322 Data Interpretation Questions B Alaska is almost 600,000 square miles, which 13 B is about i of 3,660,000 square miles ‘ is C 14 A 7,500 is in the 6,000—8 ,000 bracket so the tax will be 80 + 3% of the income over 6,000 Since 7,500 — 6,000 = 1,500, the income over 6,000 is 1,500 3% of 1500 = The tax on 26,000 is 1,070 + 7% — 25,000) Thus, the tax is 1,070 1,140 The tax on 29,000 is 1,070 (29,000 — 25,000) Thus, the tax 1,070 + 280 = 1,350 Therefore, of (26,000 + 70 = + 7% of on 29,000 is you will pay force decreased from 1947 to 1948 III cannot be inferred since there is no information about the total labor force or women as a percent of it in 1965 Thus, only I can be inferred 15 E Look in the fourth column l6 B In 1972 there were 72 million females out of 136 million persons of voting age $3,000 raise is income over 25,000, so it will be taxed at 7% Therefore, the tax on the extra $3,000 will be (0.07)(3,000) = 210 C If income is less than 6,000, then the tax is less than 80 If income is greater than 8,000, then the tax is greater than 140 Therefore, if the tax is 100, the income must be between 6,000 and 8,000 You not have to calculate Joan’s exact income 17 B 18 B 220 The tax on 10,000 is 220, so taxes are 10,000 66 million The closest answer among the choices is 65 million 11 The train’s speed increased by 70 — 40 which 30 is 30 miles per hour 40 is 75% 21 When t = QO, the speed is 40, so A and B are incorrect When ¢t = 180, the speed is 70, so C D and E are incorrect Choice D gives all the values that appear in the table 22 A The cost of food A is $1.80 per hundred grams or 1.8¢ a gram, so x grams cost (1.8x)¢ or ()< Each gram of food B costs 3ý so y In 1947, there were about 16 million women in the labor force, and about 14 — 6o0r million grams of food B will cost 3y¢ Each gram of women in the labor force who were married is food C costs 2.75¢ or he: thus, z grams of of them were married Therefore, the percent of i or 50% 12 l 20 C the labor force Using the line graph, there were about 22 million women in the labor force in 1960 So the labor force was about 3(22) or the North and West in 1964, and there were 65 25 hours is 150 minutes rounding to the nearest percent In 1960 women made up 33.4% or about of Since 78 million persons of voting age lived in 19 D = (0.022 = 2.2% of income 2.2% is 2% after 10 B age voted, and (0.7)(54,000,000) = million persons of voting age not in the 25—44 year range, there must be at least 78 — 65 = 13 million people in the North and West in the 25-44 year range X must be greater than or equal to 13 Since there were 45 million people of voting age in the 25—44 year range, X must be less than or equal to 45 1% of 3,700 or $37 Since there are 50,000 | In 1968, 70% of the 54 million males of voting 37,800,000 Each person pays the tax on $3,700, which is people in Zenith, the total taxes are (37)(50,000) = $1,850,000 = 0.529, which is 53% to the nearest percent 1,350 — 1,140 = $210 more in taxes next year A faster method is to use the fact that the I is true since the width of the band for widowed or divorced women was never more than million between 1947 and 1957 II is false since the number of single women in the labor (0.03)(1500) = 45, so the tax is 80 + 45 = 125 D the labor force By 1972 there were about 32 million Therefore, the number of women doubled, which is an increase of 100% (not of 200%) 163% so the correct answer is 15% Save time by estimating; don’t perform the calculations exactly In 1947, there were about 16 million women in Look at the possible answers first You can use your pencil and admission card as straight edges 1] food C will cost (1) cost is (2): + 3y + ý Therefore, the total 1] (1);|› Data Interpretation Questions E Since Food A is 10% protein, 500 grams of A will supply 50 grams of protein Food B is 20% protein, so 250 grams of B will supply 50 grams of protein 350 grams of Food C will supply 70 grams of protein 150 grams of Food A and 200 grams of Food B will supply 15 + 40 = 55 grams of protein 200 grams of Food B and 200 grams of Food C will supply 40 + 40 or 80 grams of protein Choice E supplies the most protein 24 E 323 The diet of Choice A will cost 2($1.80) + (3)s» = $3.60 + $4.50 = $8.10 Choice B will cost 5($3) + $1.80 = $16.80 