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Approaches to Problem Solving Copyright © 2011 Pearson Education, Inc. Approaches to Problem Solving Copyright © 2011 Pearson Education, Inc.

Unit 2 A The Problem-Solving Power of Units Copyright © 2011 Pearson Education, Inc. Unit 2 A The Problem-Solving Power of Units Copyright © 2011 Pearson Education, Inc. Slide 2 -3

Global Melting Copyright © 2011 Pearson Education, Inc. Slide 2 -4 Global Melting Copyright © 2011 Pearson Education, Inc. Slide 2 -4

The Power of Units n n There is no single best approach to solving The Power of Units n n There is no single best approach to solving quantitative problem. You will learn some strategies and guidelines to help you solve problems. Copyright © 2011 Pearson Education, Inc. Slide 2 -5

2 -A Units The units of a quantity describe what is being measured or 2 -A Units The units of a quantity describe what is being measured or counted. Operation Key word or symbol Division per Read miles hours as “miles per hour. ” Raising to second power square Read ft ft, or ft 2, as “square feet” or “feet squared. ” Raising to third power cube or cubic Read ft ft, or ft 3, as “cubic feet” or “feet cubed. ” Multiplication hyphen Read kilowatts hours as “kilowatt-hours. ” Copyright © 2011 Pearson Education, Inc. Example Slide 2 -6

2 -A Problems to try n n Your average speed on a long walk, 2 -A Problems to try n n Your average speed on a long walk, found by dividing distance traveled in miles by time elapsed in hours. The unit price of oranges, found by dividing the price in dollars by the weight in pounds. The cost of a piece of carpet, found by dividing its price in dollars by its area in square yards. The flow rate of a river in which 5000 cubic feet of water flow past a particular location every second. Copyright © 2011 Pearson Education, Inc. Slide 2 -7

2 -A Unit Conversions Convert a distance of 9 feet into inches. Copyright © 2 -A Unit Conversions Convert a distance of 9 feet into inches. Copyright © 2011 Pearson Education, Inc. Slide 2 -8

2 -A Everyday Problems n n n Everyday problems can be solved with unit 2 -A Everyday Problems n n n Everyday problems can be solved with unit conversion. There are many different ways to write the conversion of 1. The KEY is to use the correct form. Copyright © 2011 Pearson Education, Inc. Slide 2 -9

2 -A Conversion Factors A conversion factor is a statement of equality that is 2 -A Conversion Factors A conversion factor is a statement of equality that is used to convert between units. Some conversion factors: Copyright © 2011 Pearson Education, Inc. Slide 2 -10

2 -A Example 2: Feet to Inches n Convert a distance of 7 feet 2 -A Example 2: Feet to Inches n Convert a distance of 7 feet into inches (p. 85) n Convert 24 feet to inches n Convert 24 feet to yards Copyright © 2011 Pearson Education, Inc. Slide 2 -11

2 -A Example 3: Inches to Feet n Convert a length of 102 inches 2 -A Example 3: Inches to Feet n Convert a length of 102 inches to feet (p. 85) n Now try Minutes to Seconds n Convert 25 minutes to seconds n Convert 2. 5 hours to seconds Copyright © 2011 Pearson Education, Inc. Slide 2 -12

2 -A Using a Chain of Conversions Copyright © 2011 Pearson Education, Inc. Slide 2 -A Using a Chain of Conversions Copyright © 2011 Pearson Education, Inc. Slide 2 -13

2 -A Conversions with Units Raised to Powers 1 yd = 3 ft 1 2 -A Conversions with Units Raised to Powers 1 yd = 3 ft 1 yd 2 = 1 yd × 1 yd = 3 ft × 3 ft = 9 ft 2 Copyright © 2011 Pearson Education, Inc. Slide 2 -14

2 -A Hint n n We take special care when converting units raised to 2 -A Hint n n We take special care when converting units raised to powers. We may not know the conversion factor between square yards, but we know that 1 yard = 3 ft, so we can replace it to say 1 yd x 1 yd = 1 square yard, so 3 ft x 3 ft = 9 square feet Copyright © 2011 Pearson Education, Inc. Slide 2 -15

2 -A Cubic Units How many cubic yards of soil are needed to fill 2 -A Cubic Units How many cubic yards of soil are needed to fill a planter that is 20 feet long by 3 feet wide by 4 feet tall? The volume is 20 ft × 3 ft × 4 ft = 240 ft 3 1 yd = 3 ft, so (1 yd)3 = (3 ft)3 = 27 ft 3 Copyright © 2011 Pearson Education, Inc. Slide 2 -16

2 -A Hint n n n Follow the same idea with cubic yards as 2 -A Hint n n n Follow the same idea with cubic yards as you do square yards. 1 yd x 1 yd = 1 cubic yard, so 3 ft x 3 ft = 27 cubic feet. Copyright © 2011 Pearson Education, Inc. Slide 2 -17

2 -A Currency Conversions You return from a trip to Europe with 120 euros. 2 -A Currency Conversions You return from a trip to Europe with 120 euros. How many dollars do you have? Copyright © 2011 Pearson Education, Inc. Slide 2 -18

2 -A Let’s try n n n You arrive in London with $400. How 2 -A Let’s try n n n You arrive in London with $400. How many pounds can you buy? As you leave Paris, you convert 4500 euros to dollars. How many dollars do you receive? You return from Mexico with 3000 pesos. How much are they worth in U. S. dollars? Copyright © 2011 Pearson Education, Inc. Slide 2 -19

