CHAPTER ONE CRITICAL THINKING SKILLS

Transcription

1 CHAPTER ONE CRITICAL THINKING SKILLS Exercise Set a) 1,,, 4, 5, b) Counting numbers. a) If a b has a remainder of zero, then a is divisible by b. b) 4, 8, 1 c) 9, 18, 7. A conjecture is a belief based on specific observations that has not been proven or disproven. 4. Inductive reasoning is the process of reasoning to a general conclusion through observation of specific cases. 5. Deductive reasoning is the process of reasoning to a specific conclusion from a general statement. 6. A counterexample is a specific case that satisfies the conditions of the conjecture but shows the conjecture is false. 7. Inductive reasoning 8. Deductive reasoning 9. Inductive reasoning, because a general conclusion was made from observation of specific cases. 10. Inductive reasoning, because a general conclusion was made from observation of specific cases ( 1+ 4 ) 10( ) 10( ) 5( ) , , 18, 1 (Add to previous number.) 0., - 4, - 10 (Subtract 6 from previous number.) , 1, - 1 (Alternate - 1 and 1.). - 5, - 7, - 9 (Subtract from previous number.) 4. 16, - 486, 1458 (Multiply previous number by.) ,, (Multiply previous number by.) , 49, 64 (The numbers in the sequence are the squares of the counting numbers.) 6. 1,8,6 ( ,1+ 7 8, ) 1

2 CHAPTER 1 Critical Thinking Skills 7. 4, 55, 89 (Each number in the sequence is the sum of the previous two numbers.) 9. Y: There are three letters in the pattern , so the 117 th entry is the second R in the pattern. Therefore, the 118 th entry is Y. 1. a) 6, 49, 64 b) Square the numbers 6, 7, 8, 9 and 10. c) is not a square number since it falls between the two square numbers 64 and ,, (Multiply previous number by.) 0. a) Answers will vary. b) The sum of the digits is 9. c) The sum of the digits in the product when a one or two digit number is multiplied by 9 is 9.. a) 8 and 6 b) To find the 7 th triangular number, add 7 to the 6 th triangular number. To find the 8 th triangular number, add 8 to the 7 th triangular number. To find the 9 th triangular number, add 9 to the 8 th triangular number. To find the 10 th triangular number, add 10 to the 9 th triangular number. To find the 11 th triangular number, add 11 to the 10 th triangular number. c) is not a triangular number since it falls between the two triangular numbers 66 and 78.. Blue: 1, 5, 7, 10, 1 Purple:, 4, 6, 9, 11 Yellow:, 8 4. a) 19 (Each new row has two additional triangles.) b) a) 58 million b) 45 million c) We are using observation of specific cases to make a prediction. 6. a) $8,000 b) $61,000 c) We are using observation of specific cases to make a prediction. 7. P B P B B P B P P B P B B P B P a) You should obtain the original number. b) You should obtain the original number. c) Conjecture: The result is always the original number. 4n+ 8 4n 8 d) n,4 n,4n+ 8, + n+, n+ n a) You should obtain twice the original number. b) You should obtain twice the original number. c) Conjecture: The result is always twice the original number. 10n+ 5 10n 5 d) n,10 n,10n+ 5, + n+ 1, n+ 1 1 n 5 5 5

