Chapter 1: Problem Solving. Chapter 1: Problem Solving 1 / 21
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1 Chapter 1: Problem Solving Chapter 1: Problem Solving 1 / 21
2 Percents Formula percent = part whole Chapter 1: Problem Solving 2 / 21
3 Percents Formula percent = part whole part = percent whole Chapter 1: Problem Solving 2 / 21
4 1 An opinion poll finds that 64% of 1069 people surveyed said that the President is doing a good job. How many said the President is doing a good job? Chapter 1: Problem Solving 3 / 21
5 1 An opinion poll finds that 64% of 1069 people surveyed said that the President is doing a good job. How many said the President is doing a good job? = people Chapter 1: Problem Solving 3 / 21
6 2 You purchase a shirt with a labeled (pre-tax) price of $21. The local sales tax rate is 6%. What is your final cost (including tax)? Chapter 1: Problem Solving 4 / 21
7 2 You purchase a shirt with a labeled (pre-tax) price of $21. The local sales tax rate is 6%. What is your final cost (including tax)? 0.06 $21 = $1.26 Chapter 1: Problem Solving 4 / 21
8 2 You purchase a shirt with a labeled (pre-tax) price of $21. The local sales tax rate is 6%. What is your final cost (including tax)? 0.06 $21 = $1.26 $ $21 = $22.26 Chapter 1: Problem Solving 4 / 21
9 2 You purchase a shirt with a labeled (pre-tax) price of $21. The local sales tax rate is 6%. What is your final cost (including tax)? 0.06 $21 = $1.26 $ $21 = $22.26 OR (1.06) $21 = $22.26 Chapter 1: Problem Solving 4 / 21
10 3 Carol earns 50% more than William. How many times larger is her income than his? Chapter 1: Problem Solving 5 / 21
11 3 Carol earns 50% more than William. How many times larger is her income than his? Carol s salary = William s salary ( ), so Carol s salary is 150% of William s salary. Chapter 1: Problem Solving 5 / 21
12 4 In baseball, a players batting average represents the percentage of at-bats in which he got a hit. For example, a batting average of.350 means the player got a hit 35% of the times he batted. Suppose a player had a batting average of.200 during the first half of the season and.400 during the second half of the season. Can we conclude that his batting average for the entire season was.300 (the average of.200 and.400)? Chapter 1: Problem Solving 6 / 21
13 4 In baseball, a players batting average represents the percentage of at-bats in which he got a hit. For example, a batting average of.350 means the player got a hit 35% of the times he batted. Suppose a player had a batting average of.200 during the first half of the season and.400 during the second half of the season. Can we conclude that his batting average for the entire season was.300 (the average of.200 and.400)? No, Chapter 1: Problem Solving 6 / 21
14 4 In baseball, a players batting average represents the percentage of at-bats in which he got a hit. For example, a batting average of.350 means the player got a hit 35% of the times he batted. Suppose a player had a batting average of.200 during the first half of the season and.400 during the second half of the season. Can we conclude that his batting average for the entire season was.300 (the average of.200 and.400)? No, the number at-bats may not be the same each half. Chapter 1: Problem Solving 6 / 21
15 5 A stockbroker offers the following defense to angry investors: I admit that the value of your investments fell 60% during my first year on the job. This year, however, their value has increased by 75%, so you are now 15% ahead! Evaluate the stockbrokers defense. Chapter 1: Problem Solving 7 / 21
16 5 A stockbroker offers the following defense to angry investors: I admit that the value of your investments fell 60% during my first year on the job. This year, however, their value has increased by 75%, so you are now 15% ahead! Evaluate the stockbrokers defense. Suppose the value was $1,000. ( ) $1, 000 = $400 Chapter 1: Problem Solving 7 / 21
17 5 A stockbroker offers the following defense to angry investors: I admit that the value of your investments fell 60% during my first year on the job. This year, however, their value has increased by 75%, so you are now 15% ahead! Evaluate the stockbrokers defense. Suppose the value was $1,000. The stockbroker is not correct. ( ) $1, 000 = $400 ( ) $400 = $700. Chapter 1: Problem Solving 7 / 21
