Ratio & Proportion

7.1 Text HW Ratio & Proportion 1. Math 210 (Re PT5A pp71-79 Practice 5A) #1a,b. 3 : 6 = 1 : 2 6:4=3:2 #2. L:W is 16 : 12 = 4 : 3 #3. Saved: Spent = 35 : 15 #4. 2 : 7 = 4 : ? ? = 2@7 = 14 She used 14 liters of water. #5 60 m is divided into two parts, ratio 2: 3. Hmm... 5 parts altogether ... each part is 12 m.... 60 m 5 parts = 60 m ÂÅÅÅÅÅÅÅÅÅÅÅÄÅÅÅÅÅÅÅÅÅÅÅà ÆÉÉÉÈÉÉÇÆÉÉÉÉÉÉÉÉÈÉÉÉÉÉÉÉÉÇ ? (2 parts) (3 parts) 1 part = 12 m 3 parts = 36 m Checks out, adds up to 60 m 2 parts = 24 m A The shorter piece is 24 m long. #6. Aw : Jw is 6 : 5. If Aw is 48 kg, find Jw. 48 kg is 6 parts, so 1 part is 8 kg, and 5 parts is 40 kg. John’s ‘weight’ is 40 kg. #7. The ratio of B : G is 2 : 5 . If there are 100 B, how many B&G altogether? Algebraic-type solution: By the way, to solve for ?, I’d write 2 : 5 = 100 : ? x 5 ))) = ) Solve for ?, get 250. 100 2 So there are 100+250 = 350 altogether. OR: 2 : 7 = 100 : ? Solve for ? Get 350, getting the total number of children directly. “Scaling” solution: OR 2 parts = 100 OR 1 part = 50 Y 5 parts = 250 Y So there are 7 parts = 350 altogether. There are 100 + 250 = 350 altogether. “Bar-diagram” solution: ? ÂÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÄÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅà ÆÈÇÆÉÉÉÉÉÉÉÉÉÈÉÉÉÉÉÉÉÉÉÉÇ 100 2. pbf8 Once the bar diagram is drawn, the problem proceeds pretty much as shown via the “scaling method” above. (Re PT 5A, pp 80-82, Practice 5B #4-9) #4. #5. B : G is 7 : 4 If 121 total, the number of boy is 7@11 = 77 ( 11 parts total = 121, 1 part is 11, 7 parts is 7@11) Wm has $120. Steve has $20 less. What is ratio of Steve’s $ to Wm’s. $ ? 100 : 120 = 5 : 6 #6. 90 cm pole is painted G, W & B, in ratio 3 : 4 : 2 . 3:4:2 #7. Y 9 parts total. 9 parts = 90 cm 1 part = 10 cm 3 parts = 30 cm a. What length is G? b. What length is B? a. 30 cm length painted Green. b. 20 cm length painted Black. Cement, Sand & Stone are mixed in the ratio 1 : 2 : 4. If Sand & Stone total is 24 m 3 .... C = ? [In class we misread the question!!!] Sand & Stone together make 6 parts. So 6 parts = 24 m 3 and 1 part = 4 m 3 Cement is “1 part”, so we need 4 m 3 of Cement. 7.1A Text HW Ratio & Proportion 2 cont’d. Teacher’s Solution required for #8 & 9. #8. “The ratio of David’s weight to Ryan’s weight to Ali’s weight is 8 : 5 : 4. If Ryan weighs 30 kg, find the total weight of the 3 boys.” * 30kg * 5 parts = 30 kg ? ÷5 @ 1 part = 6 kg @ 17 * D * R *A 17 parts = 6@17 = 102 kg #9. 3. Ratio 12 to 16 ! """" """" """" !!!! !!!! !!!! !!!! Ratio 6: 8 ! """" """" """" !!!! !!!! !!!! !!!! Ratio 3:4 ! """" """" """" !!!! !!!! !!!! !!!! """" !!!! """" !!!! """" !!!! !!!! ² this also illustrates that 12 : 16 is 3 : 4 (Re PW 5A pp 82-90) text instructs us to not write the answers. Answers to check your work: Exercise 34 (pp 82-83) #1a. 3 : 4 #1b. #2a. 5 : 3 #2b. #3a. 2 : 3 #3b. #4a. 5 : 3 #4b. #5a. 3 : 7 #5b. #6a. 6 : 5 #6b. Exercise 35 (pp 84-85) #1a. 12 : 30 = 2 : 5 #1b. 9 : 15 = 3 : 5 #2. 2:3 3:1 1:4 3:5 5:3 2:1 5:6 4:5 1:2 5:4 #4. #5. #6. 3:4 3:5 3:2 3:5 7:3 5:6 #3a. #3b. #3c. #3d. #3e. #3f. 5 32 36 28 1 2 Length : width = 60" : 48" = 5 : 4 24 : 16 = 3 : 2 52 : 72 = 13 : 18 #3g. #3h. #3i. #3j. #3k. #3l. 1 12 4 15 1 4 pbf8 The total weight of the three boys is 102 kg. “3 boys share a sum of money in the ratio 5 : 3 : 2. If the smallest share is $30, find the biggest share.” The ratio of the smallest share to the biggest share is 2 : 5. 