ChoiceC costs 2($2.75) = $5.50 Choice D costs (3) s1.80 + $2.75 = $2.70 + $2.75 = $5.45 The diet of Choice E costs 3($1.80) or $5.40 so Choice E costs the least Mathematics Review @ Mi Mi Mi Arithmetic Algebra Geometry Tables and Graphs Mi Formulas The mathematics questions on the GRE General Test require a working knowledge of mathematical principles, including an understanding of the fundamentals of algebra, plane geometry, and arithmetic, as well as the ability to translate problems into formulas and to interpret graphs The following review covers these areas thoroughly and will prove helpful Read through the review carefully You will notice that each topic is keyed for easy reference Each of the Practice Review and Practice I I-A Arithmetic Whole Numbers The numbers 1, 2, 3, are called the posifive integers —†1,—2, —-3, are called the negative integers An integer is a positive or negative integer or the number A-2 324 Review the tactics in the preceding chapters for testtaking help lf kis a factor of m, then there is another integer n such that m= kx n; inthis case, mis called a multiple of k Since 12 = x 3, 12 is a multiple of and also 12 is a multiple of For example, 5, 10, 15, and 20 are all multi- ples of but 15 and are not multiples of 10 Any integer is a multiple of each of its factors A¬ If the integer k ble by k or k is by 4, but 12 is 2, 3, 4, 6, and Exercises in this chapter, as well as the Diagnostic and five Model Tests, are keyed in the same manner Therefore, after working the mathematics problems in each area, you should refer to the answer key and follow the mathematics reference key so that you can focus on the topics where you need improvement divides m evenly, then we say mis divisia factor of m For example, 12 is divisible not divisible by The factors of 12 are 1, 12 A-3 Any whole number is divisible by itself and by If pis a whole number greater than 1, which has onlyp and as factors, then pis called a prime number 2, 3, 5, 7, 11, 13, 17, 19 and 23 are all primes 14 is not a prime since it is divisible by and by A whole number that is divisible by is called an even number; if a whole number is not even, then it is an odd number 2, 4, 6, 8, and 10 are even numbers, and 1, 3, 5, 7, and are odd numbers Mathematics Review A-4 Any integer greater than is a prime or can be written as a product of primes To write a number as a product of prime factors: O 325 O Write kasa product of primes and j as a product of primes @ if there are any common factors delete them in one of the products © Multiply the remaining factors; the result is the least common multiple Divide the number by if possible; continue to divide by until the factor you get is not divisible by Find the L.C.M of 27 and 63 @ Divide the result from @ by if possible; continue to divide by until the factor you get is not divisible by ® Divide the result from @ by if possible; continue to divide by until the factor you get is not divisible by © Continue the procedure with 7, 11, and so on, until all the factors are primes @ 3x 3-=9 is acommon factor so delete it once © TheL.C.M.is3x3x3x7= 189 You can find the L.C.M of a collection of numbers in the Express 24 as a product of prime factors same way except that, if in step (B) the common factors are factors of more than two of the numbers, then delete the common factor in al// but one of the products O 24=2x12,12=2x6,6=2x3s024=2x2 x x Since each factor (2 and 3) is prime, 24=2x2x2x3 It takes Eric 20 minutes to inspect a car John needs only 15 minutes to inspect a car If they both Start inspecting cars at 9:00 a.m., what is the first time the two mechanics will finish inspecting a car at the same time? Aclass of 45 students will be seated in rows Every row will have the same number of students There must be at least two students in each row, and there must be at least two rows A row is parallel to the front of the room How many different arrangements are possible? Since the number of students = (the number of rows)(the number of students in each row) and the number of students is 45, the question can be answered by finding how many different ways 45 can be written as a product of two positive integers that are both greater than (The integers must be greater than because there are at