Unit Conversions • Units are Wonderful and Horrible! • Do unit conversions when you want an answer in different units than the original information you have • Familiar Units – Salary : dollars per hour [$/hr --> $/year] – Price : dollars per pound [$/lb --> $/turkey] – Speed : miles per hour [m/h, or miles/trip] • Examples of “Unity” fractions (ones that equal “1”) – Time : minutes per hour [60 min/1 hour] – Weight : ounces per pound [16 oz/1 lb]
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Unit Conversions• Units are Wonderful and Horrible!
• Do unit conversions when you want an answer in different units than the original information you have
• Familiar Units– Salary: dollars per hour [$/hr --> $/year]– Price: dollars per pound [$/lb --> $/turkey]– Speed: miles per hour [m/h, or miles/trip]
• Examples of “Unity” fractions (ones that equal “1”)– Time: minutes per hour [60 min/1 hour]– Weight: ounces per pound [16 oz/1 lb]
hour day week month year
Example: Salary
• It all boils down to multiplying your starting info by a series of “unity” fractions (1) , set up to get the final units you want
• When you add “units” (words) to fractions, you can treat them just like numbers, i.e. cancel matching units
20 $ x 8 hours x 5 days x 4 weeks x 12 months
(1) (1) (1) (1)
hour day week month year
Cancel Units
• It all boils down to multiplying your starting info by a series of “unity” fractions (1) , set up to get the final units you want
• When you add units (“words”) to fractions, you can treat them just like numbers
20 $ x 8 hours x 5 days x 4 weeks x 12 months
(1) (1) (1) (1)
hour day week month year
Cancel …
• It all boils down to multiplying your starting info by a series of “unity” fractions (1) , set up to get the final units you want
• When you add units (“words”) to fractions, you can treat them just like numbers
20 $ x 8 hours x 5 days x 4 weeks x 12 months
(1) (1) (1) (1)
hour day week month year
Cancel …
• It all boils down to multiplying your starting info by a series of “unity” fractions (1) , set up to get the final units you want
• When you add units (“words”) to fractions, you can treat them just like numbers
20 $ x 8 hours x 5 days x 4 weeks x 12 months
(1) (1) (1) (1)
hour day week month year
Cancel …
• It all boils down to multiplying your starting info by a series of “unity” fractions (1) , set up to get the final units you want
• When you add units (“words”) to fractions, you can treat them just like numbers
20 $ x 8 hours x 5 days x 4 weeks x 12 months
(1) (1) (1) (1)
Calculate
• It all boils down to multiplying your starting info by a series of “unity” fractions (1) , set up to get the final units you want
• When you add units (“words”) to fractions, you can treat them just like numbers
year20 x 8 x 5 x 4 x 12 $
hour day week month year20 $ x 8 hours x 5 days x 4 weeks x 12 months
(1) (1) (1) (1)
I’ll take the job!
• It all boils down to multiplying your starting info by a series of “unity” fractions (1) , set up to get the final units you want
• When you add units (“words”) to fractions, you can treat them just like numbers
year20 x 8 x 5 x 4 x 12 $
= 38,400 year$
hour day week month year20 $ x 8 hours x 5 days x 4 weeks x 12 months
(1) (1) (1) (1)
I’ll take the job!
• It all boils down to multiplying your starting info by a series of “unity” fractions (1) , set up to get the final units you want
• When you add units (“words”) to fractions, you can treat them just like numbers
year20 x 8 x 5 x 4 x 12 $
= 38,400 year$
hour day week month year20 $ x 8 hours x 5 days x 4 weeks x 12 months
(1) (1) (1) (1)
Anyone see the problem?
Example: Songs• How many songs are in all the iPods at school?
• Start by collecting data– 0.5 iPods/student– 250 songs/iPod– 30 students/classroom– 100 classrooms/school
Set up the Fractions
0.5 iPods x 500 songs x 30 students x 100 classroomsstudent iPod classroom school
Cancel Units
0.5 iPods x 500 songs x 30 students x 100 classroomsstudent iPod classroom school
Cancel …
0.5 iPods x 500 songs x 30 students x 100 classroomsstudent iPod classroom school
Keep going…
0.5 iPods x 500 songs x 30 students x 100 classroomsstudent iPod classroom school
Calculate
0.5 iPods x 500 songs x 30 students x 100 classroomsstudent iPod classroom school
0.5 x 500 x 30 x 100 songsschool
Answer
0.5 x 500 x 30 x 100 songsschool
= 750,000 songs/school
0.5 iPods x 500 songs x 30 students x 100 classroomsstudent iPod classroom school
Energy Unit Conversions
• Question: How important is each type of energy use in my home?
– 10,000 kilowatt-hours (kWh)/year of electricity– 200 thousand cubic feet (ccf) of natural gas– 50 gallons of heating oil
– Apples and Oranges -- How can you express these in “common units”??
Set up the Fractions
Electricity: 10,000 kWh x 3,412 BTU x 1 MMBTU year kWh 1,000,000 BTU
Cancel, Multiply --> Answer
Electricity: 10,000 kWh x 3,412 BTU x 1 MMBTU year kWh 1,000,000 BTU
= 10,000 x 3412 x 1 MBTU
Year1,000,000
= 34.1 MBTU
Year
Set up the Fractions
Natural Gas: 200 ccf x 105,000 BTU x 1 MMBTU year ccf 1,000,000 BTU
Cancel, Multiply --> Answer
Natural Gas: 200 ccf x 105,000 BTU x 1 MMBTU year ccf 1,000,000 BTU
= 200 x 105,000 x 1 MBTU
Year1,000,000
= 21.0 MBTU
Year
Set up the Fractions
Oil: 50 gallons x 138,095 BTU x 1 MMBTU year gallon 1,000,000 BTU
Cancel, Multiply --> Answer
Oil: 50 gallons x 138,095 BTU x 1 MMBTU year gallon 1,000,000 BTU
= 50 x 138,095 x 1 MBTU
Year1,000,000
= 6.9 MBTU
Year
The Answer
Electricity: 34.1 MMBTU/yearNatural Gas: 21.0 MMBTU/yearOil: 6.9 MMBTU/yearTOTAL: 62.0 MMBTU/year
ElectricityNatural Gas
Oil
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