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# Analyze Units

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1. Analyze Units 7.3 Pre-Algebra

2. Vocabulary conversion factor

3. Dimensional Analysis You can measure the speed of an object by using a strobe lamp and a camera in a dark room. Each time the lamp flashes, the camera records the object’s position. Problems often require dimensional analysis, also called unit analysis, to convert from one unit to another.

4. 1 ft 12 in Conversion Factors To convert units, multiply by one or more ratios of equal quantities called conversion factors. For example, to convert inches to feet you would use the ratio below as a conversion factor.

5. 1 ft 12 in. Conversion Factors Multiplying by a conversion factor is like multiplying by a fraction that reduces to 1, such as . 5 5 12 in. 12 in. 1 ft 1 ft = , or = 1

6. Helpful Hint • The conversion factor • must introduce the unit desired in the answer and • must cancel the original unit so that the unit desired is all that remains.

7. There are 3 feet in 1 yard. To convert feet to yards, multiply the number of feet by . 1 yd 3 ft There are 16 ounces in 1 pound. To convert pounds to ounces, multiply the number of pounds by . 16 oz 1 lb Example: Finding Conversion Factors Find the appropriate factor for each conversion. • A. feet to yards • B. pounds to ounces

8. There are 60 seconds in 1 minute. To convert minutes to seconds, multiply the number of minutes by . 60 sec 1 min There are 24 hours in 1 day. To convert hours to days, multiply the number hours by . 1 day 24 h Try This Find the appropriate factor for each conversion. A. minutes to seconds B. hours to days

9. 580 lb 1 yr 1 yr 12 mo 580 lb 12 mo = = 48.3 lb per month Example: Using Conversion Factors to Solve Problems The average American uses 580 pounds of paper per year. Find the number of pounds of paper the average American uses per month, to the nearest tenth. The problem gives the ratio 580 pounds to 1 year and asks for an answer in pounds per month. Multiply the ratio by the conversion factor Cancel yr units. Divide 580 by 12.

10. The average American uses 48.3 pounds of paper per month. Example Continued The average American uses 580 pounds of paper per year. Find the number of pounds of paper the average American uses per month, to the nearest tenth.

11. 23,000 mi 1 yr 1 yr 12 mo 23,000 mi 12 mo = = 1916.6 per month Try This Sam drives his car 23,000 miles per year. Find the number of miles he drives per month. The problem gives the ratio 23,000 miles to 1 year and asks for an answer in miles per month. Multiply the ratio by the conversion factor Cancel yr units. Divide 23,000 by 12. Sam drives his car about 1917 miles per month.

12. 1 Understand the Problem 1 h 60 min Example: Problem Solving Application A car traveled 60 miles on a road in 2 hours. How many feet per second was the car traveling? The problem is stated in units of miles and hours. The question asks for the answer in units of feet and seconds. You will need to use several conversion factors. List the important information: 5280 ft 1 mi • Miles to feet • Hours to minutes 1 min 60 s • Minutes to seconds

13. Make a Plan 2 Example Continued Multiply by each conversion factor separately, or simplify the problem and multiply by several conversion factors at once.

14. 3 Solve 1 min 60 s hours to minutes minutes to seconds 1 h 3600 s 1 h 60 min = 30 mi 1 h 5280 ft 1 mi 1 h 3600 s • • 1 h 60 min Example Continued First, convert 60 miles in 2 hours into a unit rate. 60 mi 2 h (60÷2) mi (2÷2) h 30 mi 1 h = = Create a single conversion factor to convert hours directly to seconds: 1 min 60 s hours to seconds = • Set up the conversion factors.

15. 3 ft s ft mi mi h h s Simplify. Only remains. • • 1 h 3600 s 5280 ft 1 mi 30 mi 1 h • • 30 • 5280 ft • 1 1 • 1 • 3600 s 158,400 ft 3600 s = = Example Continued Solve Do not include the numbers yet. Notice what happens to the units. Multiply. 44 ft 1 s The car was traveling 44 feet per second.

16. 4 Example Continued Look Back A rate of 44 ft/s is less than 50 ft/s. A rate of 60 miles in 2 hours is 30 min/h or 0.5 mi/min. Since 0.5 mi/min is less than 3000 ft/ 60 s or 50 ft/s and 44 ft/s is less than 50 ft/s, then 44 ft/s is a reasonable answer.