3 SECTION a) You should obtain the number 5. b) You should obtain the number 5. c) Conjecture: No matter what number is chosen, the result is always the number 5. n+ 10 n 10 d) nn, + 1, n+ ( n+ 1) n+ 1, n n+ 10, + n+ 5, n+ 5 n 5 4. a) You should obtain the number 0. b) You should obtain the number 0. c) Conjecture: No matter what number is chosen, the result is always the number 0. n+ 10 n+ 10 d) nn, + 10,,5 n+ 10, n nn, n ,001 is one example is one example. 45. Two is a counting number. The sum of and is 5. Five divided by two is 5, which is not an even number. 46. One and three are counting numbers. The product of 1 and is, which is not divisible by. 47. One and two are counting numbers. The difference of 1 and is 1 1, which is not a counting number. 48. The sum of the odd numbers 1 and 5 is 6, which is not divisible by a) The sum of the measures of the interior angles should be 180. b) Yes, the sum of the measures of the interior angles should be 180. c) Conjecture: The sum of the measures of the interior angles of a triangle is a) The sum of the measures of the interior angles should be 60. b) Yes, the sum of the measures of the interior angles should be 60. c) Conjecture: The sum of the measures of the interior angles of a quadrilateral is , the numbers in positions are found as follows: a b c a+ b+ c , 8008, 8118 (They look the same when looked at in a mirror.) 5. Counterexample 54. c Exercise Set 1. (Note: Answers in this section will vary depending on how you round your numbers. The answers may differ from the answers in the back of the textbook. However, your answers should be something near the answers given. All answers are approximate.) , , ,00,000, % of % of , , ,000, , 00,000, ,000

4 4 CHAPTER 1 Critical Thinking Skills 1. 5 $ $0.40 $0 14. hours $7.95 per hour hours $8 per hour $ mi lb 17. $.9 + $ $ $ $ $ $.49 + $5.65 $ + $1 + $5 + $7 + $50 + $ $.50 + $5.70 $ lb $44,569 $45,000 $ ,798 14,9,800 14,00 18,500 lb lb. 8% of $14,876 8% of $15, $15,000 $100.,66 4, mi $10.87 $11 $.6 $.70 per pound ( $ $9.95) 1 ( $0 + $40) 1 ( $70 ) $ Team A: Team B: lb 7. 15% of $ % of $ $40 $ grubs per square foot ( 60 ft 80. ft) 4 grubs per square foot ( square feet) grubs 19, 00 grubs Mexican pesos U.S. dollars U.S. dollars 9 U.S. dollars $50 $9 $41 0. $97+ 6 ( $41) + 6 ( $97) + 6 ( $90) $ ( $40) + 6 ( $100) + 6 ( $90) $970 + $40 + $600 + $540 $ miles. 70 miles. a) 0.98% 105 million 1% 105 million million.55 million.6 million b) 18.41% % counties c) The counties that use punch cards could be the largest counties with the most voters. 5. a) 4 million b) 98 million c) 98 million 4 million 64 million d) 19, , , , , million 4. a) 9% $40,075 40% $40,000 $16,000 b).9% $40,075 % $40,000 $ a) 19% b) 5% c) 8% of 180 lb lb

5 SECTION a) 8% b) 65% 45% 0% c) 8% of 110, , , , 771 sq mi d) No, since we are not given the area of each state. 8. a) ( 410) + 4( 545) ( 400) + 4( 550) calories b) Running: 4( 90) 4( 95) 700 calories Casual bike riding: 4( 00) 100 calories, calories c) ( 545) + ( 545) ( 550) + ( 550) calories per week, 00 calories per week ( 5 weeks ) ,000 calories berries leaves % 46. 5% square units square units feet ( 6) 10 in. or ft Answers will vary. 60. There are 118 ridges around the edge. 61. There are 6 dimples on a regulation golf ball. 6. a) Answers will vary. b) 60 seconds per minute 60 minutes per hour 4 hours per day seconds per day 86,400 seconds per day, 6. Answers will vary. The U.S. government categorized the middle class as $,000 - $50,000 in ,000, days 86,400 Exercise Set 1. 1 in..75 in mi x mi 1x ( ) x mi. 1 in. x in. 1 ft 8 ft 1x 1 8 ( ) 1x x 6 6 in. or in

6 6 CHAPTER 1 Critical Thinking Skills. ft 48.4 ft 1. ft x ft x ( ) x x 19.6 ft 4. 1 bag x bags 6000 ft 6,000 ft ( ) 6000x 1 6, x 6, , 000 x bags % of $ ( $4 ) $485.5 $4 + $485.5 $ Cost for mileage: 1 $0.0 $0.0( 1)( 5 ) $ Cost for sitting still: minutes ( 60) 10 seconds 10 $0.0 $0.0( 4 ) $1.0 0 Cost for ride: $.00 + $ $1.0 $ , 000 miles 1000 gallons 0 miles per gallon ( ) ( ) Hawaii: 1000 $.0 $00 South Carolina: 1000 $1. $10 $00 $10 $ a) or 80 hours 000 hours b) hr wk 5 weeks 1500 hours c) hr wk 5 weeks 9. Denise parks her car for eight hours per day. + ( ) [ ] ( ) 5 $.50 $ hours per day 5 $.50 + $ $9.50 $47.50 Savings: $47.50 $5.00 $ $10 + $80( 15 ) $10 + $100 $10 Savings: $10 $150 $ year mortgage: $75.40( 1)( 0 ) $180,576 0 year mortgage: $660.60( 1)( 0 ) $7,816 Savings: $7,816 $180,576 $57, $.75 + ( 1 )( $0.50 ) $ ( $0.50) $.75 + $9 $ $0,000 down payment: ( )( ) $0,000 + $ $0,000 + $51, $71, $40,000 down payment: $40,000+$559.0( 1)( 0) $40,000 + $01,1 $41,1 Savings: $71, $41,1 $0, Points needed for 80 average: 80( 5) 400 points Wallace s points so far: points Grade needed on fifth exam:

7 SECTION a) ;.54;.54; ; So, 48(.54) 1. b) Answers will vary. A close approximation can be obtained by multiplying the U.S. sizes by a) min 50 b) min a) ,000 b) 1 in 10, ,687.0 mi 8,451.4 mi 5.6 mi 5.6 mi mpg 1.6 gal c) min 5 d) min a) 11% of 7,00, ( 7,00,000) 0,06,000 b) 10% of 970, ( 970,000) 97,000 c) % of 970, ( 970,000) 9, a) 40 $ $17,680 b) Each week he makes 40 $8.50 $40. $ weeks $40 1. By mail: ( $ $ $8.56) 4 $ $67.84 Tire store: $ $4 $4 + $5.9 $49.9 Savings: $49.9 $67.84 $8.08. a) $60( 0.1 ) $74.40 b) $100( 0. ) $64 c) The store lost $100 $1000 $00 on the purchase. Store's profit: $64 $00 $64. ( ) ( ) 4. a) 1 yd ft 9 ft ,600 ft 1 gal x gal 50 ft 1,600 ft 50x 1( 1,600) 50x 1, ,600 x gal cm 60 sec 60 min 4 hr 65 days,15,600 cm b) 0 cm 0 cm 0 cm1,000 cm 0.1 cm 60 sec 60 min 4 hr , days 8640

8 8 CHAPTER 1 Critical Thinking Skills 5. Let x the amount above $1,000 $4950 $100 $ x $ x $,600 $1,000 + $.600 $4, ( ) + 5( 1) + 4( 9) + ( 01) + ( 1408) + 1( 10,5) ,5 1,906 violations 9. a) Yes, divide the total emissions by the emissions per capita. b) million 4. c) million or 1.41 billion Value after first year: $ ( $1000) $ $100 $1100 ( ) Value after second year: $ $1100 $1100 $110 $990 $990 is less than the intial investment of $ a) 1 oz 60 min 4 hr 65 days 55,600 oz 55, gal 18 b) $ $11.0 $ a) 0,000 0, gal b) 5.61 $ $56.98 c) 140,000, ,985, 400,000 gal 0. Cost after 1 year: $ ( $450) $450 + $7 $477 ( ) Cost after years: $ $477 $477 + $8.6 $ After paying the $100 deductible, Yungchen must pay 0% of the cost of x-rays. First x-ray: $ ( $540 ) $100 + $108 $08 ( ) Second x-ray: 0.0 $90 $184 Total: $08 + $184 $9. a) $ The maximum number of 10 packs is 4. $41 ( 4 $41) $00 $164 $ 6 $ 00, $ Deirdre can also buy two 4 packs. $17 10 packs 4 packs Number of rolls Cost 4 4( 10) ( 4) 48 + ( ) ( ) 4 46 $ $ $ $187 Maximum number of rolls of film is 48. b) The cost is $198 when she purchases four 10 packs and two 4 packs. 4 $41 + $17 $198