18 Definition The absolute change is the actual difference between the compared value and the reference value: absolute change = ending quantity - starting quantity Chapter 1: Problem Solving 8 / 21
19 Definition The absolute change is the actual difference between the compared value and the reference value: absolute change = ending quantity - starting quantity The relative change describes the size of the absolute change as a fraction of the reference value: relative change = absolute change starting quantity Chapter 1: Problem Solving 8 / 21
20 6 A diversified portfolio grows from $1,500 to $2,250. Describe the absolute and relative change in value. Chapter 1: Problem Solving 9 / 21
21 6 A diversified portfolio grows from $1,500 to $2,250. Describe the absolute and relative change in value. Absolute change: $2,250-$1,500 = $750 Chapter 1: Problem Solving 9 / 21
22 6 A diversified portfolio grows from $1,500 to $2,250. Describe the absolute and relative change in value. Absolute change: $2,250-$1,500 = $750 Relative change: 750 1,500 = 0.50 = 50% Chapter 1: Problem Solving 9 / 21
23 7 You bought a computer three years ago for $1000. Today, it is worth only $300. Describe the absolute and relative change in the computers value. Chapter 1: Problem Solving 10 / 21
24 7 You bought a computer three years ago for $1000. Today, it is worth only $300. Describe the absolute and relative change in the computers value. Absolute change: $300 - $1,000 = -$700 Chapter 1: Problem Solving 10 / 21
25 7 You bought a computer three years ago for $1000. Today, it is worth only $300. Describe the absolute and relative change in the computers value. Absolute change: $300 - $1,000 = -$700 Relative change: 700 1,000 = = -70% Chapter 1: Problem Solving 10 / 21
26 8 Average pay for full-time wage earners varies from state to state. Recent data (data for 2009 released in 2013) show that New York ranked first in average pay, at about $60,300 per person, and South Dakota ranked last at $32,800 per person. Compare average pay in South Dakota to that in New York in both absolute and relative terms. Chapter 1: Problem Solving 11 / 21
27 8 Average pay for full-time wage earners varies from state to state. Recent data (data for 2009 released in 2013) show that New York ranked first in average pay, at about $60,300 per person, and South Dakota ranked last at $32,800 per person. Compare average pay in South Dakota to that in New York in both absolute and relative terms. Absolute change: $60,300-$32,800 = $27,500 Chapter 1: Problem Solving 11 / 21
28 8 Average pay for full-time wage earners varies from state to state. Recent data (data for 2009 released in 2013) show that New York ranked first in average pay, at about $60,300 per person, and South Dakota ranked last at $32,800 per person. Compare average pay in South Dakota to that in New York in both absolute and relative terms. Absolute change: $60,300-$32,800 = $27,500 New Yorker s make $27,500 more than South Dakotans on average. Chapter 1: Problem Solving 11 / 21
29 8 Average pay for full-time wage earners varies from state to state. Recent data (data for 2009 released in 2013) show that New York ranked first in average pay, at about $60,300 per person, and South Dakota ranked last at $32,800 per person. Compare average pay in South Dakota to that in New York in both absolute and relative terms. Absolute change: $60,300-$32,800 = $27,500 New Yorker s make $27,500 more than South Dakotans on average. Relative change: ,800 = = 83.8% Chapter 1: Problem Solving 11 / 21
30 8 Average pay for full-time wage earners varies from state to state. Recent data (data for 2009 released in 2013) show that New York ranked first in average pay, at about $60,300 per person, and South Dakota ranked last at $32,800 per person. Compare average pay in South Dakota to that in New York in both absolute and relative terms. Absolute change: $60,300-$32,800 = $27,500 New Yorker s make $27,500 more than South Dakotans on average Relative change: 32,800 = = 83.8% New Yorker s make 83.8% more than South Dakotans on average. Chapter 1: Problem Solving 11 / 21
