1. Dimensional Analysis (aka Factor-Label)
This technique involves the use of conversion factors and writing all measurements with both numerical values and the unit of measurement
A conversion factor is where you have the same amount (entity) represented by two different units of measurement with their corresponding numerical values

2. Conversion Factors Here are some examples 1 foot = __ inches
1 kilometer = ____ meters
1 inch = 2.54 centimeters
1 gallon = __ quarts
1 acre = 4840 square yards
1 day = ___ hours 12
1000 4 24

3. Conversion factors…cont.
Conversion factors for 1 ft = 12 in
There are almost an infinite number of conversion factors that include meters:

4. Conversion Factors….cont.
One member of a dinner party orders a 16 ounce steak and another orders a one pound steak- Compare the two steaks
They are the same since
16 oz dry wt. = 1 pound

5. Conversion Factors….cont.
In grade school we learned that 1 gallon contained 4 quarts or stating that relationship as an equality:
1 gallon = 4 quarts
Since 1 gallon and 4 quarts represent the same amount, we have a Conversion Factor

6. Conversion Factors….cont.
Start with 1 gallon = 4 quarts
Dividing each side by 1 gallon
we get this equation
1 gallon = 4 quarts
1 gallon gallon
Since 1 gallon divided by 1 gallon equals 1
Our equality becomes: 1 = 4 quarts gallon

7. Conversion Factors….cont.
Again start with 1 gallon = 4 quarts
But this time we’ll divide each side of the equality by 4 quarts
The resulting equation is
1 gallon = 4 quarts
4 quarts quarts

8. Conversion Factors…. Cont.
The right side of our equation becomes one because 4 quarts divided by 4 quarts is 1
1 gallon = quarts
Rearranging this becomes 1 = 1 gallon
4 quarts

9. Conversion Factors….cont.
A mid-presentation summary
We know that 1 gallon = 4 quarts
Using a little mathematical magic
1 gallon = 1 and 4 quarts = 1
4 quarts gallon
Why is this an important concept?

10. Conversion Factors….cont.
Now a little math review…………….
What is 5 x 1?
What is 5 x ? 2
Both expressions give you the same answer- why?
Because 2/2 equals 1 and therefore the second equation is just like the first and
we did not change the initial value of 5.

11. Putting It Together Here’s An Example
How many quarts are in 15 gallons ?
Remember we do NOT want to change the amount represented by 15 gallons, only the units in quarts
So we’ll use the conversion factor between gallons and quarts; that is
1 gallon = 4 quarts

12. Our Example continued…….
We set it up like this:
15 gallons x 4 quarts
1 gallon
Cancel units
Do the math to complete the problem
15 x 4 quarts = 60 quarts

13. Every measurement must have a unit.60
quarts

14. What do I need to do? From the problem determine the following:
Known quantity (number and units) which is called the Given
Identify what the Desired units are
Conversion factor(s) needed (both universal and question specific)

15. First write down the desired quantity
Factor label example
Q - How many kilometers are in 47 miles? (note: 1 km = miles)
# km
First write down the desired quantity

16. Next, equate desired quantity to the given quantity
Factor label example
Q - How many kilometers are in 47 miles? (note: 1 km = miles)
# km
= 47 mi
Next, equate desired quantity to the given quantity

17. Now we have to choose a conversion factor
Factor label example
Q - How many kilometers are in 47 miles? (note: 1 km = miles)
# km
= 47 mi
Now we have to choose a conversion factor

18. Pick the one that will allow you to cancel out miles
Factor label example
Q - How many kilometers are in 47 miles? (note: 1 km = miles)
# km
= 47 mi
1 km
0.621 mi
0.621 mi
1 km
Pick the one that will allow you to cancel out miles

19. Multiply given quantity by chosen conversion factor
Factor label example
Q - How many kilometers are in 47 miles? (note: 1 km = miles)
# km
= 47 mi
1 km
0.621 mi
0.621 mi
1 km
Multiply given quantity by chosen conversion factor

20. Cross out common factors
Factor label example
Q - How many kilometers are in 47 miles? (note: 1 km = miles)
x 1 km
0.621 mi
# km
= 47 mi
Cross out common factors

21. Cross out common factors
Factor label example
Q - How many kilometers are in 47 miles? (note: 1 km = miles)
x 1 km
0.621
# km
= 47
Cross out common factors

22. Are the units now correct?
Factor label example
Q - How many kilometers are in 47 miles? (note: 1 km = miles)
x 1 km
0.621
# km
= 47
Are the units now correct?

