with words on top and minutes on bottom (because "per" means "divided by"): 9450 words/210 minutes = 45 words per minute

How to use the percent increase/decrease formulas:

-Identify the whole and the amount of increase/decrease -percent increase = amount of increase/original whole x 100% -percent decrease= amount of decrease/original whole x 100% -Ex: The price goes up from $80 to $100. What is the percent increase? percent increase = 20/100 x 100% = .25 x 100% = 25%

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. -Ex: If $10,000 is invested at 8 percent annual interest, compounded semi-annually, what is the balance after 1 year? -Final balance = (10,00) x (1 + 0.08^(1)(2)/20 = (10,000) x (1.04)² = $10, 816 -Semiannual interest is interest that is distributed twice a year. When an interest rate is given as an annual rate, divide by 2 to find the semiannual interest rate.

How to solve a combination problem:

-If the order of arrangement of the smaller group that's being drawn from the larger group does not matter, you are looking for the numbers of combinations, and a different formula is called for: nCk = n! / k!(n - k)! where n = the number in the larger group and k = the number you're choosing

How to find the distance between points on the coordinate plane:

-If two points have the same x-coordinates or the same y-coordinate--that is, they make a line segment that is parallel to an axis--all you have to do is subtract the numbers that are different. Just remember that distance is always positive. -Ex: What is the distance from (2,3) to (-7,3)? The y's are the same, so just subtract the x's. -If the points have different x-coordinates and different y-coordinates, make a right triangle and use the Pythagorean theorem or apply the special right triangle attributes if applicable.

How to work with new symbols:

-If you see a symbol you've never seen before, don't be alarmed. It is just a made-up symbol whose operation is uniquely defined by the problem. Everything you need to know is in the question stem. Just follow the instructions.

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 would take everyone working together equals the sum of the inverses of the times it would take each working individually. So, 1 / r + 1 / s = 1 / t where r and s are, for example, the number of hours it would take Rebecca and Sam, respectively, to complete a job working by themselves, and t is the number of hours it would take the two of them working together. Remember that 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. -Ex: 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 pain the same room, if each of them works at his respective individual rate? -Joe takes 4 hours, so Pete takes 8 hours; thus: 1/4 + 1/8 = 1/t 2/8 + 1/8 = 1/t 3/8 = 1/t t = 1/ (3/8) t = 8/3 (or 2 hours and 40 minutes)

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, alternatively, you have to determine how to combine different substances to produce a desired mixture. There are two approaches to such problems--the straightforward setup and the balancing method. -Ex: 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? -The straightforward setup: ($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. -Ex: 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? -The balancing method: Make the weaker and stronger (or cheaper and more expensive, etc.) substances balance. That is, (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). Make n the amount, in liters, of the weaker solution. n(15 - 10) = 2(50 - 15) 5n = 2(35) n = 70/5 n = 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. -If a GRE question tells you that triangles are similar, use the properties of similar triangles to find the length of a side or the measurement of an angle.

How to work with factorials:

-Indicated by a ! symbol -If n is an integer greater than 1, then n factorial, denoted by n!, is defined as the product of all the integers from 1 to n. -For example: 2! = 2 x 1 = 2 ; 3! = 3 x 2 x 1 = 6 -By definition, 0! = 1

How to work with isosceles triangles:

-Isosceles triangles have at least two equal sides and two equal angles. -If a GRE question tells you that a triangle is isosceles, you can bet that you'll need to use that information to find the length of a side or a measure of an angle.

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, and you get a:b = 14:6 and b:c = 6:15. Now that the values of b match, you can write a:b:c = 14:6:15 and then say a:c = 14:15.

How to solve an overlapping sets problem involving either/or categories:

-Other GRE word problems involve groups with distinct "either/or" categories (male/female, blue-color/white-color, etc.). The key to solving this type of problem is to organize the information on a grid. -Ex: 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 attendees are male doctors? -Make a complete a grid using the information in the problem, making each row and column add up to the corresponding total.

How to solve a remainders problem:

-Pick a number that fits the given conditions and see what happens. -Ex: 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. You can find such a number by taking any multiple of 7 and adding 5 to it. A good choice would be 12. -If n=12, then 2n = 24, which when divided by 7 leaves a remainder of 3

How to find the median:

-Put the numbers in numerical order and take the middle number. If there are two numbers in the middle, take the average of those two. -Ex: What is the median of 88, 86, 57, 94, and 73? -Line them up as: 57, 73, 86, 88, 94 -Take the middle number: 86

How to deal with tables, graphs, and charts:

-Read the question and all labels carefully. -Ignore extraneous information and zero in on what the question asks for. -Take advantage of the spread in the answer choices by approximating the answer whenever possible and choosing the answer choice closet to your approximation.

