1. DIMENSIONAL ANALYSIS
How to Change Units using Math

2. How do you change units?
Whenever you have to convert a physical measurement from one dimensional unit to another, dimensional analysis is the method used.
(It is also known as the unit-factor method or the factor-label method)

3. So what is dimensional analysis?
The converting from one unit system to another.
If this is all that it is, why make such a fuss about it?  Very simple.  Wrong units lead to wrong answers.  Scientists have thus evolved an entire system of unit conversion.   

4. Why Do Units Matter? Why is it important to always write the units?
“I know what the answer should be in, why do I have to write the unit?”

5. The loss of the the Mars Climate Orbiter on September 23, 1999, was a most unfortunate and highly avoidable event.
The cause of the mishap has been traced to a mix-up over units. Preliminary findings indicated that one team used English units (e.g., inches, feet and pounds) while the other used metric units for maneuvers required to place the spacecraft in the proper Mars orbit.
The 'root cause' of the loss of the spacecraft was the failed translation of English units into metric units.
For nearly three centuries, engineers and scientists have been struggling with English units.

6. Dimensional Analysis How does dimensional analysis work?
It will involve some easy math (Multiplication & Division)
In order to perform any conversion, you need a conversion factor.
Conversion factors are made from any two terms that describe the same or equivalent “amounts” of what we are interested in.
For example, we know that:
1 inch = centimeters
1 dozen = 12 items

7. Conversion Factors
So, conversion factors are nothing more than equalities or ratios that equal to each other. In “math-talk” they are equal to one.
In mathematics, the expression to the left of the equal sign is equal to the expression to the right. They are equal expressions.
For Example
12 inches = 1 foot
Written as an “equality” or “ratio” it looks like
= 1 or
= 1

8. Hey! These look like fractions!
Conversion Factors
Hey! These look like fractions! or
Conversion Factors look a lot like fractions, but they are not!
The critical thing to note is that the units
behave like numbers do when you multiply
fractions. That is, the inches (or foot) on top and
the inches (or foot) on the bottom can cancel
out. Just like in algebra, Yippee!!

9. (The equality that looks like a fraction)
Example Problem #1
How many feet are in 60 inches? Solve using dimensional analysis.
All dimensional analysis problems are set up the same way. They follow this same pattern:
What units you have x What units you want = What units you want
What units you have
The number & units you start with
The units you want to end with
The conversion factor
(The equality that looks like a fraction)

10. Example Problem #1 (cont)
You need a conversion factor. Something that will change inches into feet.
Remember
12 inches = 1 foot
Written as an “equality” or “ratio” it looks like
60 inches
5 feet x =
(Mathematically all you do is: 60 x 1  12 = 5)
What units you have x What units you want = What units you want
What units you have

11. Dimensional Analysis
The hardest part about dimensional analysis is knowing which conversion factors to use.
Some are obvious, like 12 inches = 1 foot, while others are not. Like how many feet are in a mile.

12. Example Problem #2
You need to put gas in the car. Let's assume that gasoline costs $3.35 per gallon and you've got a twenty dollar bill. How many gallons of gas can you get with that twenty?
Starting unit
Conversion factor
Ending unit
𝟏 𝒈𝒂𝒍𝒍𝒐𝒏 $𝟑.𝟑𝟓 X =
$ 20.00
5.97 gallons
(Mathematically all you do is: 20  3.35 = 5.97)

13. = X 5.97 gallons 143.28 miles Example Problem #3
What if you had wanted to know not how many gallons you could get, but how many miles you could drive assuming your car gets 24 miles a gallon? Let's try building from the previous problem. You know you have 5.97 gallons in the tank.
Starting unit
Conversion factor
Ending unit
Starting unit
24 𝑚𝑖𝑙𝑒𝑠 1 𝑔𝑎𝑙𝑙𝑜𝑛 X =
5.97 gallons
miles
(Mathematically all you do is: 5.97 x 24= )

14. Example Problem #3
There's another way to do the previous two problems. Instead of chopping it up into separate pieces, build it as one problem. Not all problems lend themselves to working them this way but many of them do. It's a nice, elegant way to minimize the number of calculations you have to do. Let's reintroduce the problem.

