Chapter FRACTIONS 2.1 Factors and Prime Numbers 22 Exercises A composite number is a whole number that has more than two factors 48 48 48 = 48 R0 = 24 R0 = 16 R0 48 48 = 12 R0 = R0 The factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24, and 48 24 is prime The least common multiple of two or more numbers is the smallest nonzero number that is a multiple of each number 26 24 is composite; 2, 3, 4, 6, 8, and 12 are factors The divisibility test for 10 is to check if the ones digit is 30 31 is prime 10 10 = 10 R0 = R0 The factors of 10 are 1, 2, 5, and 10 34 10 12 14 16 18 20 9 = R0 = R0 The factors of are 1, 3, and 28 75 is composite; 3, 5, 15, and 25 are factors 32 45 is composite; 3, 5, and 15 are factors 18 15 15 = 15 R0 = R0 The factors of 15 are 1, 3, 5, and 15 3 18 = × × = × 32 40 47 = 47 R0 The factors of 47 are and 47 40 20 35 35 = 35 R0 = R0 The factors of 35 are 1, 5, 7, and 35 73 = 73 R0 The factors of 73 are and 73 14 = × 10 = × 38 14 36 10 10 40 = × × × = 23 × 42 98 98 98 = 98 R0 = 49 R0 = 14 R0 The factors of 98 are 1, 2, 7, 14, 49, and 98 63 = × × = 32 × 57 44 49 46 19 57 = ì 19 Copyright â 2013 Pearson Education, Inc 7 49 = × = 13 14 Chapter Fractions 48 62 64 32 16 2 64 = × × × × × = 26 50 64 = 32 22 66 = 22 105 LCM = × = × = 12 35 68 20 = 22 × LCM = × × × = × × × = 420 LCM = 22 × × = × × = 60 13 13 169 = 13 × 13 = 132 72 30 = × × 150 = × × 52 LCM = × × 52 = × × 25 = 150 62 74 100 = 22 × 52 31 90 = × 32 × LCM = 22 × 32 × 52 = × × 25 = 900 62 = × 31 58 60 = 22 × × 70 15 = × 169 21 = × 7 105 = × × 56 6= 2×3 54 12 = 22 × LCM = 22 × 32 = × = 36 11 22 = × 11 52 400 = × × × × × = 24 × 52 76 = 200 = 23 10 = × LCM = × = × = 40 100 78 = = = LCM = × × = × = 30 50 2 12 = 22 × LCM = 23 × = × = 24 200 = × × × × = 23 × 52 60 = 23 80 = × 25 82 = 23 24 = 23 × 56 = 23 × LCM = 23 × × = × × = 168 90 84 63 is composite; 3, 7, 9, and 21 are factors 45 86 = 15 3 10 = × 12 = 22 × LCM = 22 × × = × 15 = 60 90 = × × × = × 32 × 49 R0 88 a 196 Yes, because 196 is a multiple of Copyright © 2013 Pearson Education, Inc Section 2.2 Introduction to Fractions 49 R2 b 198 of a whole shaded The mixed number is 14 There is whole shaded and No, because 198 is not a multiple of 90 Yes, an oil change would be recommended at 21,000 miles, because 21,000 is divisible by 3,000 16 11 18 10 20 11 11 22 92 LCM(4, 3) = 12 Both prizes will be given in 2006 + 12 = 2018 94 = × 3; = 3; = × LCM(6, 3, 4) = 22 × = 12 , so the bills will all fall due again in 12 months Mindstretchers a 57 = + 19 + 31 or 57 = + 17 + 37 b 81 = 11 + 23 + 47 or 81 = + 37 + 37 = = = = 22 = = × = = 23 = 32 10 = × LCM = × 23 × 32 × × = 2,520 715 × × 11 × 13 = 715, 715 2.2 Introduction to Fractions 24 26 Exercises The improper fraction can be expressed as a mixed number Divide the numerator and denominator of a fraction by the same whole number in order to simplify it The least common denominator of two or more fractions is the least common multiple of their denominators There are equal parts of which part is shaded The fraction is 10 There are equal parts of which parts are shaded The fraction is of a whole shaded The mixed number is 12 There are wholes and 28 , proper 12 30 11 , improper 10 32 12 , mixed number 34 , improper 36 38 10 , mixed number (3 × 1) + 40 = = 3 Copyright © 2013 Pearson Education, Inc , proper 15 16 Chapter Fractions 42 10 (3 × 10) + 32 = = 3 78 R1 41 = 41 44 12 (4 × 12) + 51 = = 4 80 R3 58 = 11 58 11 82 12 R2 38 = 38 84 R0 72 72 = 72 =8 9 86 19 R0 19 = 19 46 = 48 (6 × 6) + 41 = = 6 50 10 (2 × 10) + 21 = = 2 (8 × 20) + 161 52 20 − = 8 (7 × 11) + 82 54 11 = = 7 56 10 = 10 58 (13 × 2) + 33 = = 13 13 13 60 (6 × 4) + 25 = = 6 62 14 64 (10 × 14) + 141 = = 10 10 10 R1 =5 R2 12 66 = 12 12 =2 5 R0 12 68 = 12 12 12 12 =1 12 R0 100 = 100 100 70 100 72 74 76 100 =1 100 15 R1 31 = 31 31 = 15 2 20 R2 62 = 62 62 = 20 3 13 R1 40 = 40 40 = 13 3 58 =5 11 11 38 = 12 3 19 = 19 88 Possible answers: 3⋅2 = = 10 10 ⋅ 20 3⋅3 = = 10 10 ⋅ 30 90 Possible answers: 1⋅ 2 = = 10 10 ⋅ 20 1⋅ 3 = = 10 10 ⋅ 30 92 Possible answers: 5 ⋅ 10 = = 6 ⋅ 12 5 ⋅ 15 = = 6 ⋅ 18 94 Possible answers: 3⋅2 = = 5 ⋅ 10 3⋅3 = = 5 ⋅ 15 96 2⋅2 = = 9 ⋅ 18 100 = =1 5 41 =5 8 102 7 ⋅ 14 = = 10 10 ⋅ 20 104 3 ⋅ 12 = = 10 10 ⋅ 40 5 ⋅ 15 75 = = 1 ⋅ 15 15 4 ⋅ 28 = = 9 ⋅ 63 106 = 98 2 ⋅ 21 42 = = 1 ⋅ 21 21 108 7 ⋅ 21 = = 8 ⋅ 24 110 5 ⋅ 40 = = 6 ⋅ 48 112 1 ⋅ 30 30 = = 3 ⋅ 30 90 114 1 ⋅ 25 25 = = 4 ⋅ 25 100 116 7 ⋅ 49 = = 8 ⋅ 56 118 5 ⋅ 24 120 = = 6 ⋅ 24 144 122 21 ⋅ = =1 21 ⋅ 3⋅ 3 = = 120 12 ⋅ 4 Copyright © 2013 Pearson Education, Inc 1 1 Section 2.2 Introduction to Fractions 1 2⋅ = = 126 124 24 ⋅ ⋅ ⋅ 25 ⋅ 25 = = 49 ⋅ 49 156 = 3=3 17 = 22 LCM = ⋅ = ⋅ = 12 3 ⋅ 18 = = 2 ⋅ 12 3 ⋅ 12 = = 3 ⋅ 12 3⋅3 = = 4 ⋅ 12 1 75 ⋅ 25 128 = = 100 ⋅ 25 1 1 875 5⋅ 5⋅ 5⋅ 7 = = 130 1, 000 ⋅ ⋅ ⋅ 8 In increasing order, the fractions are 3 , , 1 158 = 22 132 15 ⋅ 5 = = =1 3⋅ 3 30 ⋅ 134 = = =1 18 ⋅ 3 1 138 In increasing order, the fractions are , , 19 19 = 51 51 160 = 23 1 1 21 3⋅ 7 142 = = 36 ⋅ 12 12 In increasing order, the fractions are 51 ⋅ 51 = 11 = 11 102 2 ⋅ 51 R2 162 32 146 144 =1+1= 144 164 because > > 10 10 150 20 21 < because < 24 24 166 152 = because ⋅ = 12 ⋅ 3, 36 = 36 12 R2 168 17 34 21 > because > since > 7 14 14 32 =5 =5 6 10 148 154 11 = 11 LCM = ⋅ 11 = ⋅ 11 = 88 5 ⋅ 11 55 = = 8 ⋅ 11 88 1 ⋅ 44 44 = = 2 ⋅ 44 88 4 ⋅ 32 = = 11 11 ⋅ 88 36 9⋅ = = 144 ⋅ ⋅ 4 144 11 2=2 1 140 = 23 LCM = 23 ⋅ = ⋅ = 24 3 ⋅ 18 = = 4 ⋅ 24 5 ⋅ 20 = = 6 ⋅ 24 7 ⋅ 21 = = 8 ⋅ 24 36 ⋅ 4 136 − = 45 ⋅ 5 = 2⋅3 19 = 8 =1 =1 4 There are cloves per serving Copyright © 2013 Pearson Education, Inc , , 11 18 Chapter Fractions 170 a The total number of therapists is 182,000 + 94,000 = 276,000 The fraction of therapists who are physical therapists is 182, 000 182 91 = = 276, 000 276 138 b The fraction of therapists who are respiratory 94, 000 94 47 = = therapists is 276, 000 276 138 172 The Lakers did not win 82 − 65 = 17 games 17 of the games played This is 82 1⋅ 4 = = 174 9 ⋅ 36 5 = 36 36 There is a greater probability of getting a because > 36 176 The LCM of 2, 8, and 16 is 16 1 1⋅ 2 1⋅ 8 = = = = = 16 16 8 ⋅ 16 2 ⋅ 16 2.3 Exercises To subtract unlike fractions, rewrite them as equivalent fractions with the same denominator Fractions with equal numerators and denominators are equivalent to 16 + = = or 10 10 10 10 71 79 150 50 + = =1 or 100 100 100 100 10 + + = 7 7 12 + + = = 10 10 10 10 14 a Newark Airport had the best visibility at mi b Kennedy Airport had the worst visibility at mi 16 57 + 61 + 57 + 57 + 58 + 57 347 = yr = 57 yr 6 18 178 average age = 22 Mindstretchers The shaded center triangle is equivalent to of the smaller shaded triangles, so there are shaded triangles There are unshaded triangles, so there are 16 triangles in total of the 16 triangle is shaded 24 There are an infinite number of fractions, such as = , = , = , and so on 2 3 4 a 42 = 42; 174 = 174; 406 = 406 79 b = = 158 Adding and Subtracting Fractions 26 = 20 + =+ 20 13 20 = + = 12 10 12 + 12 11 12 16 20 1 = 6 + =+ 6 21 = 28 12 + =+ 28 33 =1 28 28 9 = 10 10 + =+ 10 17 =1 10 10 7 = 20 20 15 + =+ 20 22 =1 =1 20 20 10 1 10 21 + + = + + = = 30 30 30 30 10 Copyright © 2013 Pearson Education, Inc Section 2.3 Adding and Subtracting Fractions 28 30 1 27 30 10 67 + + = + + = 10 90 90 90 90 34 +2 36 Check: + ↓ ↓ +2 = 38 42 14 = 10 20 7 + =+ 20 20 21 =5 20 20 7 Check: + 10 20 ↓ ↓ + 0=5 44 = 12 = 12 + =+ 12 13 =1 12 12 46 32 = 10 30 12 = 30 25 + =+ 30 40 10 =1 =1 30 30 12 +4 12 10 = 10 12 1 Check: + 12 12 ↓ ↓ + = 10 40 10 +6 +7 10 12 14 = 15 = 10 = 10 3 10 10 2 Check: + Check: + 3 10 10 ↓ ↓ ↓ ↓ + = 16 + = 10 48 15 = 17 40 +20 = +20 40 23 37 40 Check:17 + 20 ↓ ↓ 17 + 20 = 37 19 52 17 54 10 = 90 63 +20 = +20 10 90 73 24 90 Check: + 20 10 ↓ ↓ + 21 = 25 50 = 30 12 +3 = + 30 17 30 Check: + ↓ ↓ + 3=3 8 = 9 +5 = + 11 = 10 9 Check: + ↓ ↓ + = 10 10 = 10 12 +8 =+8 12 13 18 or 19 12 12 Check:10 + ↓ ↓ 11 + = 19 10 Copyright © 2013 Pearson Education, Inc = 20 10 +4 = +4 10 11 24 = 25 10 10 Check: 20 + ↓ ↓ 21 + = 26 20 20 56 58 60 62 Chapter Fractions 300 = 10 1, 000 321 321 +2 = +2 1, 000 1, 000 621 10 1, 000 321 Check: + 10 1, 000 ↓ ↓ + = 10 64 = 24 7 = 25 25 24 24 12 +100 = +100 24 27 = 126 125 or 126 24 24 + 100 Check: + 25 24 ↓ ↓ ↓ + 25 + 101 = 126 = 16 3 = 16 16 20 + = + 16 25 =9 16 16 Check: + + 16 ↓ ↓ ↓ + + 1= 66 − = 9 68 11 − = = 12 12 12 70 − = =1 2 72 − = = 9 74 1 − =0 8 76 = 12 10 = 12 +3 =+3 12 21 9 = 10 = 10 12 12 Check:1 + + 3 ↓ ↓ ↓ + + = 11 24 = 36 11 11 = 36 36 18 + = +1 36 53 17 =8 36 36 11 +1 Check: + 36 ↓ ↓ ↓ + + 2=9 80 84 88 12 = 30 − =− 30 30 78 20 = 24 − =− 24 17 24 82 11 11 = 12 12 − =− 12 12 −1 86 90 +1 Check: Copyright © 2013 Pearson Education, Inc 90 = 10 100 3 − =− 100 100 87 100 18 = 45 10 − =− 45 45 5 = 6 − =− 6 −2 10 Check: 10 +2 Section 2.3 Adding and Subtracting Fractions − =7 Check: 92 94 + Check: +2 −2 96 98 100 102 104 2 1 −1 = − 2 4 3 − =− 4 10 +4 10 10 =5 10 + 4 =9 Check: 11 = 10 10 3 −2 = − 10 10 =2 10 5 +1 =3 5 Check: 17 =2 10 10 9 − =− 10 10 =2 10 Check: = 10 9 + =+ 10 10 17 =3 10 10 +1 2 =2 Check: =2 5 4 − = −1 5 108 Check: 10 10 9 −4 = − 10 10 10 Check: = = 10 3 +2 =+2 10 10 11 =5 10 10 Check: 106 +1 5 =4 5 1 −1 = − 5 = = = = 1 =+ 4 =1 5 4 − =− 5 110 112 114 + 1 =2 5 Check: 1 77 = = 10 70 70 10 10 −2 = − = −2 70 70 67 70 Check: 1 −2 10 ↓ ↓ −2=5 10 110 = = 10 100 100 27 27 27 −1 = −1 = −1 100 100 100 83 100 Copyright © 2013 Pearson Education, Inc Check: 27 −1 10 1000 ↓ ↓ − 1=1 21 22 116 Chapter Fractions 13 13 113 = = 100 100 100 70 70 −6 = − = −6 10 100 100 43 100 15 118 = = 12 12 4 − =− =− 12 12 11 12 11 11 = Check: 12 12 + =+ 12 15 =1 =1 =1 12 12 120 122 124 = = 4 3 −1 = − = − 4 4 +1 1 =2 =2 4 Check: 20 = = 16 16 5 −1 = −1 = −1 16 16 16 15 16 15 Check: 16 +1 16 20 =7 =7 16 16 = +3 =+ Check: 13 −6 100 10 ↓ ↓ − 7=2 Check: 126 128 130 1 11 = = 10 10 10 5 −3 = −3 = −3 10 10 =5 10 132 134 10 10 11 =9 10 10 20 = = 16 16 5 −2 = − = − 16 16 16 15 16 15 Check: 16 +2 16 20 =3 =3 16 16 1 20 15 29 − + = − + = 30 30 30 30 20 = = 15 15 12 12 −2 = −2 = −2 15 15 15 8 =4 15 15 − = −1 15 3 =3 15 4 = 10 3 +2 −3 = −3 10 4 10 =7 10 4 9 = 10 10 − = − 10 10 35 = 18 19 = 19 30 30 21 21 −8 = −8 = −8 10 30 30 14 = 10 10 30 15 Copyright © 2013 Pearson Education, Inc Section 2.3 Adding and Subtracting Fractions 136 140 = 10 30 14 +3 =+3 15 30 17 30 = = + =+ 138 24 12 24 24 29 =1 24 24 = 10 − =− 18 7 acre = acre 142 8 20 − acre = − acre 20 13 acre 20 The area of the land not occupied by the building is 58 acre 144 a b = 32 + =+ 32 32 32 14 = 16 32 5 − =− 32 32 32 The combined amount of electricity generated by liquid fuels and nuclear power is 32 of the total world electricity The amount of electricity generated by coal is 32 greater than the combined amount generated by liquid fuel and nuclear power 150 The difference in foot length when comparing sizes and is 3 3 - = - = - = in 4 4 4 The difference in foot length when comparing sizes and 10 is 1 - = - = - = in 4 4 4 Both differences are the same 152 The total weight of the packages on the right side of the scale is: lb = lb 1 +3 lb = +3 lb 4 lb The total weight of the packages on the left side of the scale must also equal lb 4 lb −2 lb lb The small package on the left weighs lb Mindstretchers equal whole, or 88 148 1 146 The sum of the fractions of the votes would 5 + = + = 8 8 − = 8 The third candidate got 11 12 of the votes 5 = 20 8 + 10 = + 10 30 = 31 8 20 1 12 1 = + + 28 a In Method we “borrow” from the whole number in the minuend so that the fraction in the minuend is big enough to subtract the fraction in the subtrahend In Method we add a fraction to the subtrahend, making it a whole number We add the same fraction to the minuend and then subtract b Answers may vary The total weight of the boxes is 31 18 oz 23 c Answers may vary Copyright © 2013 Pearson Education, Inc 24 Chapter Fractions 2.4 Multiplying and Dividing Fractions 20 100 34 100 ⋅ = ⋅ = 40 5 Exercises To multiply mixed numbers, change each mixed number to its equivalent improper fraction 20 36 20 ⋅ = ⋅ = 16 5 To divide fractions, change the divisor to its reciprocal, and multiply the resulting fractions When multiplying fractions, we can divide any numerator and any denominator by a common factor 7 × = 16 10 ⎛ ⎞⎛1⎞ ⎜⎝ ⎟⎠ ⎜⎝ ⎟⎠ = 10 40 ⎛1⎞ ⎛1⎞ ⎛1⎞ 12 ⎜ ⎟ = ⎜ ⎟ ⎜ ⎟ = ⎝8⎠ ⎝ ⎠ ⎝ ⎠ 64 3 14 × = 2 18 16 20 40 19 × = =1 21 21 ⎛ ⎞⎛ ⎞ ⎛4⎞ ⎛1⎞ ⎜ ⎟ ⎜ ⎟ 20 ⎜ ⎟ ⎜ ⎟ = = ⎝5⎠ ⎝4⎠ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠⎝ ⎠ 4 × = × = 22 5 ⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛3⎞ ⎜ ⎟ ⎜ ⎟ = 24 ⎜ ⎟ ⎜ ⎟ = ⎝6 ⎠ ⎝8⎠ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠⎝ ⎠ 12 15 12 15 × = × 26 =9 5 1 28 30 5 25 ×5= × = =4 6 6 5 35 ×7= × = = 11 3 3 3 12 32 × 12 = × =9 4 ⎛ ⎞ ⎛ ⎞ ⎛ 13 ⎞ ⎛ ⎞ 13 40 ⎜ ⎟ ⎜ ⎟ = ⎜ ⎟ ⎜ ⎟ = ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ 15 42 11 99 49 ⋅ = =1 10 50 50 3 5 12 15 38 × 12 = × = =7 2 1 11 11 ×2 = × = 5 15 ⎛ ⎞⎛ ⎞ 13 ⎛ ⎞ ⎛ ⎞ ⎜ ⎟ ⎜ 15 ⎟ 27 = =1 44 ⎜ ⎟ ⎜ ⎟ = ⎝ 10 ⎠ ⎝ ⎠ ⎜ 10 ⎟ ⎜ ⎟ 14 14 ⎟⎠ ⎝ ⎠ ⎜⎝ 1 46 × = × = 3 1 3 16 =2 48 ⋅5 = ⋅ 8 1 ⎛ ⎞⎛ ⎞ ⎛7⎞ ⎛ 1⎞ ⎜ ⎟⎜ ⎟ = =1 50 ⎜ ⎟ ⎜ ⎟ = ⎝9⎠ ⎝ 4⎠ ⎜ ⎟⎜ ⎟ 4 ⎝ ⎠⎝ ⎠ 1 7 52 × = × = = 3 2 2 ⎛ 1⎞ ⎛3⎞ ⎛3⎞ 54 ⎜1 ⎟ = ⎜ ⎟ ⎜ ⎟ = = ⎝ 2⎠ ⎝2⎠ ⎝2⎠ 4 56 ⋅ 1 15 = × = =7 2 2 10 11 20 110 58 × 20 = × = = 36 6 3 Copyright © 2013 Pearson Education, Inc Section 2.4 Multiplying and Dividing Fractions 1 21 10 35 60 × = × = =5 4 9 6 64 ⎛ 13 ⎛ ⎞ ⎛ 4⎞ 62 ⎜1 ⎟ ⎜ ⎟ = ⎜ ⎝ 10 ⎠ ⎝ ⎠ ⎜ 10 ⎝ 17 5 ữ = ì = =2 88 6 2 ⎞ ⎛ 11 ⎞ ⎟ ⎜ 22 ⎟ = 143 = ⎟ ⎜ ⎟ 45 45 ⎟⎠ ⎠ ⎜⎝ 1 51 17 ⋅1 = ⋅ = =8 10 10 2 15 92 7 10 7 ÷ 10 = ÷ = × = 10 10 10 10 100 94 1 1 ữ2= ữ = ì = 20 20 20 40 75 66 37 ⋅ = ⋅ = 60 5 1 5 ÷ = × = 90 8 96 ÷ = ÷ = × = 36 9 1 68 1 27 11 ×2 ×6= × × = =1 8 16 16 ⎛ ⎞⎛ ⎞ ⎛ 1⎞ ⎛1⎞ ⎛5⎞ ⎜ ⎟ ⎜ ⎟ 70 ⎜1 ⎟ ⎜ ⎟ = ⎜ ⎟ = ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎜ ⎟ ⎜ ⎟ 16 ⎝ ⎠⎝ ⎠ 76 10 ÷ = × = =1 3 9 78 7 35 ÷ = × = =1 8 32 32 82 9 ữ = ì = 40 1 ữ = ì = 84 10 10 2 2 10 10 86 ÷ = × =4 1 102 ÷ 3 = ÷ = × = 24 8 1 23 ⎛ 1⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 125 = 15 74 ⎜ ⎟ = ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ = ⎝ 2⎠ ⎝2⎠ ⎝2⎠ ⎝2⎠ 8 1 2 ữ = ì = 7 10 10 ÷ = 100 10 ÷ = × = 15 3 1 25 25 72 × × × = × = =2 3 10 12 10 12 80 10 10 98 10 ữ = ữ = ì = 25 5 1 46 46 23 104 ÷ = ÷ = × = =7 9 3 1 71 71 71 ữ = ì = = 14 106 ÷ = 10 10 10 5 1 13 13 108 ữ = ữ = ì = 13 2 2 1 2 47 47 94 110 15 ÷ = ÷ = × = = 18 6 5 1 2 3 ữ1 = ữ = ì = 112 7 14 114 3 28 27 ữ3 = ữ = ì = 9 28 112 Copyright © 2013 Pearson Education, Inc 25 26 Chapter Fractions 116 ÷ 19 10 70 13 = ữ = ì = =3 10 10 19 19 19 118 41 14 41 41 ữ 14 = ữ = ì = 7 14 98 120 1 22 22 44 ữ2 = ữ = ì = =1 7 35 35 138 17 41 17 68 ÷ = × = 122 ÷ = 10 10 10 41 205 1 ⎛1 ữ ì = ữ ì ⎝3 6⎠ 2⎞ ⎛ 6⎟ ⎜ = × × ⎜3 1⎟ ⎝1 ⎠ 2 = × = =1 3 1 ⎛ 25 ⎞ ⎛ ⎞ 140 ÷ + ÷ = ⎜ ÷ ⎟ + ⎜ ÷ ⎟ ⎝ 1⎠ ⎝1 2⎠ ⎛ ⎞ 25 ⎟ ⎛ ⎞ ⎜ + ⋅ ⋅ = ⎜ ⎟ ⎜⎝ ⎟⎠ ⎜⎝ ⎟⎠ 1 49 49 49 124 ÷ = ữ = ì = =3 6 15 15 2 5 25 126 ÷ = ÷ = × = =1 5 21 21 128 9 ⎛4 8⎞ + ⋅8 = +⎜ ⋅ ⎟ 10 10 ⎝ ⎠ 32 = + 10 64 = + 10 10 73 = =7 10 10 130 ÷ 132 142 ⎛3 2⎞ −2 =⎜ ÷ ⎟−2 ⎝1 5⎠ ⎛3 5⎞ =⎜ × ⎟−2 ⎝1 2⎠ 15 1 = −2 =7 −2 3 =7 −2 =5 6 1 ⎛3 1⎞ ⋅ − =⎜ ⋅ ⎟− 10 ⎝ ⎠ 10 15 = − = − = 16 10 80 80 80 134 ữ ì 25 64 + = + 40 40 89 = =2 40 40 = 3 = × = × = 5 10 18 13 æ 18 ổ 13 ì = ỗỗ ữữ - ỗ ỗ ì ữữ 11 55 26 14 è11 55 ø è 26 14 ø ỉ 1 ổ ỗ 55 ữ ỗ 13 ữ ữ ữ-ỗ =ỗ ì ì ỗ 11 18 ữ ỗ 26 14 ữ ỗ ữ ỗ ữ ø ø è è 20 = - = 12 12 17 = =1 12 12 ỉ ư2 ỉ ử2 ổ5 ử2 ổ ử2 ỗ1 - ữữ ỗỗ1 ữữ = ỗỗ - ữữ ỗỗ ữữ 144 ỗ ố 5ứ ố 2ứ ố5 ứ ố2 ứ ổ3 ử2 ổ ử2 9 = ỗỗ ữữ ỗỗ ữữ = 25 ố5 ứ è2ø 4 = × = 25 25 2 136 ⋅ 1⎞ ⎛7⎞ 2⎞ ⎛ 1⎞ ⎛ ⎛ 146 ⎜ ⎟ + ⎜1 − ⎟ = ⎜ ⎟ + ⎜1 − ⎟ ⎝ 2⎠ ⎝ ⎝ 3⎠ ⎝ ⎠ 6⎠ = 49 ⎛1⎞ + 2⎜ ⎟ ⎝6⎠ ⎛1 ⎞ 49 ⎜ ⎟ 49 ⋅ = + = + ⎜⎜ ⎟⎟ ⎝ ⎠ ⎛4 2⎞ −1 = ⎜ ⋅ ⎟ −1 ⎝1 3⎠ 8 = −1 = −1 8 16 13 =2 −1 =1 24 24 24 = Copyright © 2013 Pearson Education, Inc 147 151 + = = 12 12 12 