How to use a ratio to determine an actual number:

-Set up a proportion using the given ratio -Ex: The ratio of bogs to girls is 3 to 4. If there are 135 boys, how many girls are there? So, 3/4 = 135/g; Solve and g = 180

How to solve an overlapping sets problem involving both/neither:

-Some GRE word problems involve two groups with overlapping members and possibly elements that belong to neither group. It's easy to identify this type of question because the words both and/or neither appear in the question. These problems are quite workable if you just memorize the following formula: group 1 + group 2 + neither - both = total -Ex: 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

How to spot special right triangles:

-Special right triangles are ones that are seen on the GRE with frequency. -Recognizing them can streamline your problem solving. - 3:4:5; 5:12:13 -These numbers (3,4,5 and 5,12,13) represent the ratio of the side lengths of these triangles. -30 ̊ -- 60 ̊ -- 90 ̊; 45 ̊ -- 45 ̊ -- 90 ̊ -In a 30 -- 60 -- 90 triangle, the side lengths are multiples of 1, √3, and 2, respectively. In a 45 -- 45 -- 90 triangle, the side lengths are multiples of 1, 1, and √2 respectively.

How to determine combined percent increase/decrease when no original value is specified:

-Start with 100 as a starting value. -Ex: 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) -Ex: 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) -Ex: 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 = 1230

How to solve probability problems where probabilities must be multiplied:

-Suppose that a random process is performed. Then there is 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 -Many more difficult probability questions involve finding the probability that several events occur. Let's consider first the case of the probability that two events occur. Call these two events A and B. The probability that both events occur is the probability that A occurs multiplied by the probability that B occurs given that even A occurred. The probability that B occurs given that A occurs is called the conditional probability that B occurs given that A occurs. 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.

How to find the mode:

-Take the number that appears most often. -If there is a tie for most often, then there is more than one mode. -If each number in a set is used equally often, then there is no mode. -Ex: If your test scores were 88, 57, 68, 85, 98, 93, 93, 84, and 81, then the mode is 93 because it appears more often than any other score.

How to find the range:

-Take the positive difference between the greatest and least values. -Ex: If your test scores were 88, 57, 68, 85, 98, 93, 93, 84, and 81, then the range is 41 because 98 - 57 = 41.

How to handle absolute values:

-The absolute value of a number n, (denoted by [n], is defined as: n if n is greater than or equal to 0 and -n if n is less than 0) is the distance from zero to the number on the number line. -The absolute value of a number or expression is always positive. -So, if the absolute value of x is 3, then x could be 3 or -3.

How to find the average of consecutive numbers:

-The average of evenly spaced numbers is simply the average of the smallest number and the largest number. -Ex: The average of all the integers from 13 to 77 is the same as the average of 13 and 77. So, 13 + 7 / 2 = 90 / 2 = 45

How to count consecutive numbers:

-The number of integers from A to B inclusive is B - A + 1 -Ex: How many integers are there from 73 through 419? 419 - 73 + 1 = 347

How to find a common multiple of two numbers:

-The product of two numbers is the easiest common multiple to find, but it is not always the least common multiple (LCM) -The 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. -Example: What is the LCM of 28 and 42? 28 = 2 x 2 x 7 42 = 2 x 3 x 7 In 28, 2 is used twice. In 42, 2 is used once. In 28, 7 is used once. In 42, 7 is used once, and 3 is used once. So, you multiple each factor the greatest number of times it appears in a prime factorization. LCM = 2 x 2 x 3 x 7 = 84

How to find the original whole before percent increase/decrease:

-Think of a 15 percent increase over x as 1.15x and set up an equation. -Ex: 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 convert fractions to decimals and decimals to fractions:

-To convert a fraction to a decimal, divide the numerator by the denominator. (ex: 4/5 = 0.8) -To convert a decimal to a fraction, write the digits in the numerator and use the decimal name in the denominator. (ex: 0.003 = 3/1000)

How to plug a number into an algebraic expression:

-To evaluate an algebraic expression, choose numbers for the variables or use the numbers assigned to the variables. (Ex: Evaluate 4np when n = -4 and p = 3; 4np + 1 = 4(-4)3 + 1 = -48 + 1 = -47)

How to solve for one variable in terms of another:

-To find "x" in terms of "y", isolate x on one side, leaving y as the only variable on the other.

How to use actual numbers to determine a ratio:

-To find a ratio, put the number associate with 'of' on the top and the number associated with 'to' on the bottom. So, ratio = of/to -Ex: The ratio of 20 oranges to 12 apples is 20/12 or 5/3. -Ratios should always be reduced to lowest terms. -Ratios can also be expressed in linear form, such as 5:3.

How to plot points on the number line:

-To plot the point 4.5 on the number line, start at 0, go right to 5.4 half way between 4 and 5. -To plot the point -2.5 on the number line, start at 0, go left to -2.5, half way between -2 and -3.