15. Example Problem #3 (cont)
You have a twenty dollar bill and you need to get gas for your car. If gas is $3.35 a gallon and your car gets 24 miles per gallon, how many miles will you be able to drive your car on twenty dollars?
Starting unit
Conversion factor
(To cancel out dollars)
Conversion factor
(To cancel out gallons)
Ending unit
𝟏 𝒈𝒂𝒍𝒍𝒐𝒏 $𝟑.𝟑𝟓
24 𝑚𝑖𝑙𝑒𝑠 1 𝒈𝒂𝒍𝒍𝒐𝒏
$ 20.00 ≈
miles X X
(Mathematically all you do is: (20 x 24)  ≈ )

16. Example Problem #4
Try this expanded version of the previous problem.
You have a twenty dollar bill and you need to get gas for your car. Gas currently costs $3.35 a gallon and your car averages 24 miles a gallon. If you drive, on average, 7.1 miles a day, how many weeks will you be able to drive on a twenty dollar fill-up?

17. Example Problem 4 (cont.)
Conversion factor
(To cancel out dollars)
Conversion factor
(To cancel out gallons)
Conversion factor
(To cancel out miles)
Conversion factor
(To cancel out days)
End
Start
𝟏 𝒈𝒂𝒍𝒍𝒐𝒏 $𝟑.𝟑𝟓
24 𝑚𝑖𝑙𝑒𝑠 1 𝒈𝒂𝒍𝒍𝒐𝒏
1 𝑑𝑎𝑦 7.1 𝑚𝑖𝑙𝑒𝑠
1 𝑤𝑘 7 𝑑𝑎𝑦𝑠
$ 20.00 X X X X ≈
2.88 wks
(Mathematically : 20 𝑥 𝑥.7.1 𝑥 7 ≈ 2.88 )

18. Review
Dimensional Analysis (DA) is a method used to convert from one unit system to another. In other words a math problem.
Dimensional Analysis uses Conversion factors . Two terms that describe the same or equivalent “amounts” of what we are interested in.
All DA problems are set the same way. Which makes it nice because you can do problems where you don’t even understand what the units are or what they mean.

19. Let’s try some examples together…
1. How old are you in days?
Given: 14 years
Want: # of days
Conversion: 365 days = one year

20. Solution
Show your work…
14 years 1
365 days
1 year
5110 days X =

21. Let’s try some examples together…
2. There are 2.54 cm in one inch. How many inches are in 17.3 cm?
Given: 17.3 cm
Want: # of inches
Conversion: 2.54 cm = one inch

22. Be careful!!! The fraction bar means divide.
Solution
Show your work…
17.3 cm 1
1 inch
2.54 cm
6.81 inches X =
Be careful!!! The fraction bar means divide.

23. Now, you try… Determine the number of eggs in 23 dozen eggs.
If one package of gum has 10 pieces, how many pieces are in packages of gum?

24. Multiple-Step Problems
Most problems are not simple one-step solutions. Sometimes, you will have to perform multiple conversions.
Example: How old are you in hours?
Given: 17 years
Want: # of days
Conversion #1: 365 days = one year
Conversion #2: 24 hours = one day

25. Solution Show your work… 14 years 1 365 days 1 year 24 hours 1 day X X=
122,640 hours

26. Combination Units
Dimensional Analysis can also be used for combination units.
Like converting km/h into cm/s.
Write the fraction in a “clean” manner:
km/h becomes km h

27. Combination Units Example: Convert 0.083 km/h into m/s.
Given: km/h
Want: # m/s
Conversion #1: 1000 m = 1 km
Conversion #2: 1 hour = 60 minutes
Conversion #3: 1 minute = 60 seconds

28. Solution in Two Parts Check your work… 83 m 1 hour 0.083 km 1000 mX =
83 m
1 hour
1 min
60 sec
1 hour
60 min X X =
0.023 m
sec

29. Solution in one Part Show your work… 0.083 km 1 hour 1000 m 1 km
60 min
1 min
60 sec X X X =
83 m
3600 sec
0.023 m
sec =