least two rows and at least two students per row.) Writing 45 as a product of primes makes this easy 45 = x 15 = x 3x Therefore, x 15, x9, x 5, and 15 x are the only possibilities, and the correct answer is (The fact that a row is parallel to the front of the room means that x 15 and 15 x are different arrangements.) A-5 Eric will finish k cars after k x 20 minutes, and John will finish j cars after / x 15 minutes Therefore, they will both finish inspecting a car at the same time when k x 20 = jx 15 Since k and / must be integers (they represent the number of cars finished) this question asks you to finda common multiple of 20 and 15 Since you are asked for the first time the two mechanics will finish at the same time, you must find the least common multiple @ 20=-4x5=2x2x5,15=3x5 € Delete from one of the products €© TheL.C.M is2 x2 x5 x 3= 60 Eric and John will finish inspecting a car at the same time 60 minutes after they start, or at 10:00 A.M A-6 Anumber mis a common multiple of two other numbers k and {if it is a multiple of each of them For example, 12 is acommon multiple of and 6, since3x4=12and2x6 = 12.15 is nota common multiple of and 6, because 15 is not a multiple of A number kis a common factor of two other numbers and nif kis a factor of m and k is a factor of n O 27=3x3x3,63=3x3x7 m The least common multiple (L.C.M.) of two numbers is the smallest number that is a common multiple of both numbers To find the least common multiple of two numbers k and /: The numbers 0, 1, 2, 3, 4, 5, 6, 7, and are called digits The number 132 is a three-digit number In the number 132, is the first or hundreds digit, is the second or tens digit, and is the last or units digit Find x ifx is a two-digit number whose last digit is The difference of the digits of x is The two-digit numbers whose last digit is are 12, 22, 32, 42, 52, 62, 72, 82, and 92 The difference of the digits of 12 is either or -—1,S0 12 is not x Since —2 1s 5, xis 72 326 Mathematics Review Ï-B Fractions B-1 A FRACTION is a number that represents a ratio or division of two numbers A fraction is written in the form number on the top, ber on the bottom, tor tells how many parts of a pie); the The a, is called the numerator; the numb, is the denominator The denominaequal parts there are (for example, numerator tells how many of these equal parts are taken For example, is a fraction whose QO 4.7=28 @ 28:+1 = 29 © zi | A fraction whose numerator is larger than its denominator can be changed into a mixed number @ Divide the denominator into the numerator; the result is the whole number of the mixed number @ Put the remainder from step @ over the denominator: this is the fractional part of the mixed number numerator is and whose denominator is 8; it represents taking of equal parts, or dividing into If a pizza pie has pieces, how many pizza pies have been eaten at a party where 35 pieces were eaten? A fraction cannot have as a denominator since division by is not defined A fraction with as the denominator is the same as the whole number that is its numerator For example, _ 29 " is 12, : is O Since there are pieces in a pie, = pies were eaten To find the number of pies, we need to change If the numerator and denominator of a fraction are identi- mixed : into a number cal, the fraction represents For example, @ Divide into 35: the result is with a remainder of 3.9 _13_ Any whole number, k, is represented 13 by a fraction with a numerator equal to k times the @ : is the fractional part of the mixed number denominator For example, © - 43 8 = 3, and = = B-2 In calculations with mixed numbers, change the mixed numbers into fractions Mixed Numbers A mixed number consists of a whole number and a fraction For example, r2 ber; it means + and is a mixed num- ; is called the fractional part of the mixed number 72 Any mixed number can be B-3 To multiply two fractions, multiMultiplying Fractions ply their numerators to form the numerator of the product Multiply their denominators to form the denominator of the product changed into a fraction as follows: @ Multiply the whole number by the denominator of the John saves of $240 How much does he save? fraction @ Add the numerator of the fraction to the result of O © Use the result of @ as the numerator, and use the denominator of the fractional part of the mixed number as the denominator This fraction is equal to the mixed number Write 75 as a fraction 1240 _ 240 _ $80, the amount John saves B-4 To divide one fraction (the diviDividing Fractions dend) by another fraction (the divisor), invert the divisor and multiply To invert a fraction turn it upside down; for example, if you invert , the result is NO © 25 Á 76 _ 25 _ 25 96 32 — œ A worker makes a basket in § of an hour If she works for 75 hours, how many baskets will she make? It takes of an hour to make divide into 71 Since 7! 2 by Since you want to work as fast as possible on the GRE, cancel whenever you can one basket, so we need to - 19 , We want to divide 15 2 — Thus ¥ C3lIF 15 —" = 111 baskets B-6 Equivalent Fractions Two fractions are equivalent or equal if they represent the same ratio or number In Section B—5, you saw that, if you multiply or divide the numerator and denominator of a fraction by the same nonzero number, the result is equivalent to the original fraction For example, B-5 If you multiply the numerator and denominator of a fraction by the same nonzero number, the value of the fraction remains the same If you divide the numerator and denominator of any fraction by the same nonzero number, the value of the fraction remains the same Consider the fraction ` lÍ we multiply by 10 and by (In 30 40 @ 1,5 2 Since is a common and 75, divide and 75 by to get Th erefore f 75 358 — *° ——— —= _— 25 3a - —- Divide the denominator of the given fraction into the known denominator @ Multiply the result of @ by the numerator of the given fraction; this is the numerator of the required equivalent fraction 10 is acommon Since is acommon factor of and 8, divide and by = since 70 = 10 x and 80 = 10 x Your answer 1s ễ One of the test answers has a Multiply 75 = so To find a fraction with a known denominator equal to a given fraction: When we multiply fractions, if any of the numerators and denominators have a common factor (see A—2 for factors) we can divide each of them by the common factor and the fraction remains the same This process Is called cancelling and can be a great time-saver 4, getting In a multiple-choice test, your answer to a problem may not be the same as any of the given choices, yet one choice may be equivalent Therefore, you may have to express your answer as an equivalent fraction Dividing and Multiplying by the Same Number 10, then 30 must be equal 40 factor of 30 and 40.) 327 would be written as follows: iI wl œi Ơn iI œ›I Ơn +>i C2 Mathematics Review = 25 — Am factor of 1,25, , Cancelling is denoted by striking or crossing out the appropriate numbers For instance, the example above denominator of 30 Find a fraction with denomina- tor 30 that is equal to @ into 30 ¡s @ 6.2= 12,50 12 _ Ê 30 Check your result Divide numerator and denominator by the same number 12 + 6=2 and 30+6=5 B-7 Reducing a Fraction to Lowest Terms A fraction has been reduced to lowest terms when the numerator and denominator have no common factors For example, : is reduced to lowest terms, but : is not because is a common factor of and 328 Mathematics Review Once you have found a common denominator, express each fraction as an equivalent fraction with the common denominator, and add as you did when the fractions had the same denominator To reduce a fraction to lowest terms, cancel all the common factors of the numerator and denominator (Canceling common factors will not change the value of the fraction.) 40 400 _2 18 @ denominator _ 12 _ 16 243 244 Since and have no common factors, e jg 100 150 reduced to lowest terms A fraction is equivalent to its reduction to lowest terms Another way to cancel common factors, and hence reduce to lowest terms, is to first write the numerator and denominator as the products of primes B-8 Adding Fractions 24 is a common @ 1,2,7_ 2°3°4° 70 _ 35 24 12 +: Nl G)I t9 For example: _ 42 24 12,16, 42 _ 12+16+42_ 14°24" 24 24 B-9 Subtracting Fractions When the fractions have the same denominator, subtract the numerators and place the result over the denominator If the fractions have the same There are five tacos in a lunch box Jim eats two of denominator, then the denominator is called a common denominator Add the numerators, and use this sum as the