Unit 1 - Solving Life’s Problems

Homework 1.1: Percents Anthony Lopez Chapter 1: Problem Solving In previous math courses, you’ve no doubt run into the infamous “word problems.” Unfortunately, these problems rarely resemble the type of problems we actually encounter in everyday life. In math books, you usually are told exactly which formula or procedure to use, and are given exactly the information you need to answer the question. In real life, problem solving requires identifying an appropriate formula or procedure, and determining what information you need (and won’t need) to answer the question. 1.1 Percents Percent literally means “per 100,” or “parts per hundred.” When we write 40%, this is equivalent to the fraction 40/100 or the decimal 0.40. Notice that 80 out of 200 and 10 out of 25 are also 40% since 80/200 = 0.40 and 10/25 = 0.40. Ex 1: Convert a Fraction to a Decimal (terminating) Ex 2: Convert a Fraction to a Decimal (terminating) https://www.myopenmath.com/assess2/ ? cid=214538&aid=15169091#/print 2/20/24, 10 : 40 PM Page 1 of 9

Question 6 0/10 pts 3 99 Employees in 2012 paid 4.2% of their gross wages towards social security (FICA tax), while employers paid another 6.2%. How much will someone earning $52,000 a year pay towards social security out of their gross wages? $ Give your answer accurate to at least the nearest dollar. Question Help: Q Video 1 Ex: Change, Absolute Change, and Relative Change https://www.myopenmath.com/assess2/ ? cid=214538&aid=15169091#/print 2/20/24, 10 : 40 PM Page 4 of 9 part =% whole x = 52 , 000 (4 . 2 % ) 2 , 184 X = 52000(0 . 042) X = $ 2 , 184

Question 14 0/10 pts 3 99 Question Help: Writing Equivalent Fractions, Decimals, and Percents Complete the table below by finding the equivalent values of the given value in the forms Simplify any fractions. Fraction or Mixed Number Decimal Per 1 4 0.08 6 3 5 1.35 22 Q Video 1 Out of 340 racers who started the marathon, 315 completed the race, 17 gave up, and 8 were disqualified. What percentage did not complete the marathon? % Give your answer accurate to at least 1 decimal place. Q 1 https://www.myopenmath.com/assess2/ ? cid=214538&aid=15169091#/print 2/20/24, 10 : 40 PM Page 7 of 9

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Sale 4200 commission rate 5.4 0.054 4200 226.8 cost of item 68 selling price 76.22 76.22 68 8.22 P 12.08 p

© David Lippman Creative Commons BY-SA Proportions and Rates If you wanted to power the city of Seattle using wind power, how many windmills would you need to install? Questions like these can be answered using rates and proportions. Rates A rate is the ratio (fraction) of two quantities. A unit rate is a rate with a denominator of one. Example 12 Your car can drive 300 miles on a tank of 15 gallons. Express this as a rate. Expressed as a rate, . We can divide to find a unit rate: , which we could also write as , or just 20 miles per gallon. Proportion Equation A proportion equation is an equation showing the equivalence of two rates or ratios. Example 13 Solve the proportion for the unknown value x . This proportion is asking us to find a fraction with denominator 6 that is equivalent to the fraction . We can solve this by multiplying both sides of the equation by 6, giving . 300 miles 15 gallons 20 miles 1gallon miles 20 gallon 5 3 6 x = 5 3 5 6 10 3 x = × =

Module 1: Proportion and Rates 4 Notice that with the miles per gallon example, if we double the miles driven, we double the gas used. Likewise, with the map distance example, if the map distance doubles, the real-life distance doubles. This is a key feature of proportional relationships, and one we must confirm before assuming two things are related proportionally. Example 17 Suppose you’re tiling the floor of a 10 ft by 10 ft room, and find that 100 tiles will be needed. How many tiles will be needed to tile the floor of a 20 ft by 20 ft room? In this case, while the width the room has doubled, the area has quadrupled. Since the number of tiles needed corresponds with the area of the floor, not the width, 400 tiles will be needed. We could find this using a proportion based on the areas of the rooms: Other quantities just don’t scale proportionally at all. Example 18 Suppose a small company spends $1000 on an advertising campaign, and gains 100 new customers from it. How many new customers should they expect if they spend $10,000? While it is tempting to say that they will gain 1000 new customers, it is likely that additional advertising will be less effective than the initial advertising. For example, if the company is a hot tub store, there are likely only a fixed number of people interested in buying a hot tub, so there might not even be 1000 people in the town who would be potential customers. Sometimes when working with rates, proportions, and percents, the process can be made more challenging by the magnitude of the numbers involved. Sometimes, large numbers are just difficult to comprehend. Example 19 Compare the 2010 U.S. military budget of $683.7 billion to other quantities. Here we have a very large number, about $683,700,000,000 written out. Of course, imagining a billion dollars is very difficult, so it can help to compare it to other quantities. If that amount of money was used to pay the salaries of the 1.4 million Walmart employees in the U.S., each would earn over $488,000. There are about 300 million people in the U.S. The military budget is about $2,200 per person. If you were to put $683.7 billion in $100 bills, and count out 1 per second, it would take 216 years to finish counting it. 2 2 100 tiles tiles 100 ft 400 ft n =

