 The FOIL Method The FOIL method can be used when multiplying bino- mials. FOIL stands for the order used to multiply the terms: First, Outer, Inner, and Last. To multiply binomi- als, you multiply according to the FOIL order and then add the like terms of the products. 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 1. Factoring out a common monomial. 10x 2 − 5x = 5x(2x − 1) and xy − zy = y(x − z) 2. 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 3. Factoring the difference between two perfect 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 dividing by the greatest common factor. 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).  Quadratic Equations A quadratic equation is an equation in which the great- est exponent of the variable is 2, as in x 2 + 2x − 15 = 0. A quadratic equation has two roots, which can be found by breaking down the quadratic equation into two simple equations. Example Solve x 2 + 5x + 2x + 10 = 0. x 2 + 7x + 10 = 0 Combine like terms. (x + 5)(x + 2) = 0 Factor. x + 5 = 0 or x + 2 = 0 ᎏ x − = 5 − − 5 5 ᎏᎏ x − = 2 − − 2 2 ᎏ Now check the answers. −5 + 5 = 0 and −2 + 2 = 0 Therefore, x is equal to both −5 and −2.  Inequalities Linear inequalities are solved in much the same way as simple equations. The most important difference is that when an inequality is multiplied or divided by a negative number, the inequality symbol changes direction. Example 10 > 5 but if you multiply by −3, (10) − 3 < (5)−3 −30 < −15 Solving Linear Inequalities To solve a linear inequality, isolate the variable and solve the same as you would in a first-degree equation. Remember to reverse the direction of the inequality sign if you divide or multiply both sides of the equation by a negative number. – ALGEBRA, FUNCTIONS, AND PATTERNS– 415 Example If 7 − 2x > 21, find x. Isolate the variable. 7 − 2x > 21 ᎏ − − 2 7 x ᎏ > ᎏ − 14 7 ᎏ Because you are dividing by a negative number, the direction of the inequality symbol changes direction. ᎏ − − 2 2 x ᎏ > ᎏ − 14 2 ᎏ x < −7 The answer consists of all real numbers less than −7.  Exponents An exponent tells you how many times the number, called the base, is a factor in the product. Example 2 5  exponent = 2 × 2 × 2 × 2 × 2 = 32  base – ALGEBRA, FUNCTIONS, AND PATTERNS– 416 T HIS SECTION WILL help you become familiar with the word problems on the GED and analyze data using specific techniques.  Translating Words into Numbers The most important skill needed for word problems is the ability to translate words into mathematical opera- tions. This list will assist you in this by giving you some common examples of English phrases and their mathe- matical equivalents. ■ Increase means add. A number increased by five = x + 5. ■ Less than means subtract. 10 less than a number = x − 10. ■ Times or product means multiply. Three times a number = 3x. CHAPTER Data Analysis, Statistics, and Probability MANY STUDENTS struggle with word problems. In this chapter, you will learn how to solve word problems with confidence by trans- lating the words into a mathematical equation. Since the GED math section focuses on “real-life” situations, it’s especially important for you to know how to make the transition from sentences to a math problem. 44 417 ■ Times the sum means to multiply a number by a quantity. Five times the sum of a number and three = 5(x + 3). ■ Two variables are sometimes used together. A number y exceeds five times a number x by ten. y = 5x + 10 ■ Inequality signs are used for at least and at most, as well as less than and more than. The product of x and 6 is greater than 2. x × 6 > 2 When 14 is added to a number x, the sum is less than 21. x + 14 < 21 The sum of a number x and four is at least nine. x + 4 ≥ 9 When seven is subtracted from a number x, the difference is at most four. x − 7 ≤ 4  Assigning Variables in Word Problems It may be necessary to create and assign variables in a word problem. To do this, first identify an unknown and a known. You may not actually know the exact value of the “known,”but you will know at least something about its value. Examples Max is three years older than Ricky. Unknown = Ricky’s age = x. Known = Max’s age is three years older. Therefore, Ricky’s age = x and Max’s age = x + 3. Lisa made twice as many cookies as Rebecca. Unknown = number of cookies Rebecca made = x. Known = number of cookies Lisa made = 2x. Cordelia has five more than three times the number of books that Becky has. Unknown = the number of books Becky has = x. Known = the number of books Cordelia has = 3x + 5.  