1. AKA How to make math ratios easy!
Dimensional analysis
AKA
How to make math ratios easy!

2. What is Dimensional Analysis?
Dimensional analysis is by far the most useful math trick you’ll ever learn. Maybe you have learned some algebra, but will you use it? Probably not after the test is done.
For a fraction of the effort needed to learn algebra, you can learn “dimensional analysis.” First, let’s get rid of the big words. We are just talking about converting one thing to another. This is real life – seriously useful stuff!

3. What is Dimensional Analysis?
This trick is about real-life math and physics – measurable stuff, stuff you can count.
Anything you measure will have a number with some sort of “unit of measure” attached. We could also call this unit of measure, a “dimension.” A unit could be seconds, miles, grams, degrees, meters per second, peas per pod, or pizza slices per person.

4. So how do I do this trick?
Example 1 : How many seconds are in one day?
(write this word question down in your notes)
First, don’t panic! Just break the problem down into small steps…
Step 1 Ask yourself, “What units of measure do I want to have in the answer and what units of measure have I been given?
If you can rephrase what you want to know using the word “per”, that means divided by in math. So for this problem, the answer we want is in the units:
sec ………………………………………………………….. sec
day

5. So how do I do this trick?
Example 1 : How many seconds are in one day?
Step 2 Ask yourself, “what do I know about these units”
What do you know about how seconds or days relate to other units of time?
You know that there are 60 seconds in 1 minute. You also know that in 1 minute there are 60 seconds. These are two ways of saying the same thing. You know that there are 24 hours in a day, or that 1 day contains 24 hours.
60 sec min 60 min 1 hr 1 day hr
1 min sec hr 60 min hr 1 day

6. So how do I do this trick?
Example 1 : How many seconds are in one day?
Step 3 Ask yourself, “Which factors will get me across the bridge?”
For every unit on top, we need a matching unit on bottom, until we can arrive at the end of the bridge.
60 sec min 60 min 1 hr 1 day hr
1 min sec hr 60 min hr 1 day
All of these “conversion factors” are equivalent. All we need to do now is pick from the ones that will get us across the bridge to where we want to end up.

7. So how do I do this trick?
Example 1 : How many seconds are in one day?
Step 4 Draw your bridge with pillars and cancel out
matching units on top and bottom
60 sec min hr sec
1 min hr day day
All of these “conversion factors” are equivalent. All we need to do now is pick from the ones that will get us across the bridge to where we want to end up.

8. So how do I do this trick?
Example 1 : How many seconds are in one day?
Step 5 Multiply
60 sec = x60x24 = 86,400 sec
day x 1 x day

9. Summarize steps: Step 1 Write units you want at the end
Step 2 Write things you know about these units
Step 3 Pick the equivalent factors to cross the bridge
Step 4 Cross out top and bottom matches
Step 5 Multiply

10. Example 2 How many inches are in the length of a football field?
Step 1
Step 2
Step 3
Step 4
Step 5

11. Example 2 How many inches are in the length of a football field?
Step 1 end with in field
Step 2
Step 3
Step 4
Step 5

12. Example 2 How many inches are in the length of a football field?
Step 1 end with in field
Step in ft yds ft 1.6 miles what do I know? ft yd field mile 1 km
Step 3
Step 4
Step 5

13. Example 2 How many inches are in the length of a football field?
Step 1 end with in field
Step in ft yds ft 1.6 miles what do I know? ft yd field mile 1 km
Step in ft yds pick equivalent factors to cross bridge ft yd field
Step 4
Step 5

14. Example 2 How many inches are in the length of a football field?
Step 1 end with in field
Step in ft yds ft yd field
Step in ft yds pick equivalent factors to cross bridge ft yd field
Step in ft yds cross out matching units
1 ft yd field
Step 5 12 x 3 x 100 = in multiply
1 x 1 x field

15. Example 3 If I measure 38.5 grams, how many milligrams do I have?
Step 1
Step 2
Step 3
Step 4
Step 5

16. Example 3 If I measure 38.5 grams, how many milligrams do I have?
Step 1 end with mg
Step 2
Step 3
Step 4
Step 5

17. Example 3 If I measure 38.5 grams, how many milligrams do I have?
Step 1 end with mg
Step mg cg g what do I know? g g kg
Step 3
Step 4
Step 5

18. Example 3 If I measure 38.5 grams, how many milligrams do I have?
Step 1 end with mg
Step mg cg g what do I know? g g kg
Step mg pick equivalent factors to cross bridge
1 g
Step 4
Step 5

19. Example 3 If I measure 38.5 grams, how many milligrams do I have?
Step 1 end with mg
Step mg cg g what do I know? g g kg
Step g 1,000 mg pick equivalent factors to cross bridge
1 g
Step g 1,000 mg eliminate top and bottom matching units
Step x 1000 = 385,000 mg

20. Example 4
If my car has a max speed of 125 miles per hour, how fast can it travel in meters per second?
(hint: there are 0.62 kilometers in 1 mile)
Step 1
Step 2
Step 3
Step 4
Step 5

21. Example 4 Did you calculate 21.53 meters per second?
If my car has a max speed of 125 miles per hour, how fast can it travel in meters per second?
(hint: there are 0.62 kilometers in 1 mile)
Did you calculate meters per second?

22. Example 5
If there are 5 peas per pod, and I can pick 32 pods per minute, how many peas will I have in 2 hours?
Step 1
Step 2
Step 3
Step 4
Step 5

23. Example 5 Did you calculate 19,200 peas?
If there are 5 peas per pod, and I can pick 32 pods per minute, how many peas will I have in 2 hours?
Did you calculate 19,200 peas?