Module 1: Proportion and Rates 5 Example 20 Compare the electricity consumption per capita in China to the rate in Japan. To address this question, we will first need data. From the CIA 1 website we can find the electricity consumption in 2011 for China was 4,693,000,000,000 KWH (kilowatt-hours), or 4.693 trillion KWH, while the consumption for Japan was 859,700,000,000, or 859.7 billion KWH. To find the rate per capita (per person), we will also need the population of the two countries. From the World Bank 2 , we can find the population of China is 1,344,130,000, or 1.344 billion, and the population of Japan is 127,817,277, or 127.8 million. Computing the consumption per capita for each country: China: ≈ 3491.5 KWH per person Japan: ≈ 6726 KWH per person While China uses more than 5 times the electricity of Japan overall, because the population of Japan is so much smaller, it turns out Japan uses almost twice the electricity per person compared to China. Try it Now Answers 4. ≈ 0.475 ounces 1 https://www.cia.gov/library/publications/the-world-factbook/rankorder/2042rank.html 2 http://data.worldbank.org/indicator/SP.POP.TOTL 4,693,000,000,000 KWH 1,344,130,000 people 859,700,000,000 KWH 127,817,277 people 1foot 19.8 pounds 16 ounces 18 inches 12 inches 1000 feet 1 pound × × ×

342 miles + 154 _ 22 2 miles 51 minutes : 6 = 8 5 mins Rates 15 4 Gallons 15 . 4 lon ⊥ miles : 6 I mile - Rate unit rate Rate unit rate Example 1 Example 2 #250 Question #1 Question #2 = I ounce 186 Newsp . 2 = 93NP 396 miles o = Clemiles 2 HOURS-2 I Hour & Hours : 6 I Hour 5 15 = 14 0 3674 I 14 Ounces : 14 I ounce Rate Unit Rate Proportions Proportion Example 3 Area = length x width cross multiply Question #3 Question #4 Dimensional = property = 340 , 000 TAX = X Analysis X = 7 tax = 4 , 760 Property = 790 , 000 4 , 760x X unit Fractions 340 , 000 790 , 000 320F/O2 * t % half gal 340 , 000 x = 3 , 760 , 400 , 000 340 , 000 340 , 000 x X = 11 , 000 Proportions 320 #02 = Shallous O Question 5 B at lo I half gol is = 2 Qt 5 & 5 Half gal 10km = 10,000m o se I 10 Ot I HOUR I Question - sec 4 Sec 1000 meters = 1 km (2 64) M 95 , 000 meters = met = 05 I Hour I sec Question 5 A # non = 000km 7500m = 20 . 8 F 3600 sec I see Questions

Question L Question #7 Zamboni traveled miles 1 gallon = 3 785 liters 2 Hours 120 gallons = 1x = 6 -gallon a I - Tiles How - = 1 60 min mill 2 gallons iters = 454 . 2 liters I 10 . 15 miles x I min I Hour X Hours Question #8 Lex = 10 15 ( x = 1 -02 ~ 3202 X = 1 69 - 2 . 55 Dollars 255 cents le miles - I mile -- 3202 1 Hour I X Horrs X = 0 16 5 ents -2x = 255 Question #9 32 32 16 Hours - I Hour - - X = 7 9 cents 204 dollars - X dollars X = 8 #X = 2204 Question &A 16 16 2902 = 2902 - X = 12 . 75 dollars 3 . 45 dollars 345 centents Question #10 29X = 345 1400 # = -400 = $ 200 29 29 I week 7 DayS I Day X = 11 8 Question # Question #13 Question #14 - I Joe 5 Hours BOB 8HRS * * 8 cartons 12 cartons = Question #12 2 Z = 199 -69 x = 36 6 gallons 16 gallons X = 12 . 97 5) 2 = 12 gallons Question #15 Question #16 $ 4 = 24 gallons MILES 2639m - $ 6 = 36 gallons Emiles I X mile gallon 18 gal - - - - To minutes 100 minutes 11263 I xm EX 00 9 T = C gal X 1 gal - T 70 10 X = 1639 G - QUESTION 16 A X = 8 - 571 10 10 79350- start of trip X = 8 . 6 X = 11 , 263 . 9 8 gallons - Full tauk 79574- end of thip