How to solve an inequality:

-Treat it much like an equation--adding, subtracting, multiplying, and dividing both sides by the same thing. -Remember to reverse the inequality sign if you multiply or divide by a negative quantity. -Ex: Rewrite 7 - 3x > 2 in its simplest form -7 -7 -3x > -5 /-3 /-3 x < 5/3

How to solve a digits problem:

-Use a little logic and some trial and error. -Ex: IF A, B, C and D represent distinct digits in the addition problem below, what is the value of D? AB + BA = CDC -Two 2 digit numbers will at up to at most something in the 100s, so C = 1. B plus A in the units column gives a 1, and since A and B in the tens column don't add up to C, it can't simply be that B + A = 1. If must be that B + A = 11, and a 1 gets carried. In fact, A and B can be any pair of digits that add up to 11 (3 and 8, 4 and 7, etc.), but it doesn't matter what they are: they always give you the same value for D, which is 2: So, 47 + 74 = 121 and 83 + 38 = 121

How to solve a simple linear equation:

-Use algebra to isolate the variable. Do the same step to both sides of the equation. Ex: 28 = -3x - 5 28 + 5 = -3x - 5 + 5 33 = -3x 33 / -3 = -3x / -3 -11 = x

How to find the third angle of a triangle, given the other two angles:

-Use the fact that the sum of the interior angles of a triangle always equals 180 ̊. (Ex: if three angles of a triangle are 35 ̊, 45 ̊, and x ̊ then 35 + 45 + x = 180, so x = 100)

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. -Ex: Michael's average score after four tests is 80. If he scores 100 on the fifth test, what is his new average? -Find the original sum: 4 x 80 = 320 -Add the fifth score to make the new sum: 320 + 100 = 420 -Find the new average from the new sum: 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 - old sum number deleted = original sum - new sum Ex: The average of five numbers is 2. After one number is deleted, the 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 ̊. -So, a ̊ = c ̊, b ̊ = d ̊, a ̊ + b ̊ = 180 ̊, and a ̊ + b ̊ + c ̊ + d ̊ = 360 ̊

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 ̊. -So, e ̊ = g ̊ = p ̊ = r ̊, f ̊ = h ̊ = q ̊ =s ̊, and e ̊ + q ̊ = g ̊ + s ̊ = 180 ̊

How to add, subtract, multiply, and divide positive and negative numbers:

-When the numbers being added (addends) have the same sign, add their absolute values; the sum has the same sign as the addends. (ex: -3 + -9 = -12) -When addends have different signs, subtract the absolute values; the sum has the sign of the greater absolute value. (ex: 3 + -9 = -6) -In multiplication and division, when the signs are the same, the product/quotient is positive. When the signs are different, the product/quotient is negative. (ex: 6 x 7 = 42 and -6 x -7 = 42) (ex: -6 x 7 = -42)

How to use PEMDAS:

-When you're given a complex arithmetic expression, it's important to know the order of operations. -PEMDAS = "Please Excuse My Dear Aunt Sally" -P = Parentheses; Clean up parentheses first (nested sets of parentheses are worked from the innermost set to the outermost set) -E = Exponents; deal with exponents or radicals -MD = Multiplication and Division; deal with these together, going from left to right -AS = Addition and Subtraction; deal with these together, going from left to right

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 -In this formula, r is defined as the interest rate plus payment period, and t is defined as the number of payment periods. -Ex: 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 (o.06) x 9/12 = $540

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. -Ex: How many three-digit numbers can be formed with the digits 1, 3, and 5 each used only once? Look at each digit individually. The first digit (or, the hundreds digit) has three possible numbers to plug in 1, 3, 5. The second digit (or, the tens digit) has two possible numbers, since one had already been plugged in. The last digit (or, the ones digit) has only one remaining possible number. Multiply the possibilities together: 3 x 2 x 1 = 6.

How to find the average or arithmetic mean:

-average = sum of terms / number of terms -Example: What is the average of 3, 4, and 8? average = 3 + 4 + 8 / 3 = 15 / 3 = 5

How to use the percent formula:

-part = percent x whole Ex: What is 12 percent of 25? part = 12/100 x 25 part = 300/100 part = 3

How to calculate a simple probability:

-probability = number of desired outcomes / number of total possible outcomes -Ex: What is the probability of throwing a 5 on a fair six-sided die? There is one desired outcome--throwing a 5. There are 6 possible outcomes--one for each side of the die. Probability = 1/6

How to add and subtract line segments:

.___A____.___B___._____C______ If AB = 6 and BC = 8, then AC = 6 + 8 = 14 If AC = 14 and BC = 8, then AB = 14 - 8 = 6

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 a 5 or 0 6: sum of digits is a multiple of 3, and the 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