2. Warm Up Find each unit rate. 1. jump rope 192 times in 6 minutes 2. four pounds of bananas for $2.36 3. 16 anchor bolts for $18.56 4. 288 movies on 9 shelves 32 jumps/min $0.59/lb $1.16/bolt 32 movies/shelf
3. MG1.1 Compare weights, capacities, geometric measures, times, and temperatures within and between measurement systems (e.g., miles per hour and feet per second, cubic inches to cubic centimeters). California Standards
5. The process of converting from one unit to another is called dimensional analysis , or unit analysis . To convert units, multiply by one or more conversion factors . A conversion factor is a ratio of two quantities that are equal but use different units. For example, to convert inches to feet you would use the ratio as a conversion factor. 1 ft 12 in.
6. Multiplying by a conversion factor is like multiplying by 1. 12 in. 12 in. 1 ft 1 ft = = = 1 1 ft 12 in.
7. The average American uses 580 pounds of paper per year. Find this rate in pounds per month, to the nearest tenth. Additional Example 1: Using Conversion Factors to Solve Problems Convert the rate 580 pounds per year to pounds per month. To convert the second amount in a rate, multiply by the conversion factor with that unit in the first quantity. Divide 580 by 12. 580 lb 1 yr 1 yr 12 mo 580 lb 12 mo 48.3 lb per month The average American uses 48.3 pounds of paper per month. Divide out like units. • yr mo = lb mo lb yr
8. Check It Out! Example 1 Sam drives his car 23,040 miles per year. Find this rate in the number of miles driven per month, to the nearest mile. Convert the rate 23,040 miles per year to miles per month. Divide 23,040 by 12. Sam drives his car 1920 miles per month. To convert the second amount in a rate, multiply by the conversion factor with that unit in the first quantity. 23,040 mi 1 yr 1 yr 12 mo 23,040 mi 12 mo = = 1920 per month Divide out units. • yr mo = mi mo mi yr
9. Additional Example 2: Problem Solving Application A car traveled 60 miles on a road in 2 hours. Find this rate in feet per second.
10. 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: • Feet to miles 5280 ft 1 mi • Hours to minutes • Minutes to seconds 1 min 60 s 1 Understand the Problem 1 h 60 min
11. You know the conversion factor that converts miles to feet. So multiply by each conversion factor separately, or simplify the problem and multiply by several conversion factors at once. 2 Make a Plan
12. Convert to miles per hour. 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: hours to seconds = • 1 min 60 s Set up the conversion factors. Solve 3 minutes to seconds 1 min 60 s hours to minutes 1 h 60 min 1 h 60 min 1 h 3600 s = 30 mi 1 h • 5280 ft 1 mi • 1 h 3600 s
13. Solve Continued 44 ft 1 s The car was traveling 44 feet per second. Divide out like units . = = 3 30 • 5280 ft • 1 1 • 1 • 3600 s = 158,400 ft 3600 s = • • 30 mi 1 h 5280 ft 1 mi 1 h 3600 s
14. A rate of 44 ft/s is less than 50 ft/s. A rate of 60 miles in 2 hours is 30 mi/h or 0.5 mi/min. Look Back 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. 4
15. Check It Out! Example 2 A train traveled 180 miles on a railroad track in 4 hours. Find this rate in feet per second.
16. 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: • Feet to miles 5280 ft 1 mi • Hours to minutes • Minutes to seconds 1 min 60 s 1 Understand the Problem 1 h 60 min
17. You know the conversion factor that converts miles to feet. So multiply by each conversion factor separately, or simplify the problem and multiply by several conversion factors at once. 2 Make a Plan
18. 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: hours to seconds = • 1 min 60 s Set up the conversion factors. Solve 3 minutes to seconds 1 min 60 s hours to minutes 1 h 60 min 1 h 60 min 1 h 3600 s = 45 mi 1 h • 5280 ft 1 mi • 1 h 3600 s
19. Solve Continued 66 ft 1 s The train was traveling 66 feet per second. Divide out like units . = = 3 45 • 5280 ft • 1 1 • 1 • 3600 s = 237,600 ft 3600 s = • • 45 mi 1 h 5280 ft 1 mi 1 h 3600 s
20. 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. Look Back 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. 4
21. Additional Example 3: Physical Science Application 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? Use dimensional analysis to check the reasonableness of your answer. 1 100 distance . time Use rate = 52 cm 1 100 s
22. It may help to eliminate the fraction first. Additional Example 3 Continued Multiply numerator and denominator by 100. 1 100 5200 cm 1 s = 52 cm 1 100 s = 100 • 52 cm 1 100 s 100 •
23. Convert centimeters to meters to see if the answer is reasonable. Additional Example 3 Continued 5200 cm 1 s Multiply by the conversion factor. The object is traveling 52 m/s. 5200 m 100 s = 52 m 1 s = 5200 cm 1 s = • 1 m 100 cm
24. Check It Out! Example 3 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
25. It may help to eliminate the fraction first. Check It Out! Example 3 Continued Multiply numerator and denominator by 100. 1 100 6500 cm 1 s = 65 cm 1 100 s = 100 • 65 cm 1 100 s 100 •
26. Convert centimeters to meters to see if the answer is reasonable. Check It Out! Example 3 Continued 6500 cm 1 s Multiply by the conversion factor. The object is traveling 65 m/s. 6500 m 100 s = 65 m 1 s = 6500 cm 1 s = • 1 m 100 cm
27. Lesson Quiz 1. 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? 2. A cheetah was timed running 200 yards in 6 seconds. What was its average speed in miles per hour? 29,920 yd gal ≈ 68 mi/h