30. Practice Conversions How many seconds are in 6 minutes? 360 seconds
How many centimeters are in 27 inches?
68.58 centimeters
If a truck weighs 15,356 pounds, how many tons is it?
7.678 tons
If you had 10.5 gallons of milk, how many pints would you have?
84 pints
Students go to school for 180 days. How many minutes is this equal to?
259,200 minutes

31. How many seconds are in 6 minutes?
Final Destination
Starting Point
6 minutes  seconds
Step 1 – Read the question and determine
what information it provides you with
(starting point & final destination)
Step 2 – Write down your
starting point and your final
destination
1 minute = 60 seconds
Step 3 – Determine how you will get
from your starting point to your final
destination (list any “connections” or conversion factors)
(6 minutes) 1
( ) 60
seconds
(6)(60 seconds) = 1
minute
(1)(1)
Step 9 – Cancel all diagonal units. Once this is done, your final destination should be the only unit left – in this case seconds
Step 5 – Multiply between fractions =
360 seconds
Step 10 – Multiply the top of the fractions; multiply the bottom of the fractions; divide the product of the top by the product of the bottom
Step 8 – Determine if this top unit is the desired unit (your final destination). In this case the answer is YES, so we move on to step 9
Step 7 – Write the appropriate conversion factor into the fraction. Your bottom unit will guide you.
Step 6 – Write in the bottom unit of the new fraction (this is the same as the top unit of your previous fraction)
Step 4 – Create a fraction by placing your starting point over one

32. How many centimeters are in 27 inches?
Final Destination
Starting Point
27 inches  centimeters
Step 1 – Read the question and determine
what information it provides you with
(starting point & final destination)
Step 2 – Write down your
starting point and your final
destination
1 inch = 2.54 centimeters
Step 3 – Determine how you will get
from your starting point to your final
destination (list any “connections” or conversion factors)
(27 inches) 1
( )
2.54 cm
(27)(2.54 cm) = 1
inch
(1)(1)
Step 9 – Cancel all diagonal units. Once this is done, your final destination should be the only unit left – in this case centimeters
Step 5 – Multiply between fractions =
68.58 centimeters
Step 10 – Multiply the top of the fractions; multiply the bottom of the fractions; divide the product of the top by the product of the bottom
Step 8 – Determine if this top unit is the desired unit (your final destination). In this case the answer is YES, so we move on to step 9
Step 7 – Write the appropriate conversion factor into the fraction. Your bottom unit will guide you.
Step 6 – Write in the bottom unit of the new fraction (this is the same as the top unit of your previous fraction)
Step 4 – Create a fraction by placing your starting point over one

33. If a truck weighs 15,356 pounds, how many tons is it?
Final Destination
Starting Point
15,356 pounds  tons
Step 1 – Read the question and determine
what information it provides you with
(starting point & final destination)
Step 2 – Write down your
starting point and your final
destination
2000 pounds = 1 ton
Step 3 – Determine how you will get
from your starting point to your final
destination (list any “connections” or conversion factors)
(15,356 lbs.) 1
( ) 1
ton
(15,356)(1 ton) =
2000
lbs.
(1)(2000)
Step 9 – Cancel all diagonal units. Once this is done, your final destination should be the only unit left – in this case tons
Step 5 – Multiply between fractions =
7.678 tons
Step 10 – Multiply the top of the fractions; multiply the bottom of the fractions; divide the product of the top by the product of the bottom
Step 8 – Determine if this top unit is the desired unit (your final destination). In this case the answer is YES, so we move on to step 9
Step 7 – Write the appropriate conversion factor into the fraction. Your bottom unit will guide you.
Step 6 – Write in the bottom unit of the new fraction (this is the same as the top unit of your previous fraction)
Step 4 – Create a fraction by placing your starting point over one

