When you find the elements that two (or more) sets have in common, you are finding the intersection of the sets. The symbol for intersection is: ∩. For example, the intersection of the integers and the whole numbers is the set of the whole numbers itself. This is because the elements (numbers) that they have in common are {0,1,2,3, }.Consider the set of positive even integers and the set of positive odd integers. The positive even integers are: {2,4,6,8, } The positive odd integers are: {1,3,5,7, } If we were to combine the set of positive even numbers with the set of positive odd numbers, we would have the union of the sets: {1,2,3,4,5, } The symbol for union is: ∪. Mean, Median, and Mode To find the average or mean of a set of numbers, add all of the numbers together and divide by the quantity of numbers in the set. Average = ᎏ qu n a u n m ti b ty er o s f e s t et ᎏ Example: Find the average of 9, 4, 7, 6, and 4. ᎏ 9+4+7 5 +6+4 ᎏ = ᎏ 3 5 0 ᎏ = 6 (because there are 5 numbers in the set) To find the median of a set of numbers, arrange the numbers in ascending order and find the middle value. ■ If the set contains an odd number of elements, then simply choose the middle value. Example: Find the median of the number set: 1, 5, 3, 7, 2. First, arrange the set in ascending order: 1, 2, 3, 5, 7, and then, choose the middle value: 3. The answer is 3. ■ If the set contains an even number of elements, simply average the two middle values. Example: Find the median of the number set: 1, 5, 3, 7, 2, 8. First, arrange the set in ascending order: 1, 2, 3, 5, 7, 8, and then, choose the middle values 3 and 5. Find the average of the numbers 3 and 5: ᎏ 3+ 2 5 ᎏ = 4. The answer is 4. The mode of a set of numbers is the number that occurs the greatest number of times. Example: For the number set 1, 2, 5, 3, 4, 2, 3, 6, 3, 7, the number 3 is the mode because it occurs the most number of times. Percent A percent is a measure of a part to a whole, with the whole being equal to 100. ■ To change a decimal to a percentage, move the decimal point two units to the right and add a percentage symbol. Examples: .45 = 45% .07 = 7% .9 = 90% ■ To change a percentage to a decimal, simply move the decimal point two places to the left and elimi- nate the percentage symbol. Examples: 64% = .64 87% = .87 7% = .07 ■ To change a fraction to a percentage, first change the fraction to a decimal. To do this, divide the numerator by the denominator. Then change the decimal to a percentage. Examples: ᎏ 4 5 ᎏ = .80 = 80% ᎏ 2 5 ᎏ = .4 = 40% ᎏ 1 8 ᎏ = .125 = 12.5% –THE SAT MATH SECTION– 117 5658 SAT2006[04](fin).qx 11/21/05 6:44 PM Page 117 ■ To change a percentage to a fraction, divide by 100 and reduce. Examples: 64% = ᎏ 1 6 0 4 0 ᎏ = ᎏ 1 2 6 5 ᎏ 75% = ᎏ 1 7 0 5 0 ᎏ = ᎏ 3 4 ᎏ 82% = ᎏ 1 8 0 2 0 ᎏ = ᎏ 4 5 1 0 ᎏ ■ Keep in mind that any percentage that is 100 or greater will need to reflect a whole number or mixed number when converted. Examples: 125% = 1.25 or 1 ᎏ 1 4 ᎏ 350% = 3.5 or 3 ᎏ 1 2 ᎏ Here are some conversions you should be famil- iar with: FRACTION DECIMAL PERCENTAGE ᎏ 1 2 ᎏ .5 50% ᎏ 1 4 ᎏ .25 25% ᎏ 1 3 ᎏ .333 . . . 33.3 % ᎏ 2 3 ᎏ .666 . . . 66.6 % ᎏ 1 1 0 ᎏ .1 10% ᎏ 1 8 ᎏ .125 12.5% ᎏ 1 6 ᎏ .1666 . . . 