unit conversions units are wonderful and horrible! do unit conversions when you want an answer in...

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