2 -A Exchange Rates n n n Current exchange rates are constantly changing, but 2 -A Exchange Rates n n n Current exchange rates are constantly changing, but you can always get current rates by typing “exchange rates” into any search engine. Find the reciprocals of the numbers in the Dollars per Foreign column of Table 2. 1. FYI: Banks offer better rates, some places charge fees. Copyright © 2011 Pearson Education, Inc. Slide 2 -20

2 -A Problem Solving with Units n Units can help us decide HOW to 2 -A Problem Solving with Units n Units can help us decide HOW to solve n Same units – use addition and subtraction ONLY n Different units – usae multiplication and division. Copyright © 2011 Pearson Education, Inc. Slide 2 -21

2 -A Problem Solving with Units 1. Identify the units involved in the problem. 2 -A Problem Solving with Units 1. Identify the units involved in the problem. Use them to n n decide how to approach the problem; and determine what units to expect in the answer. 2. Perform any operations on both the numbers and their associated units. n n n Numbers with different units cannot be added or subtracted. Combine different units through multiplication, division, or raising to powers. Replace division with multiplication by the reciprocal. 3. Make sure your answer is in the units you expected. Copyright © 2011 Pearson Education, Inc. Slide 2 -22

2 -A Using units to Check Answers n n Checking answers is one of 2 -A Using units to Check Answers n n Checking answers is one of the most important reasons for using units. You should always check your method and your calculations as well as your units!!! Copyright © 2011 Pearson Education, Inc. Slide 2 -23

2 -A Distance Time and Speed n Speed in miles per hour is distance 2 -A Distance Time and Speed n Speed in miles per hour is distance / time n You can always rearrange the equation… n n distance = speed x time = distance / speed Copyright © 2011 Pearson Education, Inc. Slide 2 -24

2 -A Let’s try n n A car is traveling 25 miles every half-hour. 2 -A Let’s try n n A car is traveling 25 miles every half-hour. How fast is it going? (p. 90) An airliner travels 45 miles in 5 minutes. What is its speed in miles per hour? Competition speed skydivers have reached record speeds of 614 miles per hour. At this speed, how many feet would you fall every second? Copyright © 2011 Pearson Education, Inc. Slide 2 -25

2 -A Problem Solving with Units You are buying 50 acres of farm land 2 -A Problem Solving with Units You are buying 50 acres of farm land at a cost of $12, 500 per acre. What is the total cost? The answer should be in dollars. We multiply the acreage by the cost per acre: Copyright © 2011 Pearson Education, Inc. Slide 2 -26

2 -A Example 10: Buying Farm Land n n You are buying 30 acres 2 -A Example 10: Buying Farm Land n n You are buying 30 acres of farm land at $12, 000 per acre. What is the total cost? (p. 90) * an acre was originally defined as the amount of land a pair of oxen could plow in a day. Copyright © 2011 Pearson Education, Inc. Slide 2 -27

2 -A Use unit conversions to answer n n n An airline travels 45 2 -A Use unit conversions to answer n n n An airline travels 45 miles in 5 minutes. What is its speed in miles per hour? What is the total cost of 1. 2 cubic yards of soil if it sells for $245 per cubic yard? A hose fills a hot tub at a rate of 3. 2 gallons per minutes. How many hours will it take to fill a 300 gallon-hot tub? Copyright © 2011 Pearson Education, Inc. Slide 2 -28

2 -A More unit conversions n n n Suppose you earn $8. 50 per 2 -A More unit conversions n n n Suppose you earn $8. 50 per hour and work 24 eight-hour days in a month. How much do you earn in that month? The median salary for the New York Yankees in 2008 was $1, 875, 00. Assuming a 160 -game season, express this salary in dollars per game. If you sleep an average of 8 hours each night, how many hours do you sleep in a year? Copyright © 2011 Pearson Education, Inc. Slide 2 -29

2 -A Analyzing Problems n n n Start analyzing a problem by looking at 2 -A Analyzing Problems n n n Start analyzing a problem by looking at its units. Gas Mileage (p. 91) Your destination is 90 miles away and your fuel gauge shows that your gas tank is only onequarter full. You know that your tank holds 12 gallons of gas and that your car averages about 25 miles per gallon. Do you need to stop for gas? Copyright © 2011 Pearson Education, Inc. Slide 2 -30

2 -A Let’s try n n You plan to take a 2000 mile trip 2 -A Let’s try n n You plan to take a 2000 mile trip in your car, which averages 32 miles per gallon. How many gallons of gasoline should you expect to use? Would a car that has only half the gas mileage(16 miles per gallon) require twice as much gasoline for the same trip? Explain Copyright © 2011 Pearson Education, Inc. Slide 2 -31

2 -A Melting Ice We can find the correct answer simply by working with 2 -A Melting Ice We can find the correct answer simply by working with units. Copyright © 2011 Pearson Education, Inc. Slide 2 -32

2 -A Homework n Quick Quiz n P. 92: 1 -12; 19 -85 odd; 2 -A Homework n Quick Quiz n P. 92: 1 -12; 19 -85 odd; 1 web 1 world Copyright © 2011 Pearson Education, Inc. Slide 2 -33