16.6 % ᎏ 1 5 ᎏ .2 20% Graphs and Tables The SAT will test your ability to analyze graphs and tables. It is important to read each graph or table very carefully before reading the question. This will help you process the information that is presented. It is extremely important to read all of the information pre- sented, paying special attention to headings and units of measure. Following is an overview of the types of graphs you will encounter. Circle Graphs or Pie Charts This type of graph is representative of a whole and is usually divided into percentages. Each section of the chart represents a portion of the whole, and all of these sections added together will equal 100% of the whole. Bar Graphs Bar graphs compare similar things with bars of differ- ent length representing different values. On the SAT, these graphs frequently contain differently shaded bars used to represent different elements. Therefore, it is important to pay attention to both the size and shad- ing of the graph. Comparison of Road Work Funds of New York and California 1990–1995 New York California KEY 0 10 20 30 40 50 60 70 80 90 1991 1992 1993 1994 1995 Money Spent on New Road Work in Millions of Dollars Year 25% 40% 35% –THE SAT MATH SECTION– 118 5658 SAT2006[04](fin).qx 11/21/05 6:44 PM Page 118 Broken-Line Graphs Broken-line graphs illustrate a measurable change over time. If a line is slanted up, it represents an increase whereas a line sloping down represents a decrease. A flat line indicates no change as time elapses. Scatterplots Scatterplots illustrate the relationship between two quantitative variables. Typically, the values of the inde- pendent variables are the x-coordinates, and the values of the dependent variables are the y-coordinates. When presented with a scatterplot, look for a trend. Is there a line that the points seem to cluster around? For example: In the scatterplot above, notice that a “line of best fit” can be created: Matrices Matrices are rectangular arrays of numbers. Below is an example of a 2 by 2 matrix: Review the following basic rules for performing operations on 2 by 2 matrices. Addition Add the given entries as shown below: += Subtraction Subtract the given entries as shown below: –= Scalar Multiplication Multiply every entry by the given constant (shown below as k): k = Multiplication of Matrices Multiply the given entries as shown below: × = [ a 1 b 1 + a 2 b 3 a 1 b 2 + a 2 b 4 ] a 3 b 1 + a 3 b 3 a 3 b 2 + a 4 b 4 [ b 1 b 2 ] b 3 b 4 [ a 1 a 2 ] a 3 a 4 [ ka 1 ka 2 ] ka 3 ka 4 [ a 1 a 2 ] a 3 a 4 [ a 1 – b 1 a 2 – b 2 ] a 3 – b 3 a 4 – b 4 [ b 1 b 2 ] b 3 b 4 [ a 1 a 2 ] a 3 a 4 [ a 1 + b 1 a 2 + b 2 ] a 3 + b 3 a 4 + b 4 [ b 1 b 2 ] b 3 b 4 [ a 1 a 2 ] a 3 a 4 [ a 1 a 2 ] a 3 a 4 HS GPA College GPA HS GPA College GPA Increase Decrease No Change In crea se Decrease Change in Time Unit of Measure –THE SAT MATH SECTION– 119 5658 SAT2006[04](fin).qx 11/21/05 6:44 PM Page 119 Algebra Review Equations An equation is solved by finding a number that is equal to an unknown variable. Simple Rules for Working with Equations 1. The equal sign separates an equation into two sides. 2. Whenever an operation is performed on one side, the same operation must be performed on the other side. 3. Your first goal is to get all of the variables on one side and all of the numbers on the other. 4. The final step often will be to divide each side by the coefficient, leaving the variable equal to a number. Cross Multiplying You can solve an equation that sets one fraction equal to another by cross multiplication. Cross multiplica- tion involves setting the products of opposite pairs of terms equal. Example: ᎏ 6 x ᎏ = ᎏ x + 12 10 ᎏ becomes 12x =6(x) + 6(10) 12x =6x + 60 –6x –6x ᎏ 6 6 x ᎏ = ᎏ 6 6 0 ᎏ Thus, x = 10. Checking Equations To check an equation, substitute the number equal to the variable in the original equation. Example: To check the equation above, substitute the number 10 for the variable x. Example: ᎏ 6 x ᎏ = ᎏ x + 12 10 ᎏ ᎏ 1 6 0 ᎏ = ᎏ 10 1 + 2 10 ᎏ = ᎏ 1 6 0 ᎏ = ᎏ 2 1 0 2 ᎏ 1 ᎏ 2 3 ᎏ = 1 ᎏ 2 3 ᎏ ᎏ 1 6 0 ᎏ = ᎏ 1 6 0 ᎏ Because this statement is true, you know the answer x = 10 must be correct. Special Tips for Checking Equations 1. If time permits, be sure to check all equations. 