34. If you had 10.5 gallons of milk, how many pints would you have?
Final Destination
Starting Point
10.5 gallons  pints
Step 1 – Read the question and determine
what information it provides you with
(starting point & final destination)
1 gallon = 4 quarts
1 quart = 2 pints
Step 2 – Write down your
starting point and your final
destination
Step 3 – Determine how you will get
from your starting point to your final
destination (list any “connections” or conversion factors)
(10.5 gallons) 1
( ) 4
quarts
( ) 2
pints 1
gallon 1
quart
(10.5)(4)(2 pints)
Step 5 – Multiply between fractions
Step 7 – Write the appropriate conversion factor into the fraction. Your bottom unit will guide you.
Step 10 – Multiply the top of the fractions; multiply the bottom of the fractions; divide the product of the top by the product of the bottom =
Step 6 – Write in the bottom unit of the new fraction (this is the same as the top unit of your previous fraction)
Step 5 – Multiply between fractions
Step 7 – Write the appropriate conversion factor into the fraction. Your bottom unit will guide you.
Step 8 – Determine if this top unit is the desired unit (your final destination). In this case the answer is YES, so we move on to step 9
84 pints
Step 8 – Determine if this top unit is the desired unit (your final destination). In this case the answer is NO, so we move back to step 5
Step 9 – Cancel all diagonal units. Once this is done, your final destination should be the only unit left – in this case pints =
(1)(1)(1)
Step 6 – Write in the bottom unit of the new fraction (this is the same as the top unit of your previous fraction)
Step 4 – Create a fraction by placing your starting point over one

35. Students go to school for 180 days. How many minutes is this equal to?
Final Destination
Starting Point
180 days  minutes
Step 1 – Read the question and determine
what information it provides you with
(starting point & final destination)
1 day = 24 hours
1 hour = 60 minutes
Step 2 – Write down your
starting point and your final
destination
Step 3 – Determine how you will get
from your starting point to your final
destination (list any “connections” or conversion factors)
(180 days) 1
( ) 24
hours
( ) 60
minutes 1
day 1
hour
Step 9 – Cancel all diagonal units. Once this is done, your final destination should be the only unit left – in this case minutes
(180)(24)(60 minutes)
Step 5 – Multiply between fractions
Step 5 – Multiply between fractions
Step 10 – Multiply the top of the fractions; multiply the bottom of the fractions; divide the product of the top by the product of the bottom =
Step 6 – Write in the bottom unit of the new fraction (this is the same as the top unit of your previous fraction)
Step 7 – Write the appropriate conversion factor into the fraction. Your bottom unit will guide you.
Step 7 – Write the appropriate conversion factor into the fraction. Your bottom unit will guide you.
259,200 minutes =
Step 8 – Determine if this top unit is the desired unit (your final destination). In this case the answer is NO, so we move back to step 5
Step 6 – Write in the bottom unit of the new fraction (this is the same as the top unit of your previous fraction)
(1)(1)(1)
Step 4 – Create a fraction by placing your starting point over one
Step 8 – Determine if this top unit is the desired unit (your final destination). In this case the answer is YES, so we move on to step 9

36. (click on “view tutorial” for dimensional analysis)
Online Tutorials
(click on “view tutorial” for dimensional analysis)
(click on examples under dimensional analysis on the left side of the page)
Interactive Quiz

37. US Conversion Steps (Dimensional Analysis)
Read the question to figure out what you have/know for information. The question will provide you with information that identifies your starting point and your final destination.
Starting point = the number and unit provided by the question
Final destination = the units desired after converting
Using the information gathered from the question, write your starting point and your final destination.
Determine the means in which you will get from your starting point to your final destination (simply find “connections” or conversion factors between your starting and final unit).
Create a fraction by placing your starting point over one.
Multiply between fractions.
Write in the bottom unit of the new fraction. This should be the same as the top unit of the previous fraction.
Write one set of “connections” or conversion factors into the fraction. Your bottom unit will guide you.
Ask yourself, “Do I have the desired unit (final destination) on the top of the new fraction?”
NO YES
Cancel any units that are diagonal. (This should leave you with only the units that represent your final destination)
Multiply the top of the fractions…multiply the bottom of the fractions…divide the top by the bottom.
(Go back to step 5)
(Proceed to step 9)