17. 1 Understand the Problem 1 h 60 min Try This A train traveled 180 miles on a railroad track in 4 hours. How many feet per second was the train traveling? The problem is stated in units of miles and hours. The question asks for the answer in units of feet and seconds. You will need to use several conversion factors. List the important information: 5280 ft 1 mi • Miles to feet • Hours to minutes 1 min 60 s • Minutes to seconds

18. Make a Plan 2 Try This Continued Multiply by each conversion factor separately, or simplify the problem and multiply by several conversion factors at once.

19. 3 Solve 1 min 60 s hours to minutes minutes to seconds 1 h 3600 s 1 h 60 min = 45 mi 1 h 5280 ft 1 mi 1 h 3600 s • • 1 h 60 min Try This Continued First, convert 180 miles in 4 hours into a unit rate. 180 mi 4 h (180 ÷ 4) mi (4 ÷ 4) h 45 mi 1 h = = Create a single conversion factor to convert hours directly to seconds: 1 min 60 s hours to seconds = • Set up the conversion factors.

20. 3 ft s ft mi mi h h s Simplify. Only remains. • • 1 h 3600 s 5280 ft 1 mi 45 mi 1 h • • 45 • 5280 ft • 1 1 • 1 • 3600 s 237,600 ft 3600 s = = Try This Continued Solve Do not include the numbers yet. Notice what happens to the units. Multiply. 66 ft 1 s The train was traveling 66 feet per second.

21. 4 Try This Continued Look Back A rate of 66 ft/s is more than 50 ft/s. A rate of 180 miles in 4 hours is 45 mi/h or 0.75 mi/min. Since 0.75 mi/min is more than 3000 ft/60 s or 50 ft/s and 66 ft/s is more than 50 ft/s, then 66 ft/s is a reasonable answer.

22. A strobe lamp can be used to measure the speed of an object. The lamp flashes every of a second. A camera records the object moving 52 cm between flashes. How fast is the object moving in m/s? 1 100 distance . time Use rate = 52 cm 1 100 s Example: Physical Science Application

23. 1 100 100 • 52 cm 1 100 = s 100 • 5200 cm 1 s 52 cm 1 100 = s Example Continued It may help to eliminate the fraction first. Multiply top and bottom by 100.

24. 1 m 100 cm 5200 cm 1 s = • 5200 m 100 s 52 m 1 s = = Example Continued Now convert centimeters to meters. 5200 cm 1 s Multiply by the conversion factor. The object is traveling 52 m/s.

25. A strobe lamp can be used to measure the speed of an object. The lamp flashes every of a second. A camera records the object moving 65 cm between flashes. How fast is the object moving in m/s? 1 100 distance . time Use rate = 65 cm 1 100 s Try This

26. 1 100 100 • 65 cm 1 100 = s 100 • 6500 cm 1 s 65 cm 1 100 = s Try This Continued It may help to eliminate the fraction first. Multiply top and bottom by 100.

27. 1 m 100 cm 6500 cm 1 s = • 6500 m 100 s 65 m 1 s = = Try This Continued Now convert centimeters to meters. 6500 cm 1 s Multiply by the conversion factor. The object is traveling 65 m/s.

28. km h Set up the units to obtain in your answer. km m nautical mi h • • m nautical mi 5 nautical mi 1 h 1852 m 1 nautical mi 1 km 1000 m • • 5 • 1852 • 1 km 1 h • 1 • 1000 9260 km 1000 h 9.26 km 1 h = = = Example: Transportation Application The rate 1 knot equals 1 nautical mile per hour. One nautical mile is 1852 meters. What is the speed in kilometers per hour of a ship traveling at 5 knots? 5 knots = 5 nautical mi/h Examine the units. The ship is traveling 9.26 km/h.

29. km h Set up the units to obtain in your answer. km m nautical mi h • • m nautical mi 9 nautical mi 1 h 1852 m 1 nautical mi 1 km 1000 m • • 9 • 1852 • 1 km 1 h • 1 • 1000 16,668 km 1000 h 16.67 km 1 h = =  Try This The rate 1 knot equals 1 nautical mile per hour. One nautical mile is 1852 meters. What is the speed in kilometers per hour of a ship traveling at 9 knots? 9 knots = 9 nautical mi/h Examine the units. The ship is traveling about 16.67 km/h.

30. Lesson Quiz Find the appropriate factor for each conversion. 1. kilograms to grams 2. pints to gallons 3. You drive 136 miles from your house to your aunt’s house at the lake. You use 8 gallons of gas. How many yards does your car get to the gallon? 4. A cheetah was timed running 200 yards in 6 seconds. What was the average speed in miles per hour? 1000 g kg 1 gal 8 pt 29,920 yd gal ≈ 68 mi/h