1. Column A Column B The number of fifths in 340% The number of eighths in 212.5% (A) The quantity in Column A is greater. (B) The quantity in Column B is greater. (C) The quantities are equal. (D) The relationship cannot be determined from the information given. The correct answer is (C). First, determine Quantity A. 340% is 3.4, or 3 2 5 . Since there are five fifths in 1, Quantity A must be (3)(5) + 2, or 17. Next, determine Quantity B. 212.5% is 2.125, or 2 1 8 . Since there are eight eighths in 1, Quantity B must be (2)(8) + 1, or 17. To guard against conversion errors, keep in mind the general magnitude of the number you’re dealing with. For example, think of .09% as just less than .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. 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 is expressed here to the nearest thousandth. Certain fraction-decimal-percent equivalents show up on the GRE 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. Chapter 9: Math Review: Number Forms, Relationships, and Sets 193 ALERT! You won’t have access to a calculator during the exam, so knowing how to convert numbers from one form to another is a crucial skill. www.petersons.com 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 SIMPLIFYING AND COMBINING FRACTIONS AGRE question might ask you to combine fractions using one or more of the four basic operations (addition, subtraction, multiplication, and division). The rules for com- bining 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. 2. 5 3 2 5 6 1 5 2 5 (A) 15 11 (B) 5 2 (C) 15 6 (D) 10 3 (E) 15 3 PART IV: Quantitative Reasoning194 www.petersons.com 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 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. The variables a and c do not equal 0. 3. 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 195 ALERT! On the GRE, pay very close attention to operation signs. You can easily flub a question by reading a plus s ign (+) as a minus sign (–), or vice versa. 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: Multiply the denominator of the fraction by the whole number. Add the product to the numerator of the fraction. Place the sum over the denominator of the fraction. For example, here’s how to rewrite the mixed number 4 2 3 as 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. 4. 4 1 2 1 1 8 2 3 2 3 is equivalent to what simple fraction? (A) 1 3 (B) 3 8 (C) 11 6 (D) 17 6 (E) 11 2 Enter an integer in the numerator box, and enter an integer in the denominator box. The correct answer is S 1 3 D . 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 Next, express each fraction using the common denominator 3; then subtract: 4 1 2 11 3 5 12 2 11 3 5 1 3 PART IV: Quantitative Reasoning196 NOTE In a GRE numeric-entry question, you don’t need to reduce a fraction to lowest terms to receive credit for a correct answer. So in Question 4, you’d receive credit for 2 6 as well as for 1 3 . 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: 1 (23.6)(0.07) Three decimal places altogether (236)(7) 5 1652 Decimals temporarily ignored (23.6)(0.07) 5 1.652 Decimal point inserted 2 (0.01)(0.02)(0.03) Six decimal places altogether (1)(2)(3) 5 6 Decimals temporarily ignored (0.01)(0.02)(0.03) 5 0.000006 Decimal point inserted Chapter 9: Math Review: Number Forms, Relationships, and Sets 197 TIP Eliminate decimal points from fractions, as well as f rom percents, to help you see more clearly the magnitude of the quantity you’re dealing with. www.petersons.com Dividing Decimal Numbers 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 GRE 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. 5. 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 1,000 . Finally, combine the two resulting fractions: 1 90 3 9 1,000 5 9 90,000 5 1 10,000 which is choice (A). SIMPLE PERCENT PROBLEMS On the GRE, 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%? PART IV: Quantitative Reasoning198 www.petersons.com 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 1,500? Set up an equation to solve for the percent: 90 1,500 5 x 100 1,500x 5 9,000 15x 5 90 x 5 90 15 or 6 PERCENT INCREASE AND DECREASE In the fourth example above, you set up a proportion. (90 is to 1,500 as x is to 100.) You’ll need to set up a proportion for other types of GRE questions as well, including questions about ratios, which you’ll look at in the next section. 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%. Chapter 9: Math Review: Number Forms, Relationships, and Sets 199 www.petersons.com 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 approximately 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 GRE 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. 6. 