270 MCGRAW-HILL’S SAT Integers and Real Numbers On the SAT, you only need to deal with two kinds of numbers: integers (the positive and negative whole numbers, . . . , Ϫ3, Ϫ2, Ϫ1, 0, 1, 2, 3, . . .) and real num- bers (all the numbers on the number line, including integers, but also including all fractions and deci- mals). You don’t have to know about wacky numbers such as irrationals or imaginaries. The SAT only uses real numbers. It will never (1) divide a number by 0 or (2) take the square root of a negative number because both these operations fail to produce a real number. Make sure that you understand why both these operations are said to be “undefined.” Don’t assume that a number in an SAT prob- lem is an integer unless you are specifically told that it is. For instance, if a question mentions the fact that x > 3, don’t automatically assume that x is 4 or greater. If the problem doesn’t say that x must be an integer, then x might be 3.01 or 3.6 or the like. The Operations The only operations you will have to use on the SAT are the basics: adding, subtracting, multiplying, divid- ing, raising to powers, and taking roots. Don’t worry about “bad boys” such as sines, tangents, or logarithms—they won’t show up. (Yay!) Don’t confuse the key words for the basic op- erations: Sum means the result of addition, dif- ference means the result of subtraction, product means the result of multiplication, and quotient means the result of division. The Inverse Operations Every operation has an inverse, that is, another oper- ation that “undoes” it. For instance, subtracting 5 is the inverse of adding 5, and dividing by Ϫ3.2 is the in- verse of multiplying by Ϫ3.2. If you perform an oper- ation and then perform its inverse, you are back to where you started. For instance, 135 ϫ 4.5 ÷ 4.5 ϭ 135. No need to calculate! Using inverse operations helps you to solve equations. For example, 3x Ϫ 7 ϭ 38 To “undo” Ϫ7, add 7 to both sides: 3x ϭ 45 To “undo” ϫ 3, divide both sides by 3: x ϭ 15 Alternative Ways to Do Operations Every operation can be done in two ways, and one way is almost always easier than the other. For instance, subtracting a number is the same thing as adding the opposite number. So sub- tracting Ϫ5 is the same as adding 5. Also, di- viding by a number is exactly the same thing as multiplying by its reciprocal. So dividing by 2/3 is the same as multiplying by 3/2. When doing arithmetic, always think about your op- tions, and do the operation that is easier! For instance, if you are asked to do 45 ÷ Ϫ1/2, you should realize that it is the same as 45 ϫϪ2, which is easier to do in your head. The Order of Operations Don’t forget the order of operations: P-E-MD- AS. When evaluating, first do what’s grouped in parentheses (or above or below fraction bars or within radicals), then do exponents (or roots) from left to right, then multiplication or division from left to right, and then do ad- dition or subtraction from left to right. What is 4 – 6 ÷ 2 ϫ 3? If you said 3, you mistakenly did the multiplication before the division. (Instead, do them left to right). If you said Ϫ3 or Ϫ1/3, you mistakenly subtracted before taking care of the multiplication and division. If you said Ϫ5, pat yourself on the back! When using your calculator, be careful to use parentheses when raising negatives to powers. For instance, if you want to raise –2 to the 4th power, type “(–2)^4,” and not just “–2^4,” be- cause the calculator will interpret the latter as –1(2)^4, and give an answer of –16, rather than the proper answer of 16. Lesson 1: Numbers and Operations CHAPTER 7 / ESSENTIAL PRE-ALGEBRA SKILLS 271 Concept Review 1: Numbers and Operations 1. What is the greatest integer less than −9.5? 1. __________ 2. Express the answer to Question 1 as a fraction in two different ways. 2. __________ 3. When is taking the square root of a number not the inverse of squaring a number? Be specific. __________________________________________________________________________________________________ __________________________________________________________________________________________________ 4. Four consecutive even integers have a sum of 76. What is the greatest of these numbers? 