tacos What fraction of the original tacos are left in the lunch box? the new numerator, retaining the common denominator as the denominator of the new fraction Reduce the new fraction to lowest terms The worker used of a box in the shipping department and 2š of a box in the accounting department used was LZ 24 „ l3 24 - 30 24 _° =11 The total boxes as ‘then (5-2) _„ —5 5 =~-= = 5 is a common + + a note that2-3-4 = 24 denominator There are many common denominators; the smallest one is called the /east common denominator For the preceding example, 12 is the least common denominator = of the Original tacos are left in the lunch box When the fractions have different denominators @ Find a common denominator € Express the fractions as equivalent fractions with the same denominator © Subtract Wn} Go If the fractions don’t have the same denominator, you must first find a common denominator One way to get a common denominator is to multiply the denominators together For example, to find are left Write1 Therefore, lÍ » light bulbs contains 24 bulbs A worker 17 bulbs in the shipping department and in the accounting department How many bulbs did he use? a A box of replaces 13 bulbs boxes of | Œ Ơn Jim took : of the original tacos, so -ã @ Acommon denominator is 5-7 = 35 @3_21 2_10 @ 35 3_2.21_ 35 35 10 35 _ 21-10 35 = 11 35 Mathematics Review B-10 What fraction does 0.503 represent? Complex Fractions A fraction whose numerator and denominator are themselves fractions is called a complex fraction For example, @ is a complex fraction A com- There are three digits to the right of the decimal point, so the denominator is 10 x 10 x 10 = 1,000 @ The numerator is 503, so the fraction is 329 plex fraction can always be simplified by dividing its numerator by its denominator 503 1,000 Find the fraction that 0.05732 represents @ There are five digits to the right of the decimal point, Simplify nar — so the denominator is 10 x 10x 10x 10x 10 = 100,000 5,372 @ The numerator is 5,732, so the fraction is 22/ 4+ 4q* + 4a — 16, these factors are Factor out a, so a? + 4a° + 4a= a(a* + 4a+ 4) Consider a*+4a+4: since 2+2=4 and x =4, the factors are (a + 2) and (a + 2) Therefore, a3 + 4a“ + 4a = a(a + 2)“ If the term with the highest exponent has a coefficient unequal to 1, divide the entire expression by that coeffi- cient For example, to factor 3a° + 12a* + 12a, factor out from each term, and the result is a? + 4a? + 4a, which is a(a + 2)* Thus, 3a? + 12a° + 12a = 3a(a+t 2)° There are some expressions that cannot be factored, for example, x* + 4x+ In general, if you can’t factor something by using the methods given above, don't waste a lot of time on the question Sometimes you may be able to find the correct factors by checking the answer choices A-6 Division of Algebraic Expressions to remember in division are: @ Factor x44 — 4x” x4y4 =(x?y*)? and 4x = (2x)*, so the factors are x*y* + 2x and x“y* — 2x You also may need to factor expressions that contain squared terms and linear terms, Such as x°+4x+3 The factors will be of the form (x+ a) and (x+ b) Since (x+ a)(x+ b) = x*+ numbers a and the expression term (the term (a+ b)x+ ab, you must look for a pair of b such that a- bis the numerical term in and a+ Dis the coefficient of the linear with exponent 1) When you divide a sum, you can get the same result by dividing each term and adding quotients For example: Jx+4xy+y x @ The main things _ 9X x 4xy You can cancel common x y x -9+4y+#, x factors, so the results on fac- toring will be helpful For example: x? 2x _ x(x-2) x2 x-a2 _ x You can also divide one algebraic expression by another using long division ... was $15 6,650 — $40,000 = $11 6,650 Therefore, the average annual income of the gˆ= 81 10° = 10 0 5° = 25 12 ? ? = 14 4 7* = 49 14 ° = 19 6 1 12 = 12 1 | 13 7 = 16 9 5ˆ = 22 5 13 =1 3= 23 =8 = $14 ,5 81 .25 64 2Z=... products of primes B-8 Adding Fractions 24 is a common @ 1 ,2, 7_ 2? ?3°4° 70 _ 35 24 12 +: Nl G)I t9 For example: _ 42 24 12 , 16 , 42 _ 12 + 16 + 42_ 14 ? ?24 " 24 24 B-9 Subtracting Fractions When the fractions... digit is are 12 , 22 , 32, 42, 52, 62, 72, 82, and 92 The difference of the digits of 12 is either or -? ?1, S0 12 is not x Since ? ?2 1s 5, xis 72  326 Mathematics Review Ï-B Fractions B -1 A FRACTION