Math Review: Number Forms, Relationships, and Sets OVERVIEW • Percents, fractions, and decimals • Simplifying and combining fractions • Decimal place values and operations • Simple percent problems • Percent increase and decrease • Ratios and proportion • Altering fractions and ratios • Ratios with more than two quantities • Proportion problems with variables • Arithmetic mean, median, mode, and range • Standard deviation • Geometric sequences • Arithmetic sequences • Permutations • Combinations • Probability • Summing it up In this chapter, you’ll focus first on various forms of numbers and relationships between numbers. Specifically, you’ll learn how to: • Combine fractions using the four basic operations • Combine decimal numbers by multiplication and division • Compare numbers in percentage terms • Compare percent changes with number changes • Rewrite percents, fractions, and decimal numbers from one form to another chapter 9 203 • Determine ratios between quantities and determine quantities from ratios • Set up equivalent ratios (proportions) Next, you’ll explore the following topics, all of which involve sets (defined groups) of numbers or other objects: • Simple average and median (two ways that a set of numbers can be described as a whole) • Arithmetic sequences (the pattern from one number to the next in a linear list of numbers) • Permutations (the possibilities for arranging a set of objects) • Combinations (the possibilities for selecting groups of objects from a set) • Probability (the statistical chances of a certain event, permutation, or combination occurring) PERCENTS, FRACTIONS, AND DECIMALS Any real number can be expressed as a fraction, a percent, or a decimal number. For instance, 2 10 , 20%, and 0.2 are all different forms of the same quantity or value. GMAT math questions often require you to rewrite one form as another as part of solving the problem at hand. You should know how to write any equivalent quickly and confidently. To rewrite a percent as a decimal, move the decimal point two places to the left (and drop the percent sign). To rewrite a decimal as a percent, move the decimal point two places to the right (and add the percent sign). 95% 5 0.95 0.004 5 0.4% To rewrite a percent as a fraction, divide by 100 (and drop the percent sign). To rewrite a fraction as a percent, multiply by 100 (and add the percent sign). Percents greater than 100 are equivalent to numbers greater than 1. 810% 5 810 100 5 81 10 5 8 1 10 3 8 5 300 8 % 5 75 2 % 5 37 1 2 % Beware: Percents greater than 100 or less than 1 (such as 457% and 0.067%) can be confusing, because it’s a bit harder to grasp their magnitude. To guard against errors when writing, keep in mind the general magnitude of the number you’re dealing with. For example, think of 0.09% as just less than 0.1%, which is one-tenth of a percent, or a thousandth (a pretty small valued number). Think of 0.45 5 as just less than 0.5 5 , which is obviously 1 10 , or 10%. Think of 668% as more than 6 times a complete 100%, or between 6 and 7. 204 PART IV: GMAT Quantitative Section ALERT! Although this is the most basic of all the math review chapters in this book, don’t skip it. The skills covered here are basic building blocks for other, more difficult types of questions covered in the following chapters. www.petersons.com To rewrite a fraction as a decimal, simply divide the numerator by the denominator, using long division. A fraction-to-decimal equivalent might result in a precise value, an approximation with a repeating pattern, or an approximation with no repeating pattern: 5 8 5 0.625 The equivalent decimal number is precise after three decimal places. 5 9 ' 0.555 The equivalent decimal number can only be approximated (the digit 5 repeats indefinitely). 5 7 ' 0.714 The equivalent decimal number can safely be approximated. Certain fraction-decimal-percent equivalents show up on the GMAT more often than others. The numbers in the following tables are the test makers’ favorites because they reward test takers who recognize quick ways to deal with numbers. Memorize these conversions so that they’re second nature to you on exam day. Percent Decimal Fraction 50% 0.5 1 2 25% 0.25 1 4 75% 0.75 3 4 10% 0.1 1 10 30% 0.3 3 10 70% 0.7 7 10 90% 0.9 9 10 33 1 3 % 0.33 1 3 1 3 66 2 3 % 0.66 2 3 2 3 Percent Decimal Fraction 16 2 3 % 0.16 2 3 1 6 83 1 3 % 0.83 1 3 5 6 20% 0.2 1 5 40% 0.4 2 5 60% 0.6 3 5 80% 0.8 4 5 12 1 2 % 0.125 1 8 37 1 2 % 0.375 3 8 62 1 2 % 0.625 5 8 87 1 2 % 0.875 7 8 Chapter 9: Math Review: Number Forms, Relationships, and Sets 205 www.petersons.com SIMPLIFYING AND COMBINING FRACTIONS A GMAT question might ask you to combine fractions using one or more of the four basic operations (addition, subtraction, multiplication, and division). The rules for combining fractions by addition and subtraction are very different from the ones for multiplication and division. Addition and Subtraction and the LCD To combine fractions by addition or subtraction, the fractions must have a common denominator. If they already do, simply add (or subtract) numerators. If they don’t, you’ll need to find one. You can always multiply all of the denominators together to find a common denominator, but it might be a big number that’s clumsy to work with. So instead, try to find the least (or lowest) common denominator (LCD) by working your way up in multiples of the largest of the denominators given. For denominators of 6, 3, and 5, for instance, try out successive multiples of 6 (12, 18, 24 ),andyou’ll hit the LCD when you get to 30. 1. 5 3 2 5 6 1 5 2 5 (A) 15 11 (B) 5 2 (C) 15 6 (D) 10 3 (E) 15 3 The correct answer is (D). To find the LCD, try out successive multiples of 6 until you come across one that is also a multiple of both 3 and 2. The LCD is 6. Multiply each numerator by the same number by which you would multiply the fraction’s denominator to give you the LCD of 6. Place the three products over this common denominator. Then, combine the numbers in the numerator. (Pay close attention to the subtraction sign!) Finally, simplify to lowest terms: 5 3 2 5 6 1 5 2 5 10 6 2 5 6 1 15 6 5 20 6 5 10 3 206 PART IV: GMAT Quantitative Section www.petersons.com Multiplication and Division To multiply fractions, multiply the numerators and multiply the denominators. The denominators need not be the same. To divide one fraction by another, multiply by the reciprocal of the divisor (the number after the division sign). Multiplication: 1 2 3 5 3 3 1 7 5 ~1!~5!~1! ~2!~3!~7! 5 5 42 Division: 2 5 3 4 5 2 5 3 4 3 5 ~2!~4! ~5!~3! 5 8 15 To simplify the multiplication or division, cancel factors common to a numerator and a denominator before combining fractions. It’s okay to cancel across fractions. Take, for instance the operation 3 4 3 4 9 3 3 2 . Looking just at the first two fractions, you can cancel out 4 and 3, so the operation simplifies to 1 1 3 1 3 3 3 2 . Now, looking just at the second and third fractions, you can cancel out 3 and the operation becomes even simpler: 1 1 3 1 1 3 1 2 5 1 2 . Apply the same rules in the same way to variables (letters) as to numbers. 2. 2 a 3 b 4 3 a 5 3 8 c 5 ? (A) ab 4c (B) 10b 9c (C) 8 5 (D) 16b 5ac (E) 4b 5c The correct answer is (E). Since you’re dealing only with multiplication, look for factors and variables (letters) in any numerator that are the same as those in any denominator. Canceling common factors leaves 2 1 3 b 1 3 1 5 3 2 c Multiply numerators and denominators and you get 4b 5c Chapter 9: Math Review: Number Forms, Relationships, and Sets 207 www.petersons.com Mixed Numbers and Multiple Operations A mixed number consists of a whole number along with a simple fraction—for example, the number 4 2 3 . Before combining fractions, you might need to rewrite a mixed number as a fraction. To do so, follow these three steps: 1. Multiply the denominator of the fraction by the whole number. 2. Add the product to the numerator of the fraction. 3. Place the sum over the denominator of the fraction. For example, here’s how to rewrite the mixed number 4 2 3 into a fraction: 4 2 3 5 ~3!~4!12 3 5 14 3 To perform multiple operations, always perform multiplication and division before you perform addition and subtraction. 3. 4 1 2 1 1 8 2 3 2 3 5 ? (A) 1 3 (B) 3 8 (C) 11 6 (D) 17 6 (E) 11 2 The correct answer is (A). First, rewrite all mixed numbers as fractions. Then, eliminate the complex fraction by multiplying the numerator fraction by the reciprocal of the denominator fraction (cancel across fractions before multiplying): 9 2 9 8 2 11 3 5 S 9 2 DS 8 9 D 2 11 3 5 S 1 1 DS 4 1 D 2 11 3 5 4 1 2 11 3 Then, express each fraction using the common denominator 3, then subtract: 4 1 2 11 3 5 12 2 11 3 5 1 3 208 PART IV: GMAT Quantitative Section www.petersons.com DECIMAL PLACE VALUES AND OPERATIONS Place value refers to the specific value of a digit in a decimal. For example, in the decimal 682.793: The digit 6 is in the “hundreds” place. The digit 8 is in the “tens” place. The digit 2 is in the “ones” place. The digit 7 is in the “tenths” place. The digit 9 is in the “hundredths” place. The digit 3 is in the “thousandths” place. So you can express 682.793 as follows: 600 1 80 12 1 7 10 1 9 100 1 3 1,000 To approximate, or round off, a decimal, round any digit less than 5 down to 0, and round any digit greater than 5 up to 0 (adding one digit to the place value to the left). The value of 682.793, to the nearest hundredth, is 682.79. The value of 682.793, to the nearest tenth, is 682.8. The value of 682.793, to the nearest whole number, is 683. The value of 682.793, to the nearest ten, is 680. The value of 682.793, to the nearest hundred, is 700. Multiplying Decimals The number of decimal places (digits to the right of the decimal point) in a product should be the same as the total number of decimal places in the numbers you multiply. So to multiply decimals quickly, follow these three steps: Multiply, but ignore the decimal points. Count the total number of decimal places among the numbers you multiplied. Include that number of decimal places in your product. Here are two simple examples: Example 1 (23.6)(0.07) 3 decimal places altogether (236)(7) 5 1652 Decimals temporarily ignored (23.6)(0.07) 5 1.652 Decimal point inserted Chapter 9: Math Review: Number Forms, Relationships, and Sets 209 www.petersons.com Example 2 (0.01)(0.02)(0.03) 6 decimal places altogether (1)(2)(3) 5 6 Decimals temporarily ignored (0.01)(0.02)(0.03) 5 0.000006 Decimal point inserted Dividing Decimals When you divide (or compute a fraction), you can move the decimal point in both numbers by the same number of places either to the left or right without altering the quotient (value of the fraction). Here are three related examples: 11.4 4 0.3 5 11.4 0.3 5 114 3 5 38 1.14 4 3 5 1.14 3 5 114 300 5 0.38 114 4 0.003 5 114 0.003 5 114,000 3 5 38,000 GMAT questions involving place value and decimals usually require a bit more from you than just identifying a place value or moving a decimal point around. Typically, they require you to combine decimals with fractions or percents. 4. Which of the following is nearest in value to 1 3 3 0.3 3 1 30 3 0.03? (A) 1 10,000 (B) 33 100,000 (C) 99 100,000 (D) 33 10,000 (E) 99 10,000 The correct answer is (A). There are several ways to convert and combine the four numbers provided in the question. One method is to combine the two fractions: 1 3 3 1 30 5 1 90 . Then, combine the two decimals: 0.3 3 0.03 5 0.009 5 9 1000 . Finally, combine the two fractions: 1 90 3 9 1000 5 9 90,000 5 1 10,000 which is choice (A). 210 PART IV: GMAT Quantitative Section TIP Eliminate decimal points from fractions, as well as from percents, to help you see more clearly the magnitude of the quantity you’re dealing with. www.petersons.com SIMPLE PERCENT PROBLEMS On the GMAT, a simple problem involving percent might ask you to perform any one of these four tasks: Find a percent of a percent Find a percent of a number Find a number when a percent is given Find what percent one number is of another The following examples show you how to handle these four tasks (task 4 is a bit trickier than the others): Finding a percent of a percent What is 2% of 2% ? Rewrite 2% as 0.02, then multiply: 0.02 3 0.02 5 0.0004, or 0.04% Finding a percent of a number What is 35% of 65? Rewrite 35% as 0.35, then multiply: 0.35 3 65 5 22.75 Finding a number when a percent is given 7 is 14% of what number? Translate the question into an algebraic equation, writing the percent as either a fraction or decimal: 7 5 14% of x 7 5 0.14x x 5 7 0.14 5 1 0.02 5 100 2 5 50 Finding what percent one number is of another 90 is what % of 1500? Set up an equation to solve for the percent: 90 1500 5 x 100 1500x 5 9000 15x 5 90 x 5 90 15 ,or6 Chapter 9: Math Review: Number Forms, Relationships, and Sets 211 www.petersons.com PERCENT INCREASE AND DECREASE In example 4, you set up a proportion. (90 is to 1500 as x is to 100.) You’ll need to set up a proportion for other types of GMAT questions as well, including questions about ratios, which you’ll look at a bit later in this chapter. The concept of percent change is one of the test makers’ favorites. Here’s the key to answering questions involving this concept: Percent change always relates to the value before the change. Here are two simple illustrations: 10 increased by what percent is 12? 1. The amount of the increase is 2. 2. Compare the change (2) to the original number (10). 3. The change in percent is S 2 10 D ~100!520, or 20%. 12 decreased by what percent is 10? 1. The amount of the decrease is 2. 2. Compare the change (2) to the original number (12). 3. The change is 1 6 ,or16 2 3 %, or approxi- mately 16.7%. Notice that the percent increase from 10 to 12 (20%) is not the same as the percent decrease from 12 to 10 S 16 2 3 % D . That’s because the original number (before the change) is different in the two questions. A typical GMAT percent-change problem will involve a story—about a type of quantity such as tax, profit or discount, or weight—in which you need to calculate successive changes in percent. For example: • An increase, then a decrease (or vice versa) • Multiple increases or decreases Whatever the variation, just take the problem one step at a time and you’ll have no trouble handling it. 5. A stereo system originally priced at $500 is discounted by 10%, then by another 10%. If a 20% tax is added to the purchase price, how much would a customer buying the system at its lowest price pay for it, including tax, to the nearest dollar? (A) $413 (B) $480 (C) $486 (D) $500 (E) $512 212 PART IV: GMAT Quantitative Section www.petersons.com . your product. Here are two simple examples: Example 1 (23. 6)(0.07) 3 decimal places altogether (236 )(7) 5 1652 Decimals temporarily ignored (23. 6)(0.07) 5 1.652 Decimal point inserted Chapter. obviously 1 10 , or 10%. Think of 668% as more than 6 times a complete 100%, or between 6 and 7. 204 PART IV: GMAT Quantitative Section ALERT! Although this is the most basic of all the math review. sign!) Finally, simplify to lowest terms: 5 3 2 5 6 1 5 2 5 10 6 2 5 6 1 15 6 5 20 6 5 10 3 206 PART IV: GMAT Quantitative Section www.petersons.com Multiplication and Division To multiply fractions,