2 parts = $30 ÷2 1 part = $15 @5 5 parts = 5@$15 = $75 The biggest share is $75. The above is shown in the style of the text. However, 4. Math 210 7.1B Text HW Ratio & Proportion Math 210 4 cont’d. Exercise 36 (pp 86-87) #1. 4 parts = 60; 7 parts = 105 There are 105 apples. #2. 3 parts = 42cm; 8 parts = 112cm The ribbon was 112 cm long. #3. 3 parts = $24; 5 parts = $40 The blouse cost $40. #4. 10 parts = $280. 4 parts = 4@$28 = $112. John received $112 more than Peter. pbf8 Exercise 37 (pp 88-89) #1. 4:8:6 = 2:4:3 #2. 9 : 6 : 12 = 3 : 2 : 4 #3. 6:2:8 = 3:1:4 #4. 12 : 10 : 8 = 6 : 5 : 4 #5. 24 : 18 :30 = 4 : 3 : 5 #6. 12 : 12 : 8 = 3 : 3 : 2 Exercise 38 (p90) #1 Ratio of B to G to W is 5 : 2 : 3. There are 90 B.... @2 #2. 5. 5 parts = 90 beads 10 parts = 180 beads There are 180 beads total. 45 cm total is divided into sides in the ratio 3 : 2 : 4 Find the length of the longest side. 9 parts = 45 cm 4 parts = 20 cm is the longest side’s length. [Study the textbook! Hey! #@$%! What’s #4 about? ] (P.S. #@$% is zrklian for “I love it.” ) (Re PT 6A Teacher’s Solutions for Practice 3A#6,7 & Practice 3B #6-9) Practice 3A #6: There are 3 times as many girls as boys in a school choir. #6a. What is the ratio of Girls to total in Choir? 3:4 G G G B #6b. What fraction of the children [in the choir] are boys? One fourth. *²² 27 ÷÷ * #6c. If there are 27 girls, how many children are there altogether? 3 parts = 27 so 1 part = 9 & 4 parts = 36. There are 36 children [in the choir]. #7. 5 “The number of men is ) 8 the number of women working in a factory. If there are 24 more women than men, how many workers are there altogether?” Women Men Notice everything in this problem is about comparison! ²The comparative bar diagram reveals all! ÆÉÈÉÉÇ 24 ÷3 @13 3 parts = 24 workers 1 part = 8 workers 13 parts = 104 workers There are 104 workers altogether. But wait! There’s more....... 7.1C Text HW Ratio & Proportion Math 210 5 cont’d. Practice 3B #6-9: #6. “A sum of money was shared between Susan and Nancy in the ratio of 2 : 5. Nancy received $36 more than Susan. How much money did Susan receive?” pbf8 The setup on this problem is very similar to that of problem #7 above. Difference: 3 parts = $36. ÷3 1 part = $12. @2 Susan: 2 parts = $24 Answer: Susan received $24. #7. “The ratio of Peter’s money to Paul’s money is 5 : 3. If Peter has $25, how much do they have altogether?” Peter: 5 parts = $25 ÷5 1 part = $5 @8 Together: 8 parts = $40 Peter and Paul have $40 altogether. Check: 5 parts = $25, 3 parts = $15. Total is $40, And ratio is 25 : 15 = 5 : 3 U #8 U “The ratio of the number of Chinese books to the number of English books in a library is 4 : 7. There are 2200 Chinese books and English books altogether. How many English books are there?” There are altogether (4+7= ) 11 parts. Those 11 parts = 2200 books. 1 part = 200 books English books make up 7 parts. Those 7 parts = 1400 books There are 1400 English books. ÷11 @7 Check: 7 parts = 1400; 4 parts = 800. Ratio of E to C is 1400 : 800 = 7@200 : 4@200 = 7 : 4 U Total of books = 1400 + 800 = 2200 U #9. “The sides of a triangle are in the ratio 4 : 5 : 6. If the perimeter of the triangle is 60 cm, find the length of the shortest side.” Perimeter of triangle is the sum of the lengths of the three sides. There are 15 parts altogether. 