2. If you get stuck on a problem with an equation, check each answer, beginning with choice c.If choice c is not correct, pick an answer choice that is either larger or smaller. This process will be further explained in the strategies for answering five-choice questions. 3. Be careful to answer the question that is being asked. Sometimes, this involves solving for a variable and then performing an operation. Example: If the question asks the value of x – 2, and you find x = 2, the answer is not 2, but 2 – 2. Thus, the answer is 0. Equations with More Than One Variable Many equations have more than one variable. To find the solution, solve for one variable in terms of the other(s). To do this, follow the rule regarding variables and numbers on opposite sides of the equal sign. Iso- late only one variable. Example: 2x + 4y = 12 To isolate the x variable, –4y = –4y move the 4y to the other side. 2x = 12 – 4y Then divide both sides by the coefficient of x. ᎏ 2 2 x ᎏ = ᎏ 12 2 –4y ᎏ The last step is to simplify your answer. x = 6 – 2y This expression for x is written in terms of y. –THE SAT MATH SECTION– 120 5658 SAT2006[04](fin).qx 11/21/05 6:44 PM Page 120 Polynomials A polynomial is the sum or difference of two or more unlike terms. Example: 2x + 3y – z The above expression represents the sum of three unlike terms 2x,3y, and –z. Three Kinds of Polynomials ■ A monomial is a polynomial with one term, as in 2b 3 . ■ A binomial is a polynomial with two unlike terms, as in 5x + 3y. ■ A trinomial is a polynomial with three unlike terms, as in y 2 + 2z – 6. Operations with Polynomials ■ To add polynomials, be sure to change all subtrac- tion to addition and the sign of the number that was being subtracted. Then simply combine like terms. Example: (3y 3 – 5y + 10) + (y 3 + 10y – 9) Change all subtraction to addition and the sign of the number being subtracted. 3y 3 + –5y + 10 + y 3 + 10y + –9 Combine like terms. 3y 3 + y 3 + –5y + 10y + 10 + –9 = 4y 3 + 5y + 1 ■ If an entire polynomial is being subtracted, change all of the subtraction to addition within the parentheses and then add the opposite of each term in the polynomial. Example: (8x – 7y + 9z) – (15x + 10y – 8z) Change all subtraction within the parentheses first: (8x + –7y + 9z) – (15x + 10y + –8z) Then change the subtraction sign outside of the parentheses to addition and the sign of each polynomial being subtracted: (8x + –7y + 9z) + (–15x +–10y +8z) Note that the sign of the term 8z changes twice because it was being subtracted twice. All that is left to do is combine like terms: 8x + –15x + –7y + –10y + 9z + 8z = –7x + –17y + 17z is your answer. ■ To multiply monomials, multiply their coeffi- cients and multiply like variables by adding their exponents. Example: (–5x 3y )(2x 2y3 ) = (–5)(2)(x 3 )(x 2 )(y)(y 3 ) = –10x 5y4 ■ To divide monomials, divide their coefficients and divide like variables by subtracting their exponents. Example: ᎏ 1 2 6 4 x x 4 3 y y 5 2 ᎏ = = ᎏ 2x 3 y 3 ᎏ ■ To multiply a polynomial by a monomial, multi- ply each term of the polynomial by the monomial and add the products. Example: 6x(10x – 5y + 7 ) Change subtraction 6x(10x + –5y + 7) to addition: Multiply: (6x)(10x) + (6x)(–5y) + (6x)(7) 60x 2 + –30xy + 42x ■ To divide a polynomial by a monomial, divide each term of the polynomial by the monomial and add the