pre-algebra 7-3 analyze units 7-3 analyze units pre-algebra warm up warm up problem of the day...
TRANSCRIPT
Pre-Algebra
7-3 Analyze Units7-3 Analyze Units
Pre-Algebra
Warm UpWarm Up
Problem of the DayProblem of the Day
Lesson PresentationLesson Presentation
Pre-Algebra
7-3 Analyze Units
Warm UpFind 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
Pre-Algebra
7-3 Analyze Units
Pre-Algebra
7-3 Analyze Units
Problem of the Day
Replace each • with a digit from 0 to 6 to make equivalent ratios. Use each digit only once. ••••
•••
= Possible answer:1365
420
=
Pre-Algebra
7-3 Analyze Units
Learn to use one or more conversion factors to solve rate problems.
Pre-Algebra
7-3 Analyze Units
Vocabulary
conversion factor
Pre-Algebra
7-3 Analyze Units
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 unit.
Pre-Algebra
7-3 Analyze Units
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.
1 ft12 in.
Pre-Algebra
7-3 Analyze Units
Multiplying by a conversion factor is like multiplying by a fraction that reduces to 1, such as .5
5
12 in.12 in.
1 ft1 ft
= , or = 11 ft12 in.
Pre-Algebra
7-3 Analyze Units
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.
Helpful Hint
Pre-Algebra
7-3 Analyze Units
Find the appropriate factor for each conversion.
Additional Example 1: Finding Conversion Factors
A. feet to yards
B. pounds to ounces
1 yd3 ft
There are 3 feet in 1 yard. To convert feet to
yards, multiply the number of feet by .
16 oz1 lb
There are 16 ounces in 1 pound. To convert
pounds to ounces, multiply the number of
pounds by .
Pre-Algebra
7-3 Analyze Units
Try This: Example 1
Find the appropriate factor for each conversion.
A. minutes to seconds
B. hours to days
60 sec1 min
There are 60 seconds in 1 minute. To convert
minutes to seconds, multiply the number of
minutes by .
1 day 24 h
There are 24 hours in 1 day. To convert
hours to days, multiply the number hours by
.
Pre-Algebra
7-3 Analyze Units
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.
Additional Example 2: Using Conversion Factors to Solve Problems
The problem gives the ratio 580 pounds to 1 year and asks for an answer in pounds per month.
580 lb 1 yr
1 yr 12 mo
580 lb 12 mo
=
= 48.3 lb per month
Multiply the ratio by the conversion factor
Cancel yr units.
Divide 580 by 12.
Pre-Algebra
7-3 Analyze Units
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.
Additional Example 2 Continued
The average American uses 48.3 pounds of paper per month.
Pre-Algebra
7-3 Analyze Units
Try This: Example 2
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.
23,000 mi 1 yr
1 yr 12 mo
23,000 mi 12 mo
=
= 1916.6 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.
Pre-Algebra
7-3 Analyze Units
11 Understand the Problem
Additional Example 3: 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:
• Feet to miles5280 ft
1 mi• Hours to minutes
• Minutes to seconds 1 min60 s
1 h60 min
Pre-Algebra
7-3 Analyze Units
Multiply by each conversion factor separately, or simplify the problem and multiply by several conversion factors at once.
22 Make a Plan
Additional Example 3 Continued
Pre-Algebra
7-3 Analyze Units
First, convert 60 miles in 2 hours into a unit rate.
Solve33
60 mi2 h
= (60÷2) mi(2÷2) h
= 30 mi1 h
Create a single conversion factor to convert hours directly to seconds:
hours to seconds = • 1 min60 s
Set up the conversion factors.
minutes to seconds 1 min60 s
hours to minutes 1 h60 min
1 h60 min
1 h3600 s
=
30 mi1 h
• 5280 ft1 mi
• 1 h 3600 s
Pre-Algebra
7-3 Analyze Units
Solve33
Do not include the numbers yet.
Notice what happens to the units.
30 • 5280 ft • 1 1 • 1 • 3600 s
= 158,400 ft3600 s
= 44 ft1 s
The car was traveling 44 feet per second.