15 parts = 60 cm 1 part = 4 cm 4 parts = 16 cm The shortest side has length 16 cm. Check: 4 parts = 16 cm; 5 parts = 20 cm; 6 parts = 24 cm. 16 cm + 20 cm + 24 cm = 60 cm U 16 : 20 : 24 = 4@4 : 5@4 : 6@4 = 4 : 5 : 6 U 9.1 Text HW Decimals 1a. Math 210 1 1 4@10 + 3@1 + 2 @ )) 1 0 + 8@ ))) 1 00 1b. 1 2 3 + 7 @ ()) 1 0) 1c. 1 5 18 @ ()) 1 0) = 1 7 + 5 @ ()) 1 0) pbf8 = 43.28 = 3.0700005 0.00018 2g. 32 ÷ 16000 = ( 32 ÷ 16) ÷ 1000 = 2 ÷ 1000 = 0.002 32 16@2 2 This is probably easier to visualize in fraction form: ))))) = ))))) = ))))) = 0.002 16000 16@1000 1000 6 6 52,000,000 52 @ 10 13 @4 @ 10 2h. 52,000,000 ÷ 130,000 = )))))))))) = )))))))) = )))))4 = 4@10 2 = 400 4 130,000 13 @ 10 13 @ 10 2i. 1 3 1 0.032 @ 0.0010001 = 32 @()) 1 0) @10001@( )) 1 0) 4a. 0.01739 to the nearest thousandth: 0.017 4b. 0.0495 to the nearest hundredth: 0.05 7 1 10 = 320032 @()) 1 0) = 0.0000320032 0.01739 to the nearest ten thousandth: 0.0174 0.0495 to the nearest thousandth: 0.050 Rounded down this would be 0.049 ‰Rounded up must be 0.049 + .001 = 0.050  6a. Chip model for 5.73 + 1.67: Units Tenths Hundredths """"" """" """ " """"" " !< !< . 7 4 """ """"" "" 0 6b. Chip model for 5.3 – 1.72 [6c & 6d not shown] 1 1 5. 73 + 1. 67 ))))) 7. 40 +10 4 2 5 . 3 10 –1.72 ))))) Units Tenths """" " """ !!!!! !!!!! !!!!! !!!!! . 5 8 3.58 –1.72 2 10. 1.78 ÷ 0.0047 11. = 17.8 ÷ 0.47 = 17.8 ÷ = 4.7 .0053 2.6.  0.0.1 3 7 8 130 )))) 78 78 ))))))))) )) 1.78 .0047 17.8 ))))) .47 17.8 ))))) 4.7 ))))) = = = 17800 47 1780 ))))) 47 178 ))))) 47 ))))) Hundredths ²largest ² middle ² smallest .304 .2 4.  0.0 7. 2 9 6 72 ))) 9 0 ) 96 96 ))))))))) ))) 9.2 Text HW Rationals & Decimals 1. Math 210 Express as fractions in simplest form: 125 25@5 a. 0.125 = )))) = )))) = 1000 25@40 875 25 @35 b. 0.0875 = ))))) = ))))) = 10000 25@400 1 8 ) 525 25@21 c. 0.525 = = )))) = )))) = 1000 25@40 2. pbf8 21 40 )) 7 80 )) 28125 625@45 9 d. 0.28125 = ))))) = ))))) = )) 100000 625@160 32 Can the fraction be written as a finite decimal? [Solutions for d & f are not given.] 137 137@16 a. )))) = )))) = 0.2192 625 10000 221 221 b. )))) = )))) and 3 does not divide 221. So this decimal does not terminate. 1500 500@3 27 9@3 3 3@5 c. )))) = )))) = )) = )))) = 0.15 180 9@ 20 20 20@5 44 4@ 11 11 e. )))) = )))) = )))) 260 4@65 13@5 3. Does not terminate. The 13 will cause it to continue. Write as finite or repeating decimals. [Parts f g h not shown. Part h takes the maximum to repeat!] 3 a. ) = 8 11 c. )) = 21 3 e. )) = 13 3@125 8@125 )))) = 0.375 )))))) .523809 ))))))))) 21 Ž 11.000000 105 ))) 50 42 )) 80 63 )) 170 168 ))) 200 189 ))) 1 1 repeats now )))))) 23 115 b. )) = ))) = 1.15 20 100 24 d. )) = 9 ) 2.6 )))) 9 Ž 24.0 18 )) 60 54 )) 6 0. 230 769 ))))))))) 13 Ž3.000000 26 )) 40 39 )) 100 91 )) 90 78 )) 120 117 )) 3 now repeats everything from the first 3 now it repeats... this part only. 9.2A Text HW Rationals & Decimals 4. Math 210 Write the following fractions in simplest form. 10 x = 1. 1 1 1 1 1 1 1 @ @ @ x = 0. 1 1 1 1 1 1 1 @ @ @ a. [Parts c, d, g & i not shown.] 