Question 17 1 mile = 5280 Et 52 m 274 , 560 76 . 26 -- 5600 see 1 Hour I sec Question 18 Area = LxW X 45 N 54x75 54 4 , 05 0 IN 301N Question 20 14 cups 1 . 25 cups - 20 cookies 20 cookies Question 21 - 1 . 25 cres ** lies property 370 , 000 TAX 7 , 400 20x = 10 370 , 000 - - > ⊥ 50 , 000 I 20 20 7 , 400 X = 0 . 5 370 , 000 x = 5 , 550 , 000 , 000 -- 370 , 000 370 , 000 x = 15 , 000

Lesson. 1.3 Geometry

I 5 Perimeter = ( + w + L + w w W P = 2l + 2w I P = 2(12) + 2(5) 12 12 P = 24 + 10 5 p = 24 - S Perimeter of a square S S p = S + S + S + S P = 45 S B 9 P = a + B + c - C = 2 π R = TD · R Depends which one S D You use based f the data 2(3 . 14) 3

Lesson 1.4 Problem Solving and Estimation

To find the actual dimensions, you need to multiply the dimensions on the drawing by the scale factor (80). Actual length = (1 7/8) * 80 = 150 inches = 12.5 feet Actual width = (3 3/8) * 80 = 270 inches = 22.5 feet Now, to find the area, multiply the actual length by the actual width: Area = 12.5 feet * 22.5 feet = 281.25 square feet So, the actual dimensions are approximately 12.5 feet by 22.5 feet, and the area is approximately 281.25 square feet. status Report - 156 pgs long Question #7 status Report : 124 pys Need to print 14 copier copies needed : 15 one ream : 500 pgs Paper is $ 3 . 55 for 500 pgs cost of ream : $ 3 . 31 14 Copies = 2 , 186 pas 3 . 3 12 EX : 4 . 5 reams = 00 = 10 . 44 ↑ 3 . 55 * 500 500 500p C = 0 . 82088 0 . 820 88x15 200 = 153 . 8 12 . 3132 500 500 12 . 31 C = 1 . 1076/copy 36 : / scale Room 1 : It "by 16" 1 . 1076 x 14 = 15 . 5004 15 . 51 New tablet has 430 Rows of pixels 740 columns of pixels -30 x 740 = 318 . 200 Bullet train traveling Anea = Lxw e323m/H How many Hours to Volume = 1xWXH travel 64 . 6 miles 6-INCH THICK lyV 34 by 24 Ft 1H XH v = (34)(24)(0 . 5) I 323m 64 . 6M 408 f + 3 Aft · * 3 323H = 64 . L 27F + 3 person traveling 165m 323 323 408 Cruise control clem/ It Yd3 = 15 . 11 H = 0 . 20 -15 . 11 H X It - O 64 . Im 3015 ft by 20ft = $ 2 , 275 165m dem 23 by 27 F + = 621 300 621 46H = 165 - S D T 5 wh lle -D = 1 , 412 , 775 # = 2 . 5 60 = 12 300 300 - To 60 m = 12 D = 4709 25

2012 4 . 2 To 6 . 2 T TV original price $ 2000 320 racers -> only 284 completed T 4 . 2 9 % sale 25 gave up 11 disqualified --- 4500 100 2000 X 0 . 09 4500 · 0 . 042 2000 - 180 25 + 11 = = 0 . 112 7 % 2 , 730 1820 ** A 396m in 6 Hours 20 HRS = $ 190 - 2x = 10 ummin X = 9 #A = 5 zom = ze 360 360 Is 200 A = 20 m = 66 D = 9 . 5 54 miles per Hour small Box to Feet per Second Joe worked 2 ARS/4 cartons measures 9 In by Gin Bob worked CARS/11 cartons and 3/4IN High 285 , 120 feet = 54 miles 3 , 600 Seconds = Hour Joe -leHRS/12 cartons 285 , 120 F - + Angela walks 5m/50 min 3 , 600 s F 5 M = - I - X 50min = 80min & & 85 , 120F = -3 , 600 50 MIN 80min M 285 , 120 285 , 120 SOM = 100 Em = 14 , 000 F = 0 . 01262626 50 5 5 Sale : $ 6700 50 m = 800 Commission : 5 1 % M = 8 6700 x 0 051 6700 - 341 . 7 Ara checked Bag : 22lbs 6 , 358 . 30 weight Before : 20lbs weight went up 2lbs -2 - 20 = 100 20 0 . 1 x 100 10 %