9 SECTION a) $ The maximum number of 4 packs of 6 exposures is 8. $5.76 ( 8 $5.76) $50 $46.08 $. 9 $ 50, Erika cannot buy any 4 exposures. 4 packs of 6 exp. 4 packs of 4 exp. Number of exposures Cost 8 0 8( 6) + 0( 4) 88 ( ) ( ) 8 $ $4.08 $ $ $ $ $ $ $ $ $48.96 packs of 4 exposures and 7 packs of 6 exposures, or 5 packs of 4 exposures and 5 packs of 6 exposures, or 8 packs of 4 exposures and packs of 6 exposures b) 00 exposures in each case c) The minimum cost is $48.48 when she purchases packs of 4 exposures and 7 packs of 6 exposures. 5. a) water/milk: (1) cups salt: cream: () 9 tbsp cup (because 16 tbsp 1 cup) b) water/milk:. 875 cups cups 8 salt: tsp tsp cream: cups cup ( 16 tbsp ) 10 tbsp c) water/milk: 1 cups salt: tsp cream: cup 9 tbsp d) Differences exist in water/milk because the amount for 4 servings is not twice that for servings. Differences also exist in Cream of Wheat because 1 cup is not twice tbsp. 8 tsp

10 10 CHAPTER 1 Critical Thinking Skills 1 1 cups water: 1 ( 4) ( 4) 5 cups water: ( ) ( ) 4 4 cups salt: ( 4) 1 tsp salt: ( ) 1 tsp 4 butter/margarine: 1 ( 4) 4 tsp butter/margarine: ( ) 4 tsp c) rice: cups water: cups 1 4 salt: + 1 tsp butter/margarine: 1 tsp + 1 tbsp 1 tsp + tsp 4 tsp d) rice: 1 cups 6. a) rice: ( 4) cups b) rice: ( ) water: 6 cups salt: 1 1 tsp butter/margarine: tbsp ( tsp) 6 tsp 6 tsp tsp 4 tsp e) Differences exist in water because the amount for 4 servings is not twice that for servings. 1 ft would be 1 in. by 1 in. Thus, 1 ft 1 in. 1 in. 144 in. 9. Area of original rectangle lw Area of new rectangle ( l)( w) 4lw Thus, if the length and width of a rectangle are doubled, the area is 4 times as large and ft 1 in. 1 in. 1 in. 178 in. 40. Volume of original cube lwh Volume of new cube ( l)( w)( h) x 10( 10) ( ) ( ) ( 6 ) 4 10 ( ) 1( 6) 6 4. Left side: Right side: Place it at 1 so the left side would total lwh Thus, if the length, width, and height of a cube are doubled, the volume is 8 times as large or increases eightfold. 10 pieces 1000 pieces 4. $ x $ x x $

11 SECTION Birds Lizards Number of Heads Number of Feet ( ) + 14( 4) ( ) + 1( 4) ( ) + 1( 4) 68 Therefore, there are 10 birds and 1 lizards ;00, 11,,, 44, 55, 66, 77, 88, a) Place the object, 1 g, and g on one side and g on the other side. b) Place the object, 9 g, and g on one side and 7 g and 1 g on the other side. 50. Eight pieces a) ( 4 4) + ( ) + ( ) + ( 1 1) b) ( 7 7) + ( 6 6) + ( 5 5) ; ; The sum of the four corner entries is 4 times the number in the center of the middle row ,1, Multiply the number in the center of the middle row by , 6, 99 Multiply the number in the center of the middle row by cubes ways 58. Each shakes with four people Other answers are possible, but 1 and 8 must appear in the center.