Ratio A ratio is a comparison of a two quantities measured in the same units. It can be symbolized by the use of a colon—x:y or ᎏ x y ᎏ or x to y. Ratio problems can be solved using the concept of multiples. Example A bag containing some red and some green can- dies has a total of 60 candies in it. The ratio of the number of green to red candies is 7:8. How many of each color are there in the bag? From the problem, it is known that 7 and 8 share a multiple and that the sum of their prod- uct is 60. Therefore, you can write and solve the following equation: 7x + 8x = 60 15x = 60 ᎏ 1 1 5 5 x ᎏ = ᎏ 6 1 0 5 ᎏ x = 4 Therefore, there are 7x = (7)(4) = 28 green candies and 8x = (8)(4) = 32 red candies.  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 = 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 The average is 6. (Divide by 5 because there are 5 numbers in the set.) sum of the number set ᎏᎏᎏ quantity of set – DATA ANALYSIS, STATISTICS, AND PROBABILITY– 418 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, 3, 5, 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 median 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 often.  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. Example .45 = 45% .07 = 7% .9 = 90% .085 = 8.5% ■ 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% ■ To change a decimal to a percentage, move the decimal point two units to the right and add a percentage symbol. ■ 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 percentage to a fraction, put the per- cent over 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 familiar with. The order is from most common to less common. 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% – DATA ANALYSIS, STATISTICS, AND PROBABILITY– 419  Calculating Interest Interest is a fee paid for the use of someone else’s money. If you put money in a savings account, you receive inter- est from the bank. If you take out a loan, you pay inter- est to the lender. The amount of money you invest or borrow is called the principal. The amount you repay is the amount of the principal plus the interest. The formula for simple interest is found on the for- mula sheet in the GED. Simple interest is a percent of the principal multiplied by the length of the loan: Interest = principal × rate × time Sometimes, it may be easier to use the letters of each as variables: I = prt Example Michelle borrows $2,500 from her uncle for three years at 6% simple interest. How much interest will she pay on the loan? Step 1: Write the interest as a decimal. 6% = 0.06 Step 2: Substitute the known values in the formula I = prt and multiply. = $2,500 × 0.06 × 3 = $450 Michelle will pay $450 in interest. Some problems will ask you to find the amount that will be paid back from a loan. This adds an additional step to problems of interest. In the previous example, Michelle will owe $450 in interest at the end of three years. However, it is important to remember that she will pay back the $450 in interest as well as the principal, $2,500. Therefore, she will pay her uncle $2,500 + $450 = $2,950. In a simple interest problem, the rate is an annual, or yearly, rate. Therefore, the time must also be expressed in years. Example Kai invests $4,000 for nine months. Her invest- ment will pay 8%. How much money will she have at the end of nine months? Step 1: Write the rate as a decimal. 8% = 0.08 Step 2: Express the time as a fraction by writing the length of time in months over 12 (the number of months in a year). 9 months = ᎏ 1 9 2 ᎏ = ᎏ 3 4 ᎏ year Step 3: Multiply. I = prt = $4,000 × 0.08 × ᎏ 3 4 ᎏ = $180 Kai will earn $180 in interest.  Probability Probability is expressed as a fraction and measures the likelihood that a specific event will occur. To find the probability of a specific outcome, use this formula: Probability of an event = Example If a bag contains 5 blue marbles, 3 red marbles, and 6 green marbles, find the probability of selecting a red marble: Probability of an event = = ᎏ 5+ 3 3+6 ᎏ Therefore, the probability of selecting a red marble is ᎏ 1 3 4 ᎏ . Helpful Hints about Probability ■ If an event is certain to occur, the probability is 1. ■ If an event is certain not to occur (impossible), the probability is 0. ■ If you know the probability of all other events occurring, you can find the probability of the remaining event by adding the known probabili- ties together and subtracting their total from 1. Number of specific outcomes ᎏᎏᎏᎏ Total number of possible outcomes Number of specific outcomes ᎏᎏᎏᎏ Total number of possible outcomes – DATA ANALYSIS, STATISTICS, AND PROBABILITY– 420 . Percentage ᎏ 1 2 ᎏ .5 50% ᎏ 1 4 ᎏ .25 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% – DATA ANALYSIS, STATISTICS, AND PROBABILITY– 419  Calculating. percentage symbol. Examples 64 % = .64 87% = .87 7% = .07 ■ To change a percentage to a fraction, put the per- cent over 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 =. the quantity of numbers in the set. Average = 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 The average is 6. (Divide by 5 because there are 5 numbers in the set.) sum