100 x = 1.010101010101 @ @ @ x = .010101010101 @ @ @ b. ))))))))))))))))))) 9x ) .1 = x ))))))))))))))))))) = 1.0000000 @ @ @ 1 = )) 9 1000 x = 189.189189189 @ @ @ x = .189189189 @ @ @ )))))))))))))))))))))) 999 x = 189 ))) 189 7 .189 = x = )) = )) 999 37 e. pbf8 99 x = )) .01 = x = 1 Multiply by 100 because the repetend is 2 digits long. 1 99 )) 100 x = 50.505050 @ @ @ x = .505050 @ @ @ ))))))))))))))))))) 99 x = 50 )) 50 .505 = x = )) 99 f. We know this is reduced because 3' * 50 & 11' * 50. 100 x = 231.56565656 @ @ @ j. 10000 x = 12345.13451345 @ @ @ x = 2.56565656 @ @ @ x = 1.23451345 @ @ @ )))))))))))))))))))))) ))))))))))))))))))))))))) 99 x = 229 9999 x = 12343.9 that decimal needs to go ! ü )) 229 12343.9 123439 ü 2.3156 = x = )) x = )))))) = )))))) 99 9999 99990 This one’s just a Reduced? So multiply numerator & little bit different. denominator by 10. Does 3*229? No worries, we’ll just Does 11*229? eliminate the decimal at the end ! 5a. [Work not shown here.] ) = 1 ... which is usually a shock, because we are not accustomed to the fact that .9 numbers can have two different decimal representations. In fact, every terminating decimal number has two different representations in decimal form. ) is 4330. 5b. As an example, in part b... you should find that 4329.9 h. PS. You have long been accustomed to the fact that a = .333333 @ @ @ & b = .666666 @ @ @ The logical extension of these observations is that 3(a) must be .999999 @ @ @ ! But 3(a) = 1. PPS. If you add any tiny decimal number, say .00000000001, to .99999999 @ @ @ you get a result that is clearly greater than 1. What does that tell you? 6. 7. This is an instructive problem but time does not permit its inclusion. (1/19 is another of those decimal expansions that takes the maximum possible time to repeat) For the decimal expansion of 3/19, you’d start with the remainder 3 in the division for 1/19. ))1 = 1/99, and .00 )))1 = 1/999, find: Mental math. Using the facts that .0 a. )) .03 = 3 99 )) = [7c and 7d are not shown.] 1 33 )) ))) b. .324 324 = ))) 999 36 = ))) 111 = 12 37 )) 7.2 Text HW Changing Ratios & Percentages 6. Math 210 pbf8 “A truck contains 1000 pounds of sand and concrete in the ratio 2 : 3. After x pounds of sand is added, the ratio of the sand [to] concrete becomes 4 : 5. Find x.” 2 parts + 3 parts = 1000 pounds, so initial amounts of sand and concrete are 400 & 600 pounds. 400 + x : 600 = 4 : 5 400 + x 4 Written in fraction form: )))))) =) ) (We multiply both sides by 600) 600 5 400 + x = 4@120 x = 480 – 400 = 80 7. (Re PT 6A pp47-53 Practice 4A ) 8 2 1 #1. a. 8% = )) = )) b. 25% = ) 100 25 4 1 c. 50% = ) 2 33 d. 66% = )) 50 #2. a. 9% = 9/100 = .09 b. 90% = .90 c. 15% = .15 d. 62% = .62 #3. 2 a. ) = .40 = 40% 5 7 b. ) = .875 = 87.5% 8 9 45 1 c. ) = )) = 45% d. )) = 5% 20 100 20 e. 0.5 = .50 = 50% f. 0.08 = 8% g. 0.15 = 15% #4. 100% – 45% = 55% #5. 45% 100% #6. #7. #8. h. .245 = 24.5% 8 20% ? 100% – (45% + 20%) = 100% – 65% = 35% 35% of the pole is painted white. “36 out of 400 seats are vacant. What percentage of the seats are vacant?” 36 The whole unit is the 400 seats; the giveaway is underlined above. ))) = 9% are vacant. 400 2 ) of the students in a school wear glasses. 5 bThe whole unit is given here. What percentage of the students in the school wear glasses? 2 ) = .40 = 40% 40% of the students in the school wear glasses. 