12 1 CHAPTER 1 Critical Thinking Skills Other answers are possible. 6. With umbrella policy: Mustang reduced premium: $1648 $90 $1558 Focus reduced premium: $ ( $150) $150 $18.60 $ Total for umbrella policy: $ $ $450 $54.40 Without umbrella policy: $ $150 $178 Net amount for umbrella policy: $54.40 $178 $ Mary is the skier Areas of the colored regions are: 1 1, 1 1,,, 5 5, 8 8, 1 1, 1 1 ; square units giraffe frogs 1 giraffe lions lions frogs Therefore, lion frog. Therefore, 1 lion frog. 1 lion ostriches Therefore, frog ostriches. frog ostriches Therefore, 1 frog ostriches. Review Exercises 1., 8, (Add 5 to previous number.). 5, 6, 49 (next three perfect squares). 48, 96, 19 (Multiply previous number by.) 4. 5,, 40 ( , 5 + 7, ) 5. 15, 9, ( ,15 6 9, 9 7 ) 6. 8, 16, (Multiply previous number by 1.)

13 REVIEW EXERCISES c 10. a) The original number and the final number are the same. b) The original number and the final number are the same. c) Conjecture: The final number is the same as the original number. n + 10 n 10 d) n, n,n + 10, + n + 5, n n n, n + 5,6 n + 5 6n + 0,6n + 0 6n n 18 n + 9 n 9 6 n + 18, + n + 9, + n +, n + n ,5 is an odd number. 11. This process will always result in an answer of. ( ) 1 (Note: Answers for Ex. 1-5 will vary depending on how you round your numbers. The answers may differ from the answers in the back of the textbook. However, your answers should be something near the answers given. All answers are approximate.) 1. 10, , ,000, % of 101 0% of Answers will vary $ $1.10 $ % of $0 6% of $ mi 1 mi mi 0. $.49 + $ $ $ $.19 + $6.75 mph min 0 min 60 min $ + $1 + $ + $0 + $ + $7 $ % 5 in. in. 0 in. 0( 0.1 ) mi mi % 4. 1 square units 5. Length 1.75 in., 1.75( 1.5) ft Height 0.65 in., 0.65( 1.5) ft 7. 4 ( $.69 ) $10.76 for four six-packs Savings: $10.76 $9.60 $ To produce the 5 Oscars he found: 5 $7 $17,004 He was awarded $50,000 $17,004 $,996 more. 6. $ ( $1.50 ) $.00 + $10.50 $ Change: $0.00 $1.50 $ Akala's: hr 10 min, 8,8 $15 $ Berkman's: hr 10 min, 4, 0 4 $5 $100 Berkman s is the better deal by $10 $100 $ $ $ ( ) 50 1 $ ( 5 ) $ $ ( 5 ) $0.0 5 $ $0.0 $ $14.70 $16.0

14 14 CHAPTER 1 Critical Thinking Skills 1. 10% of $ $50 $5 $5 7 $ mg x mg 10 lb 47 lb Savings: $71 - $60 $11 10x 47( 1.5). $ ( $800 ) $800 $1140 $660 take-home 8% of $ $660 $ P.M. 4 hr 11 A.M. July 6, 11:00 A.M. 7. Each figure has an additional two dots. To get the hundredth figure, 97 more figures must be drawn, 97( ) 194 dots added to the third figure. Thus, x x 7.05 mg 4. 9 A.M. Eastern is 6 A.M. Pacific, from 6 A.M. Pacific to 1:5 P.M. Pacific is 7 hr 5 min, 7 hr 5 min 50 min stop 6 hr 45 min 6. a) 1 in. 1 in..54 cm.54 cm cm 6.45 cm b) 1 in. 1 in. 1 in..54 cm.54 cm.54 cm cm 16.9 cm 8. c) 1 in. x in..54 cm 1 cm.54x 1() 1.54x x in min 59 sec Since it doubles every second, the jar was half full 1 second earlier than 1 hour Nothing. Each friend paid $9 for a total of $7; $5 to the hotel, $ to the clerk. $5 for the room + $ for each friend + $ for the clerk $0 4. Let x the total weight of the four women x , x 50, 140 lb Yes; quarters and 4 dimes, or 1 half dollar, 1 quarter and 4 dimes, or 1 quarter and 9 dimes. Other answers are possible cm 6 cm 6 cm 16 cm