quotients: Example: ᎏ 5x –10 5 y +20 ᎏ = ᎏ 5 5 x ᎏ – ᎏ 1 5 0y ᎏ + ᎏ 2 5 0 ᎏ = x – 2y + 4 FOIL The FOIL method is used when multiplying binomi- als. FOIL stands for the order used to multiply the terms: First, Outer, Inner, and Last. To multiply bino- mials, you multiply according to the FOIL order and then add the products. (16)(x 4 ) (y 5 ) ᎏᎏ (24) (x 3 ) (y 2 ) –THE SAT MATH SECTION– 121 5658 SAT2006[04](fin).qx 11/21/05 6:44 PM Page 121 Example: (3x + 1)(7x + 10) 3x and 7x are the first pair of terms, 3x and 10 are the outermost pair of terms, 1 and 7x are the innermost pair of terms, and 1 and 10 are the last pair of terms. Therefore, (3x)(7x) + (3x)(10) + (1)(7x) + (1)(10) = 21x 2 + 30x + 7x + 10. After we combine like terms, we are left with the answer: 21x 2 + 37x + 10. Factoring Factoring is the reverse of multiplication: 2(x + y) = 2x + 2y Multiplication 2x + 2y = 2(x + y) Factoring THREE BASIC TYPES OF FACTORING ■ Factoring out a common monomial: 10x 2 – 5x = 5x(2x – 1) and xy – zy = y(x – z) ■ Factoring a quadratic trinomial using the reverse of FOIL: y 2 – y – 12 = (y – 4) (y – +3) and z 2 – 2z + 1 = (z – 1)(z – 1) = (z – 1) 2 ■ Factoring the difference between two squares using the rule: a 2 – b 2 = (a + b)(a – b) and x 2 – 25 = (x + 5)(x – 5) REMOVING A COMMON FACTOR If a polynomial contains terms that have common fac- tors, the polynomial can be factored by using the reverse of the distributive law. Example: In the binomial 49x 3 + 21x,7x is the greatest common factor of both terms. Therefore, you can divide 49x 3 + 21x by 7x to get the other factor. ᎏ 49x 3 7 + x 21x ᎏ = ᎏ 4 7 9 x x 3 ᎏ + ᎏ 2 7 1 x x ᎏ = 7x 2 + 3 Thus, factoring 49x 3 + 21x results in 7x(7x 2 + 3). I SOLATING VARIABLES USING FRACTIONS It may be necessary to use factoring in order to isolate a variable in an equation. Example: If ax – c = bx + d, what is x in terms of a, b, c, and d? 1. The first step is to get the x terms on the same side of the equation. ax – bx = c + d 2. Now you can factor out the common x term on the left side. x(a – b) = c + d 3. To finish, divide both sides by a – b to isolate the variable of x. ᎏ x( a a – – b b) ᎏ = ᎏ c a + – d b ᎏ 4. The a – b binomial cancels out on the left, result- ing in the answer: x = ᎏ c a + – d b ᎏ Quadratic Trinomials A quadratic trinomial contains an x 2 term as well as an x term. x 2 – 5x + 6 is an example of a quadratic trino- mial. It can be factored by reversing the FOIL method. ■ Start by looking at the last term in the trinomial, the number 6. Ask yourself, “What two integers, when multiplied together, have a product of posi- tive 6?” ■ Make a mental list of these integers: 1 × 6 –1 × –6 2 × 3 –2 × –3 ■ Next, look at the middle term of the trinomial, in this case, the –5x. Choose the two factors from the above list that also add up to –5. Those two factors are: –2 and –3 ■ Thus, the trinomial x 2 – 5x + 6 can be factored as (x – 3)(x – 2). –THE SAT MATH SECTION– 122 5658 SAT2006[04](fin).qx 11/21/05 6:44 PM Page 122 . 25% ᎏ 1 3 ᎏ .333 . . . 33.3 % ᎏ 2 3 ᎏ .66 6 . . . 66 .6 % ᎏ 1 1 0 ᎏ .1 10% ᎏ 1 8 ᎏ .125 12.5% ᎏ 1 6 ᎏ . 166 6 . . . 16. 6 % ᎏ 1 5 ᎏ .2 20% Graphs and Tables The SAT will test your ability to analyze. =6x + 60 –6x –6x ᎏ 6 6 x ᎏ = ᎏ 6 6 0 ᎏ Thus, x = 10. Checking Equations To check an equation, substitute the number equal to the variable in the original equation. Example: To check the equation. the numerator by the denominator. Then change the decimal to a percentage. Examples: ᎏ 4 5 ᎏ = .80 = 80% ᎏ 2 5 ᎏ = .4 = 40% ᎏ 1 8 ᎏ = .125 = 12.5% THE SAT MATH SECTION– 117 565 8 SAT2 0 06[ 04](fin).qx