4. __________ 5. If −2 < x < 2, how many possible values may x have? 5. __________ 6. The result of an addition is called a __________. 7. The result of a subtraction is called a __________. 8. The result of a multiplication is called a __________. 9. What is the difference between the product of 12 and 3 and their sum? 9. __________ What is the alternative way to express each of the following operations? 10. − (6) __________ 11. ÷ (4) __________ 12. __________ 13. __________ What is the inverse of each of the following operations? 14. − (6) __________ 15. ÷ (4) __________ 16. __________ 17. __________ Simplify without a calculator: 18. 4 − 6 ÷ 2 × 3 + 1 − (2 + 1) 2 = __________ 19. = __________ 20. = __________ 21. = __________ 22. Circle the real numbers and underline the integers: 23. The real order of operations is _____________________________________________________________________. 24. Which two symbols (besides parentheses) are “grouping” symbols? _____________________________________ 25. If are all positive integers, what is the least possible value of x? 25. __________ 26. List the three operations that must be performed on each side of this equation (in order!) to solve for x: 3x 2 + 7 = 34 Step 1 __________ Step 2 __________ Step 3 __________ x = __________ xxx x 568 12 ,,,and 3 75 1 333 25 7 2 5 56 7 0. . −− 1212 22 2 −−− () () − () − 1 2 62 12 2 + () 925162 2 −−÷ () ÷ 6 7 ×− 5 3 ÷ 6 7 ×− 5 3 There’s a lot of detail to learn and understand to do well on the SAT. For more tools and resources that will help, visit our Online Practice Plus at www.MHPracticePlus.com/SATmath. 272 MCGRAW-HILL’S SAT 1. Which of the following is NOT equal to 1 ⁄3 of an integer? (A) 1 ⁄ 3 (B) 1 (C) 5 ⁄ 2 (D) 16 ⁄ 2 (E) 10 2. Which of the following can be expressed as the product of two consecutive even integers? (A) 22 (B) 36 (C) 48 (D) 60 (E) 72 3. (A) −3 (B) −2 (C) −1 (D) 2 (E) 3 4. If , what is the value of k? (A) 11 (B) 64 (C) 67 (D) 121 (E) 132 5. In the country of Etiquette, if 2 is a company and 3 is a crowd, then how many are 4 crowds and 2 1 ⁄2 companies? (A) 14 (B) 17 (C) 23 (D) 28 1 ⁄2 (E) 29 6. For what integer value of x is 3x + 7 > 13 and x − 5 < − 1? k −=38 1113 1112−−− () () () −−−− () () () = 8. Dividing any positive number by 3 ⁄4 and then multiplying by −2 is equivalent to (A) multiplying by − 8 ⁄3 (B) dividing by 3 ⁄2 (C) multiplying by − 3 ⁄2 (D) dividing by 3 ⁄8 (E) multiplying by − 3 ⁄8 9. When 14 is taken from 6 times a number, 40 is left. What is half the number? SAT Practice 1: Numbers and Operations 0 0 0 1 1 1 1 2 2 2 2 3 3 3 3 4 4 4 4 5 5 5 5 6 6 6 6 7 7 7 7 8 8 8 8 9 9 9 9 . . . . 0 0 0 1 1 1 1 2 2 2 2 3 3 3 3 4 4 4 4 5 5 5 5 6 6 6 6 7 7 7 7 8 8 8 8 9 9 9 9 . . . . 10. If the smallest positive four-digit integer without repeated digits is subtracted from the greatest four-digit integer without repeated digits, the result is (A) 8,642 (B) 1,111 (C) 8,853 (D) 2,468 (E) 8,888 11. If x > 1, the value of which of the following expressions increases as x increases? I. II. III. (A) II only (B) III only (C) I and II only (D) II and III only (E) I, II, and III 10 1 − x x 1 2 x 7. For all real numbers x, let {x be defined as the least integer greater than x. {−5.6 = (A) −6 (B) −5.7 (C) −5.5 (D) −5 (E) 1 CHAPTER 7 / ESSENTIAL PRE-ALGEBRA SKILLS 273 Concept Review 1 1. −10 (Remember: “greatest” means farthest to the right on the number line.) 2. −10/1, −20/2, −30/3, etc. (Fractions can be integers.) 3. If the original number is negative, then taking a square root doesn’t “undo” squaring the number. Imagine that the original number is −3. The square of −3 is 9, but the square root of 9 is 3. This is the absolute value of the original number, but not the original number itself. 4. 22 (16 + 18 + 20 + 22 = 76. Just divide 76 by 4 to get the “middle” of the set = 19.) 5. Infinitely many (If you said 3, don’t assume that unknowns are integers!) 6. sum 7. difference 8. product 9. 21 ((12 × 3) − (12 + 3) = 36 − 15 = 21) 10. + (−6) 11. × (1/4) 12. ÷ (−3/5) 13. × (7/6) 14. + (6) 15. × (4) 16. ÷ (−5/3) 17. × (6/7) 18. −13 (If you said −5, remember to do multiplica- tion/division from left to right.) 19. 0 20. 50 21. −4 22. Circle all numbers except and underline only 0, , and 56/7 (= 8). 23. PG-ER-MD-AS (Parentheses/grouping (left to right), exponents/roots (left to right), multiplica- tion/division (left to right), addition/subtraction (left to right)) 24. Fraction bars (group the numerator and denom- inator), and radicals (group what’s inside) 25. 120 (It is the least common multiple of 5, 6, 8, and 12.) 26. Step 1: subtract 7; step 2: divide by 3; step 3: take the square root; x = 3 or −3 (not just 3!) 25 5= () −7 Answer Key 1: Numbers and Operations SAT Practice 1 1. C 5 ⁄2 is not 1 ⁄3 of an integer because 5 ⁄2 × 3 = 15 ⁄2 = 7.5, which is not an integer. 2. C 48 = 6 × 8 3. 4. D Add 3: Square: k = 121 5. B (4 × 3) + (2 1 ⁄2 × 2) = 12 + 5 = 17 6. 3 3x + 7 > 13 and x − 5 < −1 3x > 6 and x < 4 x > 2 and x < 4 k =11 k −=38 C 1113 1112 11 2 1 −−− ()()() −−−− ()()() =−−− ()()() −−−−− ()()() =− () −− () =− − − () =− + =− 11 13 12 21 21 1 7. D −5 is the least (farthest to the left on the num- ber line) of all the integers that are greater than (to the right on the number line of) −5.6. 8. A Dividing by 3 ⁄4 is equivalent to multiplying by 4 ⁄3: x ÷ 3 ⁄4 ×−2 = x × 4 ⁄3 ×−2 = x ×− 8 ⁄3 9. 4.5 6x − 14 = 40 Add 14: 6x = 54 Divide by 6: x = 9 (Don’t forget to find half the number!) 10. C 9,876 − 1,023 = 8,853 Don’t forget that 0 is a digit, but it can’t be the first digit of a four-digit integer. 11. D You might “plug in” increasing values of x to see whether the expressions increase or decrease. 1 and 4 are convenient values to try. Also, if you can graph y = 1/x 2 , , and y = 10 − 1 ⁄x quickly, you might notice that , and y = 10 − 1/x “go up” as you move to the right of 1 on the x-axis. yx= yx= The Laws of Arithmetic When evaluating expressions, you don’t always have to follow the order of operations strictly. Sometimes you can play around with the expression first. You can commute (with addition or multiplication), associate (with addition or multiplication), or distribute (multi- plication or division over addition or subtraction). Know your options! When simplifying an expression, consider whether the laws of arithmetic help to make it easier. Example: 57(71) + 57(29) is much easier to simplify if, rather than using the order of operations, you use the “distributive law” and think of it as 57(71 + 29) = 57(100) = 5,700. The Commutative and Associative Laws Whenever you add or multiply terms, the order of the terms doesn’t matter, so pick a convenient arrangement. To commute means to move around. (Just think about what commuters do!) Example: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 1 + 9 + 2 + 8 + 3 + 7 + 4 + 6 + 5 (Think about why the second arrangement is more convenient than the first!) Whenever you add or multiply, the grouping of the terms doesn’t matter, so pick a convenient grouping. To associate means to group together. (Just think about what an association is!) Example: (32 × 4) × (25 × 10) × (10 × 2) = 32 × (4 × 25) × (10 × 10) × 2 (Why is the second grouping more convenient than the first?) Whenever you subtract or divide, the grouping of the terms does matter. Subtraction and divi- sion are neither commutative nor associative. Example: 15 − 7 − 2 ≠ 7 − 15 − 2 (So you can’t “commute” the numbers in a difference until you convert it to addition: 15 +−7 +−2 =−7 + 15 +−2.) 24 ÷ 3 ÷ 2 ≠ 3 ÷ 2 ÷ 24 (So you can’t “commute” the numbers in a quotient until you convert it to multiplication: The Distributive Law When a grouped sum or difference is multiplied or divided by something, you can do the multipli- cation or division first (instead of doing what’s in- side parentheses, as the order of operations says) as long as you “distribute.” Test these equations by plugging in numbers to see how they work: Example: a(b + c) = ab + ac Distribution is never something that you have to do. Think of it as a tool, rather than a re- quirement. Use it when it simplifies your task. For instance, 13(832 + 168) is actually much easier to do if you don’t distribute: 13(832 + 168) = 13(1,000) = 13,000. Notice how annoy- ing it would be if you distributed. Use the distributive law “backwards” when- ever you factor polynomials, add fractions, or combine “like” terms. Example: 9x 2 − 12x = 3x(3x − 4) Follow the rules when you distribute! Avoid these common mistakes: Example: (3 + 4) 2 is not 3 2 + 4 2 (Tempting, isn’t it? Check it and see!) 3(4 × 5) is not 3(4) × 3(5) 57 27 37−= 33 b a b a b += + bc a b a c a + () =+ 24 1 3 1 2 1 3 1 2 24××=××.) Lesson 2: Laws of Arithmetic 274 MCGRAW-HILL’S SAT CHAPTER 7 / ESSENTIAL PRE-ALGEBRA SKILLS 275 Concept Review 2: Laws of Arithmetic Simplify the following expressions, and indicate what law(s) of arithmetic you use to do it. Write D for distrib- ution, CA for commutative law of addition, CM for commutative law of multiplication, AA for associative law of addition, and AM for associative law of multiplication. 1. 3 + x + 9 _____________________________________ 2. −2x(x − 3) ____________________________________ 3. (5m 2 n)(2mn 3 ) ________________________________ 4. 5 + (7 + 2y) + 5y _______________________________ Look carefully at the following equations. If the equation is always true, write the law of arithmetic that justi- fies it (D, CA, CM, AA, or AM). If it is false, rewrite the right side of the equation to make it true. 5. 5. _________________________ 6. 15(67) + 15(33) = 15(100) 6. _________________________ 7. (a + b) 2 = a 2 + b 2 7. _________________________ 8. 5c(c × 3x) = 5c 2 × 15cx 8. _________________________ 9. 3(2 6 + 3 4 ) = 6 6 + 9 4 9. _________________________ 10. 10. _________________________ Rewrite the expression 3x 2 + 12x 3 according to the following laws: 11. distributive law: 12. commutative law of addition: 13. commutative law of multiplication: ____________________ _____________________________ ________________________________ Do the following calculations mentally (no calculator!) by using the appropriate laws of arithmetic. 14. 25 + 48 + 75 + 60 + 52 + 40 = __________________ 15. (4y)(6y)(25)(y 2 )(5) = __________________________ 16. 19(550) + 19(450) = __________________________ 17. (25 × 5x)(4x × 20) = __________________________ If a and b are not 0: 18. What’s the relationship between a ÷ b and b ÷ a? 18. ______________________________ 19. What’s the relationship between a × b and b × a? 19. ______________________________ 20. What’s the relationship between a − b and b − a? 20. ______________________________ 21. The distributive law says that only ____________________ or ____________________ can be distributed over grouped __________ or __________. 22. Which operations are not commutative? 22. ______________________________ 23. Are powers commutative? That is, is (x m ) n always equal to (x n ) m ? 23. ______________________________ 6 3 21 3 + =+ y yy 23 2 5 3 x y x y x y += 276 MCGRAW-HILL’S SAT 1. The difference of two integers is 4 and their sum is 14. What is their product? (A) 18 (B) 24 (C) 36 (D) 45 (E) 56 2. For all real numbers x and y, 4x(x) − 3xy(2x) = (A) 12x 2 y(x − 2x) (B) 2x 2 (2 − 3y) (C) xy(−x) (D) 2x 2 (2 + 3y) (E) 4x 2 (x − 3y) 3. If 3x 2 + 2x = 40, then 15x 2 + 10x = (A) 120 (B) 200 (C) 280 (D) 570 (E) 578 4. The expression −2(x + 2) + x(x + 2) is equivalent to which of the following expressions? I. x 2 − 4 II. (x − 2)(x + 2) III. x 2 − 4x − 4 (A) none (B) II only (C) I and II only (D) II and III only (E) I, II, and III 5. If (x + y) + 1 = 1 − (1 − x), what is the value of y? (A) −2 (B) −1 (C) 0 (D) 1 (E) 3 6. For all real numbers x, 1 − (1 − (1 − x) − 1) = (A) x (B) x − 1 (C) x − 2 (D) 1 − x (E) 2 − x 7. If a = 60(99) 99 + 30(99) 99 , b = 99 100 , and c = 90(90) 99 , then which of the following ex- presses the correct ordering of a, b, and c? (A) c < a < b (B) b < c < a (C) a < b < c (D) c < b < a (E) b < a < c 8. Which of the following is equivalent to 5x(2x × 3) − 5x 2 for all x? (A) 5x 2 + 15x (B) 5x 2 × 15x (C) 10x 2 × 15x − 5x 2 (D) 145x (E) 25x 2 9. Which of the following statements must be true for all values of x, y, and z? I. (x + y) + z = (z + y) + x II. (x − y) − z = (z − y) − x III. (x × y) × z = (z × y) × x (A) I only (B) I and II only (C) I and III only (D) II and III only (E) I, II, and III 10. The symbol ◊ represents one of the fundamen- tal arithmetic operators: +, −, ×, or ÷. If (x ◊ y) × (y ◊ x) = 1 for all positive values of x and y, then ◊ can represent (A) + only (B) × only (C) + or × only (D) − only (E) ÷ only SAT Practice 2: Laws of Arithmetic CHAPTER 7 / ESSENTIAL PRE-ALGEBRA SKILLS 277 Concept Review 2 1. x + 12 (commutative law of arithmetic) 2. −2x 2 + 6x (distributive law) 3. 10m 3 n 4 (commutative law of multiplication and associative law of multiplication) 4. 7y + 12 (associative law of arithmetic and com- mutative law of arithmetic) 5. false: (Find a common denominator, then add numerators.) 6. true: distributive law 7. false: a 2 + 2ab + b 2 (Don’t ever distribute a power over addition. Expand and “FOIL” (First + Out- side + Inside + Last).) 8. false: 15c 2 x (Don’t ever distribute multiplication over multiplication. Use the associative and com- mutative laws of multiplication.) 9. false: 3(2) 6 + 3 5 (You can distribute, but don’t mul- tiply before doing the powers!) 7 2 x y 10. true: distributive law (Remember you can dis- tribute division over addition!) 11. 3x 2 (1 + 4x) 12. 12x 3 + 3x 2 13. x 3 × 12 + x 2 × 3 14. 300 (Reorder: (25 + 75) + (48 + 52) + (60 + 40)) 15. 3,000y 4 (Reorder: (4 × 25)(6 × 5)(y × y × y 2 )) 16. 19,000 (Use the distributive law.) 17. 10,000x 2 (You can’t “FOIL” here! Reorder: (4 × 25) (5 × 20)(x)(x) = (100)(100)(x 2 )) 18. They’re reciprocals. (Their product is 1.) 19. They’re the same. 20. They’re opposites. (Their sum is 0.) 21. multiplication or division over addition or subtraction 22. subtraction and division 23. yes Answer Key 2: Laws of Arithmetic SAT Practice 2 1. D m − n = 4 m + n = 14 2m = 18 m = 9 n = 59 × 5 = 45 2. B 4x(x) − 3xy(2x) = 4x 2 − 6x 2 y = 2x 2 (2 − 3y) 3. B Don’t solve for x. It’s too hard, and it’s unnecessary. 15x 2 + 10x = 5(3x 2 + 2x) = 5(40) = 200 4. C −2(x + 2) + x(x + 2) = (−2 + x)(x + 2) Distributive law = (x − 2)(x + 2) Commutative law of addition = x 2 − 4 Distributive law (“FOILing”) 5. B (x + y) + 1 = 1 − (1 − x) associate: x + y + 1 = 1 − (1 − x) distribute: x + y + 1 = 1 − 1 + x simplify: x + y + 1 = x subtract x: y + 1 = 0 subtract 1: y =−1 6. E 1 − (1 − (1 − x) − 1) distribute: = 1 − (1 − 1 + x − 1) distribute: = 1 − 1 + 1 − x + 1 commute: = 1 − 1 + 1 + 1 − x simplify: = 2 − x 7. A Each number has 100 factors, to make them simpler to compare: a = 60(99) 99 + 30(99) 99 = 90(99) 99 b = 99 100 = 99(99) 99 c = 90(90) 99 = 90(90) 99 8. E 5x(2x × 3) − 5x 2 parentheses: = 5x(6x) − 5x 2 multiplication: = 30x 2 − 5x 2 subtraction: = 25x 2 9. C Just remember that only addition and multi- plication are commutative and associative. If you’re not convinced, you might plug in 1, 2, and 3 for x, y, and z, and notice that equation II is not true. 10. E xy yx x y y x xy xy ÷ ( ) ×÷ () = × = = 1 Adding and Subtracting Fractions Just as 2 apples + 3 apples = 5 apples, so 2 sevenths + 3 sevenths = 5 sevenths! So it’s easy to add fractions if the denominators are the same. But if the denominators are different, just “convert” them so that they are the same. When “converting” a fraction, always multiply (or divide) the numerator and denominator by the same number. Example: If the denominator of one fraction is a multi- ple of the other denominator, “convert” only the fraction with the smaller denominator. Example: One easy way to add fractions is with “zip-zap- zup”: cross-multiply for the numerators, and multiply denominators for the new denomina- tor. You may have to simplify as the last step. Example: Multiplying and Dividing Fractions To multiply two fractions, just multiply straight across. Don’t cross-multiply (we’ll dis- cuss that in the next lesson), and don’t worry about getting a common denominator (that’s just for adding and subtracting). Example: To multiply a fraction and an integer, just mul- tiply the integer to the numerator (because an integer such as 5 can be thought of as 5/1). y x y x y x5 33 5 3 5 ×= × × = 5 6 7 8 5 6 7 8 40 48 42 48 82 48 41 24 +=+= + = = xx x x 3 2 53 2 5 5 15 6 15 56 15 +=+= + = + 5 18 4 9 5 18 42 92 5 18 8 18 13 18 += + × × =+= 12 18 12 6 18 6 2 3 = ÷ ÷ = 2 5 25 55 10 25 = × × = Example: To divide a number by a fraction, remember that dividing by a number is the same as multi- plying by its reciprocal. So just “flip” the second fraction and multiply. Example: Simplifying Fractions Always try to simplify complicated-looking frac- tions. To simplify, just multiply or divide top and bottom by a convenient number or expression. If the numerator and the denominator have a common factor, divide top and bottom by that common factor. If there are fractions within the fraction, multiply top and bottom by the com- mon denominator of the “little” fractions. Example: (Notice that, in the second example, 60 is the common multiple of all of the “little denominators”: 5, 3, and 4.) Be careful when “canceling” in fractions. Don’t “cancel” anything that is not a common factor. To avoid the common canceling mistakes, be sure to factor before canceling. Example: Right: x x xx x x 2 1 1 11 1 1 − − = + () − () − () =+ Wrong: x x x x x 22 1 1 − − == 2 5 2 3 1 4 60 2 5 2 3 60 1 4 24 40 15 6 + = ×+ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ × ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ = + = 44 15 42 2 22 1 2 21 xx x + = + () =+ 3 7 5 2 3 7 2 5 6 35 2 m m mmm ÷=×= 4 7 5 4 7 5 1 45 7 20 7 ×= × = × = Lesson 3: Fractions 278 MCGRAW-HILL’S SAT 15 5x 6 48 40 42 CHAPTER 7 / ESSENTIAL PRE-ALGEBRA SKILLS 279 Simplify the following expressions: 1. __________ 2. __________ 3. __________ 4. __________ 5. __________ 6. __________ 7. __________ 8. __________ 9. __________ 10. __________ 11. __________ 12. __________ Convert each expression to a fraction: 13. 25% = __________ 14. 0.20 = __________ 15. 10% = __________ 16. 0.333 . . . = __________ 17. How do you divide a number by a fraction? __________________________________________________________________________________________________ 18. How do you add two fractions by “zip-zap-zup”? __________________________________________________________________________________________________ 19. What can be canceled to simplify a fraction? __________________________________________________________________________________________________ 20. How do you convert a fraction to a decimal? __________________________________________________________________________________________________ 21. If a class contains 12 boys and 15 girls, then what fraction of the class is boys? 21. __________ 22. If 2/3 of a class is girls, and there are 9 boys in the class, what is the total number of students in the class? 22. __________ 23. If m and n are positive, and m < n, then what is true about ? 23. __________ m n 69 12 n n + = x x 2 25 5 − − = 3 42 −= x 1 2 1 3 1 4 ++= −÷= 2 9 4 3 2 x x 12 4 84 m m + + = 3 2 9 8 x z = 4 7 2 3 += x 2 9 5 4 ÷= 56 21 = 5 2 3 8 = 1 7 2 5 += Concept Review 3: Fractions . understand to do well on the SAT. For more tools and resources that will help, visit our Online Practice Plus at www.MHPracticePlus.com/SATmath. 272 MCGRAW-HILL’S SAT 1. Which of the following. 270 MCGRAW-HILL’S SAT Integers and Real Numbers On the SAT, you only need to deal with two kinds of numbers: integers (the positive. −5, remember to do multiplica- tion/division from left to right.) 19. 0 20. 50 21. −4 22. Circle all numbers except and underline only 0, , and 56/7 (= 8). 23. PG-ER-MD-AS (Parentheses/grouping