23. Yes. Both sides have km as units.
Factor label example
Q - How many kilometers are in 47 miles? (note: 1 km = miles)
x 1 km
0.621
# km
= 47
Yes. Both sides have km as units.

24. Yes. Both sides have km as units.
Factor label example
Q - How many kilometers are in 47 miles? (note: 1 km = miles)
x 1 km
0.621
# km
= 47
Yes. Both sides have km as units.

25. Factor label example Now finish the math.
Q - How many kilometers are in 47 miles? (note: 1 km = miles)
x 1 km
0.621
= 75.7 km
# km
= 47
Now finish the math.

26. The final answer is 76 km (correct sig fig)
Factor label example
Q - How many kilometers are in 47 miles? (note: 1 km = miles)
x 1 km
0.621
= 75.7 km
# km
= 47
The final answer is 76 km (correct sig fig)

27. Summary The previous problem was not that hard.
In other words, you probably could have done it faster using a different method.
However, for harder problems the factor label method is easiest.

28. Let’s answer the beginning questions
The fastest human is reported to be able to run at a rate of 27 mph, while the fastest fish can swim at a rate of 31 m/s.
Which one is faster?
Both must be in the same units, so we must convert one.
Does it matter which one?
NO.

29. First write down the desired quantity
Factor label example
Question: 27 mph is equal to how many m/s? factors needed: 1 mi = km
1 hr = 60 min
1000 m = 1 km
1 min = 60 sec
# m/s
First write down the desired quantity

30. Next, equate desired quantity to the given quantity
Factor label example
Q – 27 mph is equal to how many m/s? factors needed: 1 mi = km
1 hr = 60 min
1000 m = 1 km
1 min = 60 sec
# m/s = 27 mi/hr
Next, equate desired quantity to the given quantity

31. Now we have to choose conversion factors
Factor label example
Q – 27 mph is equal to how many m/s? factors needed: 1 mi = km
1 hr = 60 min
1000 m = 1 km
1 min = 60 sec
# m = 27 mi x s
1 hr
Now we have to choose conversion factors

32. Pick the one that will allow you to cancel out miles
Factor label example
Q – 27 mph is equal to how many m/s? factors needed: 1 mi = km
1 hr = 60 min
1000 m = 1 km
1 min = 60 sec
1.609 km
1 mi
1 mi
1.609 km
# m = 27 mi
1 hr s
Pick the one that will allow you to cancel out miles

33. Multiply given quantity by chosen conversion factor
Factor label example
Q – 27 mph is equal to how many m/s? factors needed: 1 mi = km
1 hr = 60 min
1000 m = 1 km
1 min = 60 sec
# m = 27 mi
1 hr s
1.609 km
1 mi X
Multiply given quantity by chosen conversion factor

34. Cross out common factors
Factor label example
Q – 27 mph is equal to how many m/s? factors needed: 1 mi = km
1 hr = 60 min
1000 m = 1 km
1 min = 60 sec
# m = 27 mi
1 hr s
1.609 km
1 mi X
Cross out common factors

35. NO, both sides aren’t equal
Factor label example
Q – 27 mph is equal to how many m/s? factors needed: 1 mi = km
1 hr = 60 min
1000 m = 1 km
1 min = 60 sec
# m =
1 hr s
1.609 km 1 X
NO, both sides aren’t equal
Are the units now correct?

36. Must choose another factor
Factor label example
Q – 27 mph is equal to how many m/s? factors needed: 1 mi = km
1 hr = 60 min
1000 m = 1 km
1 min = 60 sec
# m =
1 hr s
1.609 km 1
1000 m
1 km X X
Cross out common factors
Must choose another factor

37. NO, must choose another factor
Factor label example
Q – 27 mph is equal to how many m/s? factors needed: 1 mi = km
1 hr = 60 min
1000 m = 1 km
1 min = 60 sec
# m =
1 hr s
1.609 1
1000 m 1 X X
NO, must choose another factor
Do units match?