Simplify. Only remains.fts
Multiply.
mih
•ftmi
• hs
• •30 mi1 h
5280 ft1 mi
1 h 3600 s
Pre-Algebra
7-3 Analyze Units
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.
44 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.
Pre-Algebra
7-3 Analyze Units
11 Understand the Problem
Try This: Example 3A 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:
• Feet to miles5280 ft
1 mi• Hours to minutes
• Minutes to seconds 1 min60 s
1 h60 min
Pre-Algebra
7-3 Analyze Units
Multiply by each conversion factor separately, or simplify the problem and multiply by several conversion factors at once.
22 Make a Plan
Try This: Example 3 Continued
Pre-Algebra
7-3 Analyze Units
First, convert 180 miles in 4 hours into a unit rate.
Solve33
180 mi4 h
= (180 ÷ 4) mi(4 ÷ 4) h
= 45 mi1 h
Create a single conversion factor to convert hours directly to seconds:
hours to seconds = • 1 min60 s
Set up the conversion factors.
minutes to seconds 1 min60 s
hours to minutes 1 h60 min
1 h60 min
1 h3600 s
=
45 mi1 h
• 5280 ft1 mi
• 1 h 3600 s
Pre-Algebra
7-3 Analyze Units
Solve33
Do not include the numbers yet.
Notice what happens to the units.
45 • 5280 ft • 1 1 • 1 • 3600 s
= 237,600 ft3600 s
= 66 ft1 s
The train was traveling 66 feet per second.
Simplify. Only remains.fts
Multiply.
mih
•ftmi
• hs
• •45 mi1 h
5280 ft1 mi
1 h 3600 s
Pre-Algebra
7-3 Analyze Units
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.
44 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.
Pre-Algebra
7-3 Analyze Units
Additional Example 4: 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?
1 100
distance .time
Use rate =52 cm1
100s
Pre-Algebra
7-3 Analyze Units
It may help to eliminate the fraction first.
Additional Example 4 Continued
1 100
Multiply top and bottom by 100.
5200 cm1 s
=
52 cm1
100s
= 100 • 52 cm 1
100 s 100 •
Pre-Algebra
7-3 Analyze Units
Now convert centimeters to meters.
Additional Example 4 Continued
5200 cm1 s
Multiply by the conversion factor.
5200 m100 s
=52 m1 s
=
The object is traveling 52 m/s.
5200 cm1 s
= • 1 m100 cm
Pre-Algebra
7-3 Analyze Units
Try This: Example 4
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 cm1
100s
Pre-Algebra
7-3 Analyze Units
It may help to eliminate the fraction first.
Try This: Example 4 Continued
1 100
Multiply top and bottom by 100.
6500 cm1 s
=
65 cm1
100s
= 100 • 65 cm 1
100 s 100 •
Pre-Algebra
7-3 Analyze Units
Now convert centimeters to meters.
Try This: Example 4 Continued
6500 cm1 s
Multiply by the conversion factor.
6500 m100 s
=65 m1 s
=
The object is traveling 65 m/s.
6500 cm1 s
= • 1 m100 cm
Pre-Algebra
7-3 Analyze Units
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?
Additional Example 5: Transportation Application
5 knots = 5 nautical mi/h
Set up the units to obtain in your answer.kmh
Examine the units.
5 • 1852 • 1 km1 h • 1 • 1000
= = 9260 km1000 h
=9.26 km1 h
The ship is traveling 9.26 km/h.
nautical mih
• mnautical mi
• kmm
5 nautical mi1 h
• •1852 m1 nautical mi
1 km1000 m
Pre-Algebra
7-3 Analyze Units
Try This: Example 5
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
Set up the units to obtain in your answer.kmh
Examine the units.
9 • 1852 • 1 km1 h • 1 • 1000
= =16,668 km1000 h
16.67 km1 h
The ship is traveling about 16.67 km/h.
nautical mih
• mnautical mi
• kmm
9 nautical mi1 h
• •1852 m1 nautical mi
1 km1000 m
Pre-Algebra
7-3 Analyze Units
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 gkg
1 gal8 pt
29,920 ydgal
≈ 68 mi/h