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 pay who is buying the system at its lowest price, including tax? (A) $413 (B) $480 (C) $486 (D) $500 (E) $512 The correct answer is (C). After the first 10% discount, the price is $450 ($500 minus 10% of $500). After the second discount, which is calculated based on the $450 price, the price of the stereo is $405 ($450 minus 10% of $450). A 20% tax on $405 is $81. Thus, the customer has paid $405 1 $81 5 $486. RATIOS AND PROPORTION A ratio expresses proportion or comparative size—the size of one quantity relative to the size of another. As with fractions, you can simplify ratios by dividing common factors. For example, given a class of 28 students—12 freshmen and 16 sophomores: • The ratio of freshmen to sophomores is 12:16, or 3:4. • The ratio of freshmen to the total number of students is 12:28, or 3:7. • The ratio of sophomores to the total number of students is 16:28, or 4:7. PART IV: Quantitative Reasoning200 NOTE GRE problems involving percent and percent change are often accompanied by a chart, graph, or table. www.petersons.com Finding a Ratio A GRE question might ask you to determine a ratio based on given quantities. This is the easiest type of GRE ratio question. 7. A class of 56 students contains only freshmen and sophomores. If 21 of the students are sophomores, what is the ratio of the number of freshmen to the number of sophomores in the class? (A) 3:5 (B) 5:7 (C) 5:3 (D) 7:4 (E) 2:1 The correct answer is (C). Since 21 of 56 students are sophomores, 35 must be freshmen. The ratio of freshmen to sophomores is 35:21. To simplify the ratio to simplest terms, divide both numbers by 7, giving you a ratio of 5:3. Determining Quantities from a Ratio (Part-to-Whole Analysis) You can think of any ratio as parts adding up to a whole. For example, in the ratio 5:6, 5 parts 1 6 parts 5 11 parts (the whole). If the actual total quantity were 22, you’d multiply each element by 2: 10 parts 1 12 parts 5 22 parts (the whole). Notice that the ratios are the same: 5:6 is the same ratio as 10:12. You might be able to solve a GRE ratio question using this part-to-whole approach. 8. A class of students contains only freshmen and sophomores. If 18 of the students are sophomores, and if the ratio of the number of freshmen to the number of sophomores in the class is 5:3, how many students are in the class? students Enter a number in the box. The correct answer is (48). Using a part-to-whole analysis, look first at the ratio and the sum of its parts: 5 (freshmen) 1 3 (sophomores) 5 8 (total students). These aren’t the actual quantities, but they’re proportion- ate to those quantities. Given 18 sophomores altogether, sophomores account for 3 parts—each part containing 6 students. Accordingly, the total number of students must be 6 3 8 5 48. Determining Quantities from a Ratio (Setting Up a Proportion) Since you can express any ratio as a fraction, you can set two equivalent, or propor- tionate, ratios equal to each other, as fractions. So the ratio 16:28 is proportionate to the ratio 4:7 because 16 28 5 4 7 . If one of the four terms is missing from the equation (the proportion), you can solve for the missing term using algebra. So if the ratio 3:4 is Chapter 9: Math Review: Number Forms, Relationships, and Sets 201 www.petersons.com proportionate to 4:x, you can solve for x in the equation 3 4 5 4 x . Using the cross-product method, equate products of numerator and denominator across the equation: ~3!~x!5~4!~4! 3x 5 16 x 5 16 3 or 5 1 3 If the quantities in a proportion problem strike you as “unround,” it’s a good bet that doing the math will be easier than you might first think. 9. If 3 miles are equivalent to 4.8 kilometers, then 14.4 kilometers are equivalent to how many miles? (A) 18.2 (B) 12.0 (C) 10.6 (D) 9.0 (E) 4.8 The correct answer is (B). The question essentially asks, “3 is to 4.83 as what is to 14.4?” Set up a proportion, then solve for x by the cross-product method: 3 4.8 5 x 14.4 ~4.8!~x!5~3!~14.4! x 5 ~3!~14.4! 4.8 5 14.4 1.2 =12 Notice that, despite all the intimidating decimal numbers, the solution turns out to be a tidy number. That’s typical of the GRE. Altering Fractions and Ratios An average test taker might assume that adding the same positive quantity to a fraction’s numerator (p) and to its denominator (q) leaves the fraction’s value S p q D unchanged. But this is true if and only if the original numerator and denominator were equal to each other. Otherwise, the fraction’s value will change. Remember the following three rules, which apply to any positive numbers x, p, and q: If p 5 q, then p q 5 p 1 x q 1 x . (The fraction’s value remains unchanged and is always 1.) If p . q, then p q . p 1 x q 1 x . (The fraction’s value will decrease.) If p , q, then p q , p 1 x q 1 x . (The fraction’s value will increase.) As you might suspect, this concept makes great fodder for GRE Quantitative Com- parison questions. PART IV: Quantitative Reasoning202 TIP On the GRE, what look like unwieldy numbers typically boil down to simple ones. In fact, your ability to recognize this feature is one of the skills being tested on the GRE. www.petersons.com . Ratio (Part- to-Whole Analysis) You can think of any ratio as parts adding up to a whole. For example, in the ratio 5:6, 5 parts 1 6 parts 5 11 parts (the. whole). If the actual total quantity were 22, you’d multiply each element by 2: 10 parts 1 12 parts 5 22 parts (the whole). Notice that the ratios are the same: