Solution Manual for Thinking Mathematically, 7th Edition

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SOLUTIONSMANUALDANIELS.MILLERNiagara County Community CollegeTHINKINGMATHEMATICALLYSEVENTHEDITIONRobert BlitzerMiami Dade College

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TABLE OF CONTENTSTHINKINGMATHEMATICALLY,7EChapter 1:Problem Solving and Critical Thinking1Chapter 2:Set Theory25Chapter 3:Logic51Chapter 4:Number Representation and Calculation125Chapter 5:Number Theory and the Real Number System159Chapter 6:Algebra: Equations and Inequalities227Chapter 7:Algebra: Graphs, Functions, and Linear Systems285Chapter 8:Personal Finance353Chapter 9:Measurement411Chapter 10:Geometry429Chapter 11:Counting Methods and Probability Theory469Chapter 12:Statistics527Chapter 13:Voting and Apportionment587Chapter 14:Graph Theory643

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Chapter 1Problem Solving and Critical Thinking1Check Points 1.11.Counterexamples will vary. Example:404016002.a.Add 6 each time.b.Multiply by 5 each time.3 + 6 = 925 = 109 + 6 = 15105 = 5015 + 6 = 21505 = 25021 + 6 = 272505 = 125027 + 6 = 332, 10, 50, 250, 12503, 9, 15, 21, 27, 33c.Cycle multiplying by 2, 3, 4.d.Cycle adding 8, adding 8, subtracting 14.32 = 61 +8 = 963 = 189 +8 = 17184 = 7217 –14 = 3722 = 1443 +8 = 111443 = 43211 +8 = 194324 = 172819 –14 = 517282 = 34565 +8 = 136, 18, 72, 144, 432, 1728, 345613 +8 = 2121 –14 = 79, 17, 3, 11, 19, 5, 13, 21, 73.a.Starting with the third number, each number is the sum of the previous two numbers, 29 + 47 = 76b.Starting with the second number, each number one less than twice the previous number,2(129)12574.The shapes alternate between rectangle and triangle.The number of little legs cycles from 1 to 2 to 3 and then back to 1.Therefore the next figure will be a rectangle with 2 little legs.5.a.Conjecture based on results: The original number is doubled.Select a number.41003Multiply the number by 4.4416104400403412Add 6 to the product.166224064606612618Divide this sum by 2.22211462236231829Subtract 3 from the quotient.113823320330936Summary of results:4810200036b.Select a number:nMultiply the number by 4:4nAdd 6 to the product:46nDivide this sum by 2:464623222nnnSubtract 3 from the quotient:2332nn

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Chapter 1Problem Solving and Critical Thinking2Concept and Vocabulary Check 1.11.counterexample2.deductive3.inductive4.trueExercise Set 1.11.Counterexamples will vary. Example: PresidentObama was younger than 65 at the time of hisinauguration.2.Counterexamples will vary. Example: BeyoncéKnowles is a singer who appears in movies.3.Counterexamples will vary. Example: 3 multipliedby itself is 9, which is not even.4.Counterexamples will vary. Example: 100 is athree-digit number and100100200,which isnot a four-digit number.5.Counterexamples will vary. Example: Adding 1 tothe numerator and denominator of12results in23which is not equal to12.6.Counterexamples will vary. Example:835,which is odd, but 8 and 3 are not both odd.7.Counterexamples will vary. Example: When1isadded to itself, the result is2,which is less than1.8.Counterexamples will vary. Example: When 1 isdivided by2,the result is12,which is greaterthan2.9.Pattern: Add 424 + 4 = 288, 12, 16, 20, 24, 2810.Pattern: Add 539 + 5 = 4419, 24, 29, 34, 39, 4411.Pattern: Subtract 517 – 5 = 1237, 32, 27, 22, 17, 1212.Pattern: Subtract 417 – 4 = 1333, 29, 25, 21, 17, 1313.Pattern: Multiply by 32433 = 7293, 9, 27, 81, 243, 72914.Pattern: Multiply by 45124 = 20482, 8, 32, 128, 512, 204815.Pattern: Multiply by 2162 = 321, 2, 4, 8, 16, 3216.Pattern: Multiply by 51255 = 6251, 5, 25, 125, 62517.Pattern: 1 alternates with numbers that aremultiplied by 2162 = 321, 4, 1, 8, 1, 16, 1, 3218.Pattern: 1 alternates with numbers that are increasedby 310 + 3 = 131, 4, 1, 7, 1, 10, 1, 1319.Pattern: Subtract 24– 2 =64, 2, 0,2,4,620.Pattern: Subtract 36– 3 =96, 3, 0,3,6,921.Pattern: Add 4 to the denominator111842212,16,110,114,118,12222.Pattern: Add 1 to the denominator115161,12,13,14,15,16

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Section 1.1Inductive and Deductive Reasoning323.Pattern: Multiply the denominator by 311273811,13,19,127,18124.Pattern: Multiply the denominator by 21182161,12,14,18,11625.Pattern: The second number is obtained by adding 4to the first number. The third number is obtained byadding 5 to the second number. The number beingadded to the previous number increases by 1 eachtime.3394226.Pattern: The second number is obtained by adding 3to the first number. The third number is obtained byadding 4 to the second number. The number beingadded to the previous number increases by 1 eachtime.2783527.Pattern: The second number is obtained by adding 3to the first number. The third number is obtained byadding 5 to the second number. The number beingadded to the previous number increases by 2 eachtime.38135128.Pattern: The second number is obtained by adding 3to the first number. The third number is obtained byadding 5 to the second number. The number beingadded to the previous number increases by 2 eachtime.37135029.Pattern: Starting with the third number, eachnumber is the sum of the previous two numbers.27447130.Pattern: Starting with the third number, eachnumber is the sum of the previous two numbers.19315031.Pattern: Cycle by adding 5, adding 5, thensubtracting 7.1351832.Pattern: Cycle by adding 6, adding 6, thensubtracting 10.1361933.Pattern: The second number is obtained bymultiplying the first number by 2. The third numberis obtained by subtracting 1 from the secondnumber. Then multiply by 2 and then subtract 1,repeatedly.3413334.Pattern: The second number is obtained bymultiplying the first number by 3. The third numberis obtained by subtracting 1 from the secondnumber. Then multiply by 3 and then subtract 1,repeatedly.123112235.Pattern: Divide by411( 4)4 64,16, 4,1,1436.Pattern: Divide by511( 5)5 125,25, 5,1,1537.Pattern: The second value of each pair is 4 less thanthe first.3 – 4 =11122(6, 2), (0,4), (7, 3), (2,2), (3,1)38.Pattern: The second value of each pair is the squareof the first.2164749 525162411439525636749,,,, (7, 49),,,,39.The figure cycles from square to triangle to circleand then repeats. So the next figure is40.The figure rotates90counterclockwise. So thenext figure is

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Chapter 1Problem Solving and Critical Thinking441.The pattern is to add one more letter to theprevious figure and use the next consecutive letterin the alphabet. The next figure is shown at right.42.The figure alternates from triangle to square and gains one line on the bottom. The next figure is.43.a.Conjecture based on results: The original number is doubled.Select a number.41003Multiply the number by 4.4416104400403412Add 8 to the product.168244084808812820Divide this sum by 2.242124822482420210Subtract 4 from the quotient.1248244204401046Summary of results:4810200036b.4n484848242222442nnnnnn44.a.Conjecture based on results: The result is always 2.Select a number.41003Multiply the number by 3.431210330030339Add 6 to the product.12618306360669615Divide this sum by 3.1836363126321535Subtract the original from the quotient.64212102202532Summary of results:421020232b.3n363636233322nnnnnn45.a.Conjecture based on results: The result is always 3.Select a number.41003Add 5 to the number.45910515055358Double the result.92181523052108216Subtract 4.1841430426104616412Divide the result by 2.1427262136231226Subtract the original number.74313103303633Summary of results:431030333b.5n2(5)2102104262626322233nnnnnnnnn

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Section 1.1Inductive and Deductive Reasoning546.a.Conjecture based on results: The result is always 5.Select a number.41003Add 3 to the number.43710313033336Double the result.7214132263266212Add 4.1441826430641012416Divide the result by 2.18293021510251628Subtract the original number.94515105505835Summary of results:451050535b.3n2(3)26264210210210522255nnnnnnnnn47.Using inductive reasoning we predict671234562.Arithmetic verifies this result: 21 = 2148.Using inductive reasoning we predict1873691215182.Arithmetic verifies this result: 63 = 6349.Using inductive reasoning we predict135791166.Arithmetic verifies this result: 36 = 3650.Using inductive reasoning we predict5 9(59)59.Arithmetic verifies this result:5 9(59)595 9(14)5945(14)59595951.Using inductive reasoning we predict9876593888,888.Arithmetic verifies this result:9876593888,888888,8853888,888888,888888,88852.Using inductive reasoning we predict5432191488,888.Arithmetic verifies this result:5432191488,888488,8891488,888488,888488,88853.The first multiplier increases by 33.132 + 33 = 165The second multiplier is 3367.The product increases by 111,111.1653367 = 555,555 is correct.

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Chapter 1Problem Solving and Critical Thinking654.The pattern implies we should attach a 6 to the right of the first multiplier. The second multiplier is always 8. Thepattern implies we should add 6 to that product to obtain 987,654.123,4568 + 6 = 987,654 is correct.55.b; The resulting exponent is always the first exponent added to twice the second exponent.56.c; The resulting exponent is always half the sum of the three exponents.57.deductive; The specific value was based on a general formula.58.inductive; The general conclusion for all HMO patients was based on specific observations.59.inductive; The general conclusion for all full-time four-year colleges was based on specific observations.60.deductive; The specific grade was based on a general course policy.61.a.1, 3, 6, 10, 15, and 21 are followed by21 + 7 = 2828 + 8 = 3636 + 9 = 4545 + 10 = 5555 + 11 = 661, 3, 6, 10, 15, 21, 28, 36, 45, 55, and 66.b.4 – 1 = 39 – 4 = 516 – 9 = 725 – 16 = 9The successive differences increase by 2.25 + 11 = 3636 + 13 = 4949 + 15 = 6464 + 17 = 8181 + 19 = 100c.The successive differences are 4, 7, and 10. Since these differences are increasing by 3 each time. The next fivenumbers will be found by using differences of 13, 16, 19, 22, and 25.22 + 13 = 3535 + 16 = 5151 + 19 = 7070 + 22 = 9292 + 25 = 117d.Ifa triangular number is multiplied by 8 and then 1 is added to the product, a square number is obtained.62.Each row begins and ends with 1. Other numbers are the sum of the two values that are diagonally above.66.does not make sense; Explanations will vary. Sample explanation: Such conclusions would be certain.67.makes sense

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Section 1.1Inductive and Deductive Reasoning768.does not make sense; Explanations will vary. Sample explanation: Though this sample was 51%, it is not certain that thisexact percentage will hold for the entire population.69.makes sense70.The pattern suggests that the compatible expression is the square of the first number minus twice the product of the twonumbers, plus the square of the second number.2(117)1211544971.a.The sums are all 30:16311510159174b.The sums are all 36:175149121510197c.For any values ofa,b, andc, the sums of all rows, all columns, and both diagonals are the same.d.The sums of the expressions in each row, each column, and each diagonal is 3a.e.Finding each sum verifies the conjecture that they are all 3a.First row:()()()3ababcacaSecond row:()( )()3abcaabcaThird row:()()()3acabcabaFirst Column:()()()3ababcacaSecond Column:()( )()3abcaabcaThird Column:()()()3acabcabaFirst Diagonal:()( )()3abaabaSecond Diagonal:()( )()3acaaca72.Answers will vary. Possible answer: 5, 10, 15 or 5, 10, 20.51 = 5052552 = 101521053 = 152522073.a.The result is a three- or four- digit number in which the thousands and hundreds places represent the month of thebirthday and the tens and ones places represent the day of the birthday.b.5[4(56)9]1655[20249]1655[2033]165100165165100MDMDMDMDMD

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Chapter 1Problem Solving and Critical Thinking874.a.66 = 366666 = 4356666666 = 443,55666666666 = 44,435,556b.An additional digit of 6 is attached to thenumbers being multiplied. An additional digitof 4 is attached to the left of the result and anadditional digit of 5 is placed between the 3 andthe 6.c.6666666666 = 4,444,355,556666,666666,666 = 444,443,555,556d.Inductive reasoning; it uses an observed patternand draws conclusions from that pattern.75.a.33673 = 1010133676 = 2020233679 = 30303336712 = 40404b.The first multiplier is always 3367. The secondmultipliers are successive multiples of 3. Theproduct increases by 10101.c.336715 = 50505336718 = 60606d.Inductive reasoning; it uses an observed patternand draws conclusions from that pattern.Check Points 1.21.a.The digit to the right of the billions digit is lessthan 5. Thus, replace all the digits to the rightwith zeroes. 7,476,242,056 rounded to thenearest billion is 7,000,000,000.b.The digit to the right of the ten millions digit is5 or greater. Thus, add 1 to the ten millionsdigit and replace all the digits to the right withzeroes. 7,48,242,056 rounded to the nearesthundred millions is 7,480,000,000.2.a.The digit to the right of the tenths digit is lessthan 5. Thus, 3.141593 rounded to the nearesttenth is 3.1.b.The digit to the right of the ten-thousandthsdigit is greater than 5. Thus, 3.141593 roundedto the nearest ten-thousandth is 3.1416.3.a.$3.40$2.25$5.60$5.40$3.40$2.853.95$3$2$6$5$3$34$26b.The given bill is not reasonable. It is too high.4.a.Round $52 per hour to $50 per hour andassume 40 hours per week.40 hours$50$2000weekhourweekThe architect’s salary is$2000per week.b.Round 52 weeks per year to 50 weeks per year.$200050 weeks$100,000weekyearyearThe architect’s salary is$100, 000per year.5.a.0.482148.72b.0.521001050Your family spent approximately $1050 onheating and cooling last year.6.a.The yearly increase in life expectancy can beapproximated by dividing the change in lifeexpectancy by the change in time from 1950 to2020.81.971.110.80.15 yr2020195070for eachsubsequent birth year.b.number of yearslife expectancyyearlyfrom 1950 to 2050in 1950increase71.10.15 ( 20501950 )71.10.15(100)71.11586.1 yr7.a.about 22%b.The greatest rate of increase in the percentageof college students who smoked cigarettes canbe found by identifying the portion of the graphwith the largest upward slope. This occursbetween 1994 and 1998.c.Approximately 24% of college students smokedcigarettes in 1982 and 1994.d.2014 was the year in which the least percentageof college students smoked cigarettes. Thepercentage that year was 13%.

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Section 1.2Estimation, Graphs, and Mathematical Models98.a.The yearly increase in tuition and fees can beapproximated by dividing the change in tuitionand fees by the change in time from 2000 to2016.$33, 480$15,518$17, 962$11232016200016b.Cost inyearly2000increase15,5181123Txc.2020 is 20 years after 2000. Thus,15,518112315,5181123(20)$37,978TxConcept and Vocabulary Check 1.21.estimation2.circle graph3.mathematical model4.true5.true6.falseExercise Set 1.21.a.39,144,200b.39,145,000c.39,140,000d.39,100,000e.39,000,000f.40,000,0002.a-f.Answers will vary depending upon statechosen.3.2.7184.2.71835.2.718286.2.7182827.2.7182818288.2.71828182859.350600950Actual answer of 955 compares reasonably well10.2508001050Actual answer of 1045 compares reasonably well11.911929Actual answer of 29.23 compares quite well12.832435Actual answer of 35.34 compares quite well13.321121Actual answer of 20.911 compares quite well14.461531Actual answer of 30.893 compares quite well15.406240Actual answer of 218.185 compares not so well16.807560Actual answer of 512.98 compares not so well17.0.8400320Actual answer of 327.06 compares reasonably well18.0.7200140Actual answer of 141.37 compares quite well19.48316Actual answer of 16.49 compares quite well20.55511Actual answer of 11.62 compares quite well21.30% of 200,000 is 60,000Actual answer of 59,920.96 compares quite well22.40% of 300,000 is 120,000Actual answer of 122,432.52 compares reasonablywell23.$3.47$5.89$19.98$2.03$11.85$0.23$3$6$20$2$12$0$4324.$4.23$7.79$28.97$4.06$13.03$0.74$4$8$29$4$13$1$59

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Chapter 1Problem Solving and Critical Thinking1025.Round $19.50 to $20 per hour.40 hours per week(40$20) per week = $800/weekRound 52 weeks to 50 weeks per year.50 weeks per year(50$800) per year = $40,000$19.50 per hour$40,000 per year26.Round $29.85 to $30 per hour.40 hours per week(40$30) per week = $1200/weekRound 52 weeks to 50 weeks per year.50 weeks per year(50$1200) per year = $60,000$29.85 per hour$60,000 per year27.Round the $605 monthly payment to $600.3 years is 36 months.Round the 36 months to 40 months.$60040 months = $24,000 total cost.$605 monthly payment for 3 years$24,000 totalcost.28.Round the $415 monthly payment to $400.4 years is 48 months.Round the 48 months to 50 months.$40050 months = $20,000 total cost.$415 monthly payment for 3 years$20,000 totalcost.29.Round the raise of $310,000 to $300,000.Round the 294 professors to 300.$300,000 ÷ 300 professors = $1000 per professor.$310,000 raise$1000 per professor.30.Round the raise of $310,000 to $300,000.Round the 196 professors to 200.$300,000 ÷ 200 professors = $1500 per professor.$310,000 raise$1500 per professor.31.Round $61,500 to $60,000 per year.Round 52 weeks per year to 50 weeks per year.50 weeks40 hours per week = 2000 hours$60,000 ÷ 2000 hours = $30 per hour$61,500 per year$30 per hour32.Round $38,950 to $40,000 per year.Round 52 weeks per year to 50 weeks per year.50 weeks40 hours per week = 2000 hours$40,000 ÷ 2000 hours = $20 per hour$38,950 per year$20 per hour33.8036524700,800 hr34.4036524350, 400 hr35.0.210020400.50.5Actual answer of 42.03 compares quite reasonable.36.0.590451800.250.25Actual answer of 169.62 compares somewhatreasonable.37.The given information suggests $30 would be agood estimate per calculator.$3010$300which is closest to choice b.38.The given information suggests $7 would be a goodestimate per calculator.$710$70which is closest to choice c.39.The given information suggests 65 mph would be agood rate estimate and 3.5 would be a good timeestimate.653.5227.5which is closest to choice c.40.The given information suggests 45 mph would be agood rate estimate and 3.5 would be a good timeestimate.453.5157.5which is closest to choice b.41.The given information suggests you can count 1number per second.100002.77 or 3 hours606042.The given information suggests you can count 1number per second.1,000, 00011.57 or 12 days60602443.0.1016,000,0001,600, 00010% of 16,000,000 is 1,600,000 high schoolteenagers.44.0.2016,000,0003, 200, 00020% of 16,000,000 is 3,200,000 high schoolteenagers.45.a.about 85 people per 100b.(8523)87540046.a.about 66 people per 100b.(6627)722800

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Solution Manual for Thinking Mathematically, 7th Edition - Page 15 preview image

Section 1.3Problem Solving1147.a.26.39.716.60.52012198032The annual increase is about 0.5%.b.9.70.5(20201980)9.70.5(40)29.7In 2020 the percentage will be approximately29.7%.48.a.21.76.715.90.52012198032The annual increase is about 0.5%.b.6.70.5(20201980)6.70.5(40)26.7In 2020 the percentage will be approximately26.7%.49.a.The percent body fat in 45-year-old women isabout 37%.b.The percent body fat in women reaches amaximum at age 55 of about 38%c.Women have 34% body fat at age 25.50.a.The percent body fat in 25-year-old men isabout 23%.b.The percent body fat in men reaches amaximum at age 65 of about 26%c.Men have 24% body fat at age 35.51.a.401310911.4 ppm per year2015195065b.3101.4Cxc.2050 is 100 years after 1950.3101.4(100)450 ppmC52.a.58.4456.981.460.022015195065Fb.56.980.02Txc.2050 is 100 years after 1950.56.980.02(100)58.98TF66.makes sense67.does not make sense; Explanations will vary.Sample explanation: Very large numbers and verysmall numbers often must be estimated when usinga calculator.68.makes sense69.does not make sense; Explanations will vary.Sample explanation: Some mathematical modelscan break down over time.70.Since there are infinitely many digits, the digits cannot be reversed.71.a72.d73.b74.c75.20165016, 000 hours.16, 000667 days246671.8 yr36576.Round days in a year to 400.$1, 000, 000,0001, 000,000 days$1000 / day1,000,000 days400 days/year2500 yearsCheck Points 1.31.The amount of money given to the cashier isunknown.2.Step 1: Understand the problem.Bottles: 128 ounces costs $5.39Boxes: a 9-pack of 6.75 ounce boxes costs $3.15We must determine whether bottles or boxes are thebetter value.Step 2: Devise a plan.Dividing the cost by the number of ounces will giveus the cost per ounce. We will need to multiply 9 by6.75 to determine the total number of ounces theboxes contain. The lower cost per ounce is the bestvalue.Step 3: Carry out the plan and solve the problem.Unit price for the bottles:$5.39$0.042 per ounce128 ouncesUnit price for the boxes:$3.15$3.15$0.052 per ounce96.75 ounces60.75 ouncesBottles have a lower price per ounce and are thebetter value.Step 4: Look back and check the answer.This answer satisfies the conditions of the problem.

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Solution Manual for Thinking Mathematically, 7th Edition - Page 16 preview image

Chapter 1Problem Solving and Critical Thinking123.Step 1: Understand the problem.We are given the cost of the computer, the amountof cash paid up front, and the amount paid eachmonth. We must determine the number of months itwill take to finish paying for the computer.Step 2: Devise a plan.Subtract the amount paid in cash from the cost ofthe computer. This results in the amount still to bepaid. Because the monthly payments are $45, dividethe amount still to be paid by 45. This will give thenumber of months required to pay for the computer.Step 3: Carry out the plan and solve the problem.The balance is$980$350$630.Now divide the$630 balance by $45, the monthly payment.$45month630 months$630$63014 months.month$4545Step 4: Look back and check the answer.This answer satisfies the conditions of the problem.14 monthly payments at $45 each gives14$45$630.Adding in the up front cashpayment of $350 gives us$630$350$980.$980 is the cost of the computer.4.Step 1: Understand the problem.The total change must always be 30 cents. Onepossible coin combination is six nickels. Another isthree dimes. We need to count all suchcombinations.Step 2: Devise a plan.Make a list of all possible coin combinations. Beginwith the coins of larger value and work toward thecoins of smaller value.Step 3: Carry out the plan and solve the problem.QuartersDimesNickels101030022014006There are 5 combinations.Step 4: Look back and check the answer.Check to see that no combinations are omitted, andthat those given total 30 cents. Also double-checkthe count.5.Step 1: Understand the problem.We must determine the number of jeans/T-shirtcombinations that we can make.For example, one such combination would be towear the blue jeans with the beige shirt.Step 2: Devise a plan.Each pair of jeans could be matched with any of thethree shirts. We will make a tree diagram to showall combinations.Step 3: Carry out the plan and solve the problem.There are 6 different outfits possible.Step 4: Look back and check the answer.Check to see that no combinations are omitted, anddouble-check the count.6.Step 1: Understand the problem.There are many possible ways to visit each cityonce and then return home. We must find a routethat costs less than $1460.Step 2: Devise a plan.From cityAfly to the city with the cheapestavailable flight. Repeat this until all cities have beenvisited and then fly home. If this cost is above$1460 then use trial and error to find otheralternative routes.Step 3: Carry out the plan and solve the problem.AtoDcosts $185,DtoEcosts $302,EtoCcosts$165,CtoBcosts $305,Bback toAcosts $500$185 + $302 + $165 + $305 + $500 = $1457The routeA,D,E,C,B,Acosts less than $1460Step 4: Look back and check the answer.This answer satisfies the conditions of the problem.Trick Questions 1.31.The farmer has 12 sheep left since all but 12 sheep died.2.All 12 months have [at least] 28 days.3.The doctor and brother are brother and sister.4.You should light the match first.Concept and Vocabulary Check 1.31.understand2.devise a plan3.false4.false

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