38. Cross out common factors
Factor label example
Q – 27 mph is equal to how many m/s? factors needed: 1 mi = km
1 hr = 60 min
1000 m = 1 km
1 min = 60 sec
# m =
1 hr s
1.609 1
1000 m 1
1 hr
60 min X X X
Do units match
Cross out common factors

39. NO – must choose another factor NO, must choose another factor
Factor label example
Q – 27 mph is equal to how many m/s? factors needed: 1 mi = km
1 hr = 60 min
1000 m = 1 km
1 min = 60 sec
# m = 1 s
1.609 1
1000 m 1 1
60 min X X X
NO – must choose another factor
Cross out common factors
NO, must choose another factor
Do units match?

40. NO – must choose another factor NO, must choose another factor
Factor label example
Q – 27 mph is equal to how many m/s? factors needed: 1 mi = km
1 hr = 60 min
1000 m = 1 km
1 min = 60 sec
1.609 1
# m = s X
1000 m
60 min
1 min
60 s X
Cross out common factors
NO – must choose another factor
Do units match?
Cross out common factors
NO, must choose another factor

41. NO – must choose another factor NO, must choose another factor
Factor label example
Q – 27 mph is equal to how many m/s? factors needed: 1 mi = km
1 hr = 60 min
1000 m = 1 km
1 min = 60 sec
1.609 1
# m = s X
1000 m 60 1
60 s X
Do units match?
NO – must choose another factor
Do units match?
Cross out common factors
NO, must choose another factor

42. NO, must choose another factor NO – must choose another factor
Factor label example
Q – 27 mph is equal to how many m/s? factors needed: 1 mi = km
1 hr = 60 min
1000 m = 1 km
1 min = 60 sec
1.609 1
# m = s X
1000 m 60 1
60 s X
Do units match?
YES !
Cross out common factors
Do units match?
NO, must choose another factor
NO – must choose another factor

43. Factor label example = 12.0675 m/s Do the math
Q – 27 mph is equal to how many m/s? factors needed: 1 mi = km
1 hr = 60 min
1000 m = 1 km
1 min = 60 sec
1.609 1
# m = s X
1000 m 60 1
60 s X
= m/s
Do the math
= 12 m/s (correct sig fig)

44. The fastest human is reported to be able to run at a rate of 27 mph, while the fastest fish can swim at a rate of 31 m/s. Which one is faster? How much faster?
Human: 27 mph = 12 m/s
Fish: 31 m/s
Which one is fastest?
How much faster?
31m/s – 12 m/s = 19 m/s

45. Working with metric/SI
quantity base unit
length meter
mass gram
volume liter

46. SI Base Units Base Quantity Name Symbol Length meter m Mass kilogramkg
Time
seconds s
Electric current
ampere A
Thermodynamic temperature
Kelvin K
Amount of substance
mole
mol
Luminous intensity
candela cd

47. Working with metric/SI
Base Unit
gram
meter
liter m

48. Working with metric/SI
When converting within the metric system it is helpful to remember:
“1 always goes with the prefix”
the value of the prefix goes with base unit

49. Conversions with metric/SI
Example: How many meters are in 12 km?
# m = 12 km
1 km
x m m m
= 1.2 x 104 m
Base Unit
gram
meter
liter
Use chart to get conversion

50. Conversions with metric/SI
Example: How many cm are in 1.3 m?
# cm = 1.3 m x
1 cm
0.01 m
Base Unit
gram
meter
liter
Use chart to get conversion

51. Conversions with metric/SI
Example: How many cm are in 1.3 m?
# cm = 1.3 m x
1 cm
0.01 m
= 1.3 x 102 cm

52. Conversions with metric/SI
When given a problem with 2 “prefixes” always go from prefix to base/base to prefix.
Example: converting cm to km – convert cm (prefix) to meters (base), then meters (base) to km (prefix).
These are often referred to as “2 step problems”

53. Conversions with metric/SI
Example: How many km are in 2.7 x 104mm?
This would be considered a “2-step problem”.
There are 2 prefixes – km to mm

54. Conversions with metric/SI
Example: How many km are in 2.7 x 104mm?
# km = 2.7 x 104mm
1 mm
x m
Base Unit
gram
meter
liter
Use chart to get conversion

55. Conversions with metric/SI
Example: How many km are in 2.7 x 104mm?
# km = 2.7 x 104mm
Use chart to get conversion
1 mm
x m
x 1 km
1000 m
Base Unit
gram
meter
liter

56. Conversions with metric/SI
Example: How many km are in 2.7 x 104mm?
# km = 2.7 x 104mm
1 mm
x m
x 1 km
1000 m =
0.027 km

57. Take time now to work on the practice problems
Ask questions if you need help!!