5 “Eva had 3m of cloth. She used 75 cm of it to make a dress for her doll. The whole unit What percentage of the cloth did she use for the dress?” 9 100cm 75 cm is what % of the cloth... 75 = ?? of 3m. First rewrite 3m as 3m ))))) = 300 cm . 1m 75 25 75 = X @ 300 So X = ))) = )) = 25% Eva used 25% of the cloth for the dress. 300 100 #9. “45 medals were given out[...]. There were 22 bronze medals and 14 silver medals. The rest were gold medals. What percentage of the medals were gold medals?” The whole unit is the 45 medals that were given out. 9 1 20 20% of the medals 45 22 14 ? 45 – (22+14) = 9 )) = ) = )) 8 45 5 100 were gold medals. #10. K spends 30%of savings & 60% of the remainder. What is left? Whole unit = original savings. She spent 30% + 60% of 70% = 30% + 42%, for a total of 72%. 100% – 72% = 28%. 28% of her savings is left. 30% 60% of 70% 7.2A Text HW Changing Ratios & Percentages 8. Math 210 pbf8 tells us the whole unit (100%) is (Re PT 6A, pp 55-59, Practice C) the 1.5L, aka 1500mL. #1. “Express 480 mL as a percentage of 1.5L.” 1000 mL ? 100% First we convert L: 1.5 L = 1.5L@ ))))) = 1500 mL. L 480 Our job is to compare 480mL to 1500mL. 1500 1500mL 6 100% 1 mL 6 1/15 % Direct 480 160 32 ))) = ))) = )) = 32% 480 mL6 480/15 % = 160/5 % = 32% method 1500 500 100 Above is the Primary Text’s method. 480 mL is 32% of 1500 mL #2. “What percentage of 2 hours [120 min] is 30 minutes?” 30 minutes is 25% of 2 hours. #3. “Express the length of A [36m] as a percentage of the length of B [24m].” 36 Compare the length of A to the length of B and express as a percentage of B. )) = 150% 24 “How many percent longer is A than B?” How much longer is A than B? Express result as percentage of B. A is 12 m longer than B. 12 m is half of 24m. Half is 50%. A is 50% longer than B. #4. “MB had 2.5 kg of sugar. She used 650g of it to make syrup. What percentage of the sugar was used for making the syrup?” The whole unit is 2.5 kg (which is 2500 g). 650g 650 g is what part of 2500 g? ))))) = 2500g 26% of the 2.5 kg of sugar was used to make syrup. #5. #6. #7. #8. #9. “The price of a [TV] was reduced from $200 to $150. By what percentage was the price reduced?” The Q asks us to compare the reduction to the original price. The price was reduced 25%. 26 100 ))) 50 200 ))) = = 26% 25 100 ))) = 25% “A club had 80 members last year. This year it has 96 members. By what percentage was the membership increased? “ The Q asks us to compare the increase to the old membership. The membership increased by 20%. “The price of beef increased from $12 per kg to $15 per kg. [Express as percent increase.]” Percent increase or decrease always compares the CHANGE in value to the OLD value. Change $3 1 ))))) = )) = ) = 25% The price of beef increased 25%. Old value $12 4 “Kyle bought a pair of shoes for $51. The usual price of the shoes was $60. How many percent discount was given to Kyle?” Change $9 3 15 ))))) = )) = )) = ))) = 15% Kyle was given a 15% discount. Old value $60 20 100 “A factory has 600 workers. 250 ... are men and the rest are women. How many percent more women than men are there?” 350 are women. 100 more women than men. Compare the “more women” to the men. 100 10 40 ))) = )) = ))) = 40% There are 40% more women than men. 250 25 100 [number 10 is not shown.]