31 8 Average pay for full-time wage earners varies from state to state. Recent data (data for 2009 released in 2013) show that New York ranked first in average pay, at about $60,300 per person, and South Dakota ranked last at $32,800 per person. Compare average pay in South Dakota to that in New York in both absolute and relative terms. Absolute change: $60,300-$32,800 = $27,500 New Yorker s make $27,500 more than South Dakotans on average Relative change: 32,800 = = 83.8% New Yorker s make 83.8% more than South Dakotans on average. Alternately, $32,800 - $60,300=-$27,500 and = = 45.6%, Chapter 1: Problem Solving 11 / 21
32 8 Average pay for full-time wage earners varies from state to state. Recent data (data for 2009 released in 2013) show that New York ranked first in average pay, at about $60,300 per person, and South Dakota ranked last at $32,800 per person. Compare average pay in South Dakota to that in New York in both absolute and relative terms. Absolute change: $60,300-$32,800 = $27,500 New Yorker s make $27,500 more than South Dakotans on average Relative change: 32,800 = = 83.8% New Yorker s make 83.8% more than South Dakotans on average. Alternately, $32,800 - $60,300=-$27,500 and = = 45.6%, so South Dakotans typically make $27,500 less than New Yorkers and make 45.6% less than New Yorkers. Chapter 1: Problem Solving 11 / 21
33 Proportions and Rates Definition 1 A rate is the ratio (fraction) of two quantities. A unit rate is a rate with a denominator of one. Chapter 1: Problem Solving 12 / 21
34 Proportions and Rates Definition 1 A rate is the ratio (fraction) of two quantities. A unit rate is a rate with a denominator of one. 2 A proportion equation is an equation showing the equivalence of two rates or ratios. Chapter 1: Problem Solving 12 / 21
35 Examples Unit Conversions Length 1 foot (ft) = 12 inches (in) 1 yard (yd) = 3 feet (ft) 1 mile = 5,280 feet 1000 millimeters (mm) = 1 meter (m) 100 centimeters (cm) = 1 meter 1000 meters (m) = 1 kilometer (km) 2.54 centimeters (cm) = 1 inch Weight and Mass 1 pound (lb) = 16 ounces (oz) 1 ton = 2000 pounds 1000 milligrams (mg) = 1 gram (g) 1000 grams = 1kilogram (kg) 1 kilogram = 2.2 pounds (on earth) Capacity 1 cup = 8 fluid ounces (fl oz) 1 pint = 2 cups 1 quart = 2 pints = 4 cups 1 gallon = 4 quarts = 16 cups 1000 milliliters (ml) = 1 liter (L) Chapter 1: Problem Solving 13 / 21
36 Rates Question 9 Convert 15,000 centimeters to miles. Chapter 1: Problem Solving 14 / 21
37 Rates Question 9 Convert 15,000 centimeters to miles. 1 mi = 5,280 ft,1 ft = 12 in, and 1in = 2.54 cm Chapter 1: Problem Solving 14 / 21
38 Rates Question 9 Convert 15,000 centimeters to miles. 1 mi = 5,280 ft,1 ft = 12 in, and 1in = 2.54 cm 15, 000 cm 1 in 2.54 cm 1 ft 12 in 1 mi 5, 280 ft Chapter 1: Problem Solving 14 / 21
39 Rates Question 9 Convert 15,000 centimeters to miles. 1 mi = 5,280 ft,1 ft = 12 in, and 1in = 2.54 cm 15, 000 cm 1 in 2.54 cm 1 ft 12 in 1 mi 5, 280 ft mi Chapter 1: Problem Solving 14 / 21
40 Rates Question 10 Convert 8 square yards to square inches. Chapter 1: Problem Solving 15 / 21
41 Rates Question 10 Convert 8 square yards to square inches. 1 yd = 3 ft and 1 ft = 12 in Chapter 1: Problem Solving 15 / 21
42 Rates Question 10 Convert 8 square yards to square inches. 1 yd = 3 ft and 1 ft = 12 in ( ) 3 ft 2 8 yd 2 1 yd ( ) 12 in 2 1 ft Chapter 1: Problem Solving 15 / 21
43 Rates Question 10 Convert 8 square yards to square inches. 1 yd = 3 ft and 1 ft = 12 in ( ) 3 ft 2 8 yd 2 1 yd ( ) 12 in 2 1 ft = 8 yd 2 9 ft2 144 in2 2 1 yd 1 ft 2 Chapter 1: Problem Solving 15 / 21
44 Rates Question 10 Convert 8 square yards to square inches. 1 yd = 3 ft and 1 ft = 12 in ( ) 3 ft 2 8 yd 2 1 yd ( ) 12 in 2 1 ft = 8 yd 2 9 ft2 144 in2 2 1 yd 1 ft 2 = 10, 368 in 2 Chapter 1: Problem Solving 15 / 21
45 Rates Question 11 A car is driving at 100 kilometers per hour. How far does it travel in 2 seconds? Chapter 1: Problem Solving 16 / 21