12 12 Section 2.4 Multiplying and Dividing Fractions 162 ÷ = 16 ⎞ ⎛4⎞ ⎛ 148 14 − ÷ ⎜ ⎟ = 14 − ⎜ ÷ ⎟ ⎝5⎠ ⎝ 25 ⎠ º F three times 10 1⎞ 10 ⎛ 70 − ⎜ × ⎟ = 70 − = 69 − = 69 ⎝ 10 ⎠ 10 10 10 10 The temperature drops by 75 ⎛ 25 ⎞ = 14 − ⎜ ⋅ ⎟ = 14 − ⎝ 16 ⎠ 16 = 14 − 11 16 11 = 13 − = 16 16 16 16 °F The temperature after is 69 10 9 150 ÷ = ⋅ = =2 10 10 4 164 ÷ = × = 12 3 He can administer 12 doses 10 ⎛ ⎞ ⎛ ⎞ 20 152 ⎜ ⎟ ⎜ ⎟ = ⋅ = 30 ⎝ 2⎠ ⎝ 3⎠ 1 3 166 a ữ = ữ = ì = 2 1 154 The average amount collected per year is $2 million ⋅ = 20 20 27 b If an additional $1 million were collected, then the total collected is $4 million 4 ữ1 = ữ = ì = = 2 3 2 −2= 3 of the emergency room visits were due to motor vehicle accidents 156 $24, 000 ÷ 12 = $2, 000 Their monthly income is $2,000 500 2, 000 1 × 2, 000 = × = 500 4 1 They should spend no more than $500 per month on rent 16 23 368 158 16 × 11 = × = = 184 2 31 12 372 15 × 12 = × = = 186 2 The average would increase by $ 23 million Mindstretchers The reciprocal of would have to be 10 , which is impossible because the product of any number and its reciprocal is 1, but times any number is The area of the 15 12 ft × 12 ft room is larger since its area is 186 sq ft and the area of the 16 ft × 11 12 ft room is 184 sq ft 9 21 189 × 10 = × = =9 160 10 10 20 20 There are 20 gallons of gasoline in 10 12 gallons of gasohol So there are 10 10 − = 10 −9 =1 gallons of 20 20 20 20 ethyl alcohol in gasohol There are 9 −1 =8 = more gallons of 20 20 20 gasoline than ethyl alcohol in gasohol 1 1 ⋅ ⋅ ⋅ 1 = ⋅ ⋅ ⋅ 1 101 = = 50 2 Copyright © 2013 Pearson Education, Inc ⋅1 1 1 ⋅1 99 100 100 101 ⋅ ⋅ 99 100 1 ... change the divisor to its reciprocal, and multiply the resulting fractions When multiplying fractions, we can divide any numerator and any denominator by a common factor 7 × = 16 10 ⎛ ⎞⎛1⎞ ⎜⎝ ⎟⎠ ⎜⎝... There are an infinite number of fractions, such as = , = , = , and so on 2 3 4 a 42 = 42; 174 = 174; 406 = 406 79 b = = 158 Adding and Subtracting Fractions 26 = 20 + =+ 20 13 20 = + = 12 10 12... Adding and Subtracting Fractions 136 140 = 10 30 14 +3 =+3 15 30 17 30 = = + =+ 138 24 12 24 24 29 =1 24 24 = 10 − =− 18 7 acre = acre 142 8 20 − acre = − acre 20 13 acre 20 The area of the land