18cm A " A2 9 Az = (14cm)(acm) cu Az = 126 cm2 14 am A 1 = (4 cm) (4cm) Total Area = 126 + 56 P 182 unz 1 Al = 56cm2 4 an Fan 2cm 10 8 A , = (10cm) (2 cm) = 20 cm - 3 - 2 Az = (6cm) (3 cm = 18 cm2 Tom - - 38 cm2 T im A , secu uum " Az ⊥ Gun

Lesson 2.1: US Measurement System and Conversion

25m = 82 . 03 ft - 25m I f C . 0 305m : oft 8 . 97 + T C # . = 25 (3 . 281) + =82 . 637t I m 11 . 5 Ft ~ 3 : 81 m 12 . 5 + · m = 12 . 5 (0 . 305) m = 3 . 81 - - I 1 FT # . I

Suki is making lemonade to bring to the beach. She has two containers. One holds one gallon and the other holds 6 quarts. If she fills both containers, how many cups of lemonade will she have? 16 in ~ um 12 in 16 (2 . 540) am = 40 . 64 on - example 1 - . 2 540CM T I in video 2 - 25m ~ 82 ft 1 FT zem 0 305 N (255) fr = 82ft -Example 2 video 2 zm . 3 . 281 ft ~ 25 (3 . 281) FT82FT Im Question # 3 180lbs-k9 2yd = Ft 180mbS · 11 kg8 : Ex unit equivalent : lyd = 3f + I 2 21b 2 E 3ft = 2(3)f+ = left Kilo = THOUSAND L Iyd EX2 - 2kg = 424 . 0 02 Question 4 112 19 1000 I · 10 = 12000 o - 9 - 1lb = 1602 - I kg 28 3 - 9 1/ lbs to 174 02 11 lbs 1602 = 11(14)02 42402 - -x- I It Question L 17602 Question 5 116 = 1602 98 ounces to 6 . 1lbs 2 containers : Igal & Lat Look · 116 F I gal = 16 cups 1602 I - I (lb = 6 . 1 lbs , = 1 . 5 gal Question 7 S 144 cups = 99 e I #44 BCs /gal = 16 cups 24 cups 10 Question #8 1 day = 24 Hours 1 Hour = 60 run · Hours to 1 min = 60 sees SECONDS 8 Hours · Gon * 60 sees = (8 - 60 · 60) secs = 28 , 800 sees - I I HOUR InmN

Question 9 Question 10 Codays = 144 Hours 48 IN = 4 Ft dys . 24 HRS = G(24)HRS I day I . = 4 I I Question 11 - lyd = 3f + Ift = 12 in Question 12 I day = 24 Hours 6 yd = 216 IN L days = Sec 1 Hour = 60 run I 1 min = 60 sees 6 days & & & COMM · ⑭ . . = (6 . 3 . 12) in - 24 HAS 60 secs Iyd I I day I - - - HOUR I MIN Question 13 66 . 24 . 60 · 60) Secs = 518 , 400 S 19iN/sec = F + /MIN

Lesson 2.2: Metric System and Conversions

44 . L 2 . 68 1633 0 . 912

Lesson 3.1: Voting Theory and Methods

Preference ballot A preference ballot is a ballot in which the voter ranks the choices in order of preference. b =4424 3 00 21 00 16 - 3 % 321 #5 3x14 = 42 O G 2x3 = 4 60 1x8 = X91v

-- WINNER IS Brown 34 1StpI voter ⊥ Adamy 37 + 29 - votes & - monotificity Carter wins 49 vate criteria by 51 votes is violated G 32 voters I - Noel : 15 - 14 NO WINNER Canon : 13-the- Skyler , I - CANDIDATEA : 13 + 8 : 21 CANDIDATED : 35 Candidate C : 24

42 483 153 Ferg s Candidate 20 + 17 + 8 + y = - B - 51 voleng D = A T = B S = C B- 17 + 2 = 19 T S-13 + 6 = 19 #5

Lesson 3.2: Apportionment

467000117 , 000 8500 124 3 · Jefferson method 10 5 4 · Hamilton Method 31 U 3 14 3 2 13/15 370 31013570 - S 521 7