15 CHAPTER TEST Place six coins in each pan with one coin off to the side. If it balances, the heavier coin is the one on the side. If the pan does not balance, take the six coins on the heavier side and split them into two groups of three. Select the three heavier coins and weigh two coins. If the pan balances, it is the third coin. If the pan does not balance, you can identify the heavier coin. 47. n ( n + 1) 500( 501) 50,500 15, blue: 4 green 8 blue, yellow 5 blue, white blue : 101, 111, 11, 11, 141, 151, 161, 171, 181, 191, 50. The fifth figure will be an octagon with sides of equal length. Inside the octagon will be a seven sided figure with each side of equal length. The figure will have one antenna : The sixth figure will have 6 rows of 6 tiles and 5 rows of 5 tiles ( ). 5. Some possible answers are given below. There are other possibilities. 5. a) b) There are choices for the first spot. Once that person is standing, there are choices for the second spot and 1 for the third. Thus, 1 6. c) d) n n 1 n 1, (or n!), where n the number of people in line e) ( )( ) Chapter Test 1. 18, 1, 4 (Add to previous number.). 1 1,, (Multiply previous number by 1.). a) The result is the original number plus 1. b) The result is the original number plus 1. c) Conjecture: The result will always be the original number plus 1. 5n n 10 d) n, 5n,5n + 10, + n +, n + 1 n

16 16 CHAPTER 1 Critical Thinking Skills (Note: Answers for #4 - #6 will vary depending on how you round your numbers. The answers may differ from the answers in the back of the textbook. However, your answers should be something near the answers given. All answers are approximate.) , , , ,000,,.,000, square units 7. a) 10 lb in in b) He is in the at risk range. 8. $1.1 $9.6 $11.50 $ therms $ therms + first therms therms 9. $ $.59 The maximum number of 6 packs is 5. $15.00 ( 5 $.59 ) $15.00 $1.95 $.05 $ $0.80 Thus, two individual cans can be purchased. 6 packs Indiv. cans Number of cans The maximum number of cans is cut yields equal pieces. Cut each of these equal pieces to get 4 equal pieces. cuts (.5 min) 7.5 min in. by in by in. by 9.65 in. 9.5 in. by 9.6 in. (The actual dimensions are cm by 76.5 cm.) 1. $ $510 $ $191.5 $510 + $191.5 $701.5 $ $65.5 $ Christine drove the first 15 miles at 60 mph which took hr, and the second 15 miles at 0 mph which took hr for a total time of hr. If she drove the entire 0 miles at 45 mph, the trip would take hr (40 min) which is less than hr (45 min). 45 4

17 GROUP PROJECTS , ; 11 does not divide 11, jelly beans; , , a) $.99 $11.97 b) 9 ( $ ) $11.81 c) $11.97 $11.81 $0.16 Using the coupon is least expensive by $ : $ on * off $$$$, $$$*, $$*$, $*$$, *$$$, *$*$, *$$*, $*$* Group Projects $5 1. a) $108. b) Let x the amount before tax x+ 0.07x x x $0.74 $ $101.5 c) Inductive reasoning - arriving at a general conclusion from specific cases d) Combination set: $6.00 ( $ ) $6.00 $6.0 $55.80 Individual sets: $6.00 $7.00,$7.00 ( $ ) $7.00 $14.40 $57.60 Therefore, the combination set is cheaper. e) Combintion with tax: $ $59.71 Individual set with tax: $ $61.6 $ $59.71 $1.9. a) d) Answers will vary. e) 400 mi 50mi hr 8 hrs, 9 A.M. + 8 hrs 5 P.M. f) h) Answers will vary.. Order Name Apparel 1 Ernie holster Zeke vest Jed chaps 4 Tex stetson