46 Rates Question 11 A car is driving at 100 kilometers per hour. How far does it travel in 2 seconds? 100 km = 1 hour, 1 hour = 60 min, and 1 min = 60 sec Chapter 1: Problem Solving 16 / 21
47 Rates Question 11 A car is driving at 100 kilometers per hour. How far does it travel in 2 seconds? 100 km = 1 hour, 1 hour = 60 min, and 1 min = 60 sec 2 sec 1 min 60 sec 1 hr 100 km 60 min 1 hr Chapter 1: Problem Solving 16 / 21
48 Rates Question 11 A car is driving at 100 kilometers per hour. How far does it travel in 2 seconds? 100 km = 1 hour, 1 hour = 60 min, and 1 min = 60 sec 2 sec 1 min 60 sec 1 hr 60 min km 100 km 1 hr Chapter 1: Problem Solving 16 / 21
49 Rates Question 12 A highway had a landslide, where 3,000 cubic yards of material fell on the road, requiring 200 dump truck loads to clear. On another highway, a slide left 40,000 cubic yards on the road. How many dump truck loads would be needed to clear this slide? Chapter 1: Problem Solving 17 / 21
50 Rates Question 12 A highway had a landslide, where 3,000 cubic yards of material fell on the road, requiring 200 dump truck loads to clear. On another highway, a slide left 40,000 cubic yards on the road. How many dump truck loads would be needed to clear this slide? 3,000 yd 3 = 200 dtl Chapter 1: Problem Solving 17 / 21
51 Rates Question 12 A highway had a landslide, where 3,000 cubic yards of material fell on the road, requiring 200 dump truck loads to clear. On another highway, a slide left 40,000 cubic yards on the road. How many dump truck loads would be needed to clear this slide? 3,000 yd 3 = 200 dtl 40, 000 yd dtl = 2, 667 dtl 3 3, 000 yd Chapter 1: Problem Solving 17 / 21
52 Problem Solving Process 1 Identify the question you re trying to answer. Chapter 1: Problem Solving 18 / 21
53 Problem Solving Process 1 Identify the question you re trying to answer. 2 Work backwards, identifying the information you will need and the relationships you will use to answer that question. Chapter 1: Problem Solving 18 / 21
54 Problem Solving Process 1 Identify the question you re trying to answer. 2 Work backwards, identifying the information you will need and the relationships you will use to answer that question. 3 Continue working backwards, creating a solution pathway. Chapter 1: Problem Solving 18 / 21
55 Problem Solving Process 1 Identify the question you re trying to answer. 2 Work backwards, identifying the information you will need and the relationships you will use to answer that question. 3 Continue working backwards, creating a solution pathway. 4 If you are missing necessary information, look it up or estimate it. If you have unnecessary information, ignore it. Chapter 1: Problem Solving 18 / 21
56 Problem Solving Process 1 Identify the question you re trying to answer. 2 Work backwards, identifying the information you will need and the relationships you will use to answer that question. 3 Continue working backwards, creating a solution pathway. 4 If you are missing necessary information, look it up or estimate it. If you have unnecessary information, ignore it. 5 Solve the problem, following your solution pathway. Chapter 1: Problem Solving 18 / 21
57 Problem Solving Question 13 How many hours will you spend commuting to work over your lifetime? Chapter 1: Problem Solving 19 / 21
58 Problem Solving Question 13 How many hours will you spend commuting to work over your lifetime? 30 min = 1 trip to work, 2 trips to work = 1 day, 5 days = 1 week, 50 weeks = 1 year, 40 years = 1 lifetime, 60 min = 1 hr Chapter 1: Problem Solving 19 / 21
59 Problem Solving Question 13 How many hours will you spend commuting to work over your lifetime? 30 min = 1 trip to work, 2 trips to work = 1 day, 5 days = 1 week, 50 weeks = 1 year, 40 years = 1 lifetime, 60 min = 1 hr 30 min 1 ttw 2 ttw 1 day 5 day 50 wk 40 yr 1 wk 1 yr 1 lt = 600, 000 min lt 1 hr 60 min = 10, 000 hr lt Chapter 1: Problem Solving 19 / 21
60 Apple Interview Question 14 How many babies are born every day in the United States? Chapter 1: Problem Solving 20 / 21
61 Google Interview Question 15 How many tennis balls can fit in an airplane? Chapter 1: Problem Solving 21 / 21
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