Dimensional analysis is a method of solving problems using units to guide the process.
Dimensional analysis is a method of solving problems using units to guide the process.

In dimensional analysis we can use conversion factors, which relate two equivalent quantities with different units.
In dimensional analysis we can use conversion factors, which relate two equivalent quantities with different units.

In this example we're going to use dimensional analysis to convert 15.5 feet to meters.
In this example we're going to use dimensional analysis to convert 15.5 feet to meters.

We'll start by mapping out how we want to get from feet to meters using the conversion factors were given.
We'll start by mapping out how we want to get from feet to meters using the conversion factors were given.

Starting with feet, we can convert to inches using 1 foot equals 12 inches.
Starting with feet, we can convert to inches using 1 foot equals 12 inches.

Then we can convert from inches to centimeters using 1 inch equals 2.54 centimeters.
Then we can convert from inches to centimeters using 1 inch equals 2.54 centimeters.

Then we'll convert from centimeters to meters using 1 meter equals 100 centimeters.
Then we'll convert from centimeters to meters using 1 meter equals 100 centimeters.

Let's try it. We're starting with 15.5 feet.
Let's try it. We're starting with 15.5 feet.

If we want to cancel out our units of feet, we need to multiply by a conversion factor that has feet in the denominator.
If we want to cancel out our units of feet, we need to multiply by a conversion factor that has feet in the denominator.

Since 1 foot equals 12 inches, multiplying by 12 inches divided by 1 foot is like multiplying by the quantity 1.
Since 1 foot equals 12 inches, multiplying by 12 inches divided by 1 foot is like multiplying by the quantity 1.

It doesn't change our length of 15.5 feet; it only changes the units from feet to inches.
It doesn't change our length of 15.5 feet; it only changes the units from feet to inches.

Here our units a feet cancel and we're left with just inches.
Here our units a feet cancel and we're left with just inches.

Next we want to cancel out our units of inches.
Next we want to cancel out our units of inches.

Inches are in the numerator so we need to multiply by a factor that has inches in the denominator.
Inches are in the numerator so we need to multiply by a factor that has inches in the denominator.

So we multiply by 2.54 centimeters divided by 1 inch.
So we multiply by 2.54 centimeters divided by 1 inch.

Now inches cancel and we're left with centimeters.
Now inches cancel and we're left with centimeters.

To cancel out centimeters we can multiply by 1 meter divided by 100 centimeters and we'll be left with meters, which is the unit we want.
To cancel out centimeters we can multiply by 1 meter divided by 100 centimeters and we'll be left with meters, which is the unit we want.

If you wanted to put this into your calculator to get the answer,
If you wanted to put this into your calculator to get the answer,

you would multiply everything on top (in the numerators) and divide by everything in the bottom (in the denominators).
you would multiply everything on top (in the numerators) and divide by everything in the bottom (in the denominators).

So it would be 15.5 times 12 times 2.54 times 1
So it would be 15.5 times 12 times 2.54 times 1

divided by the quantity--here you would use parentheses--1 times 1 times 100.
divided by the quantity--here you would use parentheses--1 times 1 times 100.

If you want, since multiplying or dividing by one doesn't change the number, you could leave them out of your calculation.
If you want, since multiplying or dividing by one doesn't change the number, you could leave them out of your calculation.

You should get the number 4.72, so 15.5 feet is 4.72 meters.
You should get the number 4.72, so 15.5 feet is 4.72 meters.

Now let's try a word problem.
Now let's try a word problem.

To solve a word problem with dimensional analysis
To solve a word problem with dimensional analysis

we're going to start by reading through the problem and determining if each piece of information we're given
we're going to start by reading through the problem and determining if each piece of information we're given

is a conversion factor or standalone piece of information.
is a conversion factor or standalone piece of information.

Let's get started. Your kitten is very sick and your vet prescribed medicine.
Let's get started. Your kitten is very sick and your vet prescribed medicine.

You need to give your kitten 550 milligrams of medicine every 8 hours for the next 3 weeks.
You need to give your kitten 550 milligrams of medicine every 8 hours for the next 3 weeks.

From this sentence we know that 550 milligrams of medicine equals 8 hours; this is a conversion factor.
From this sentence we know that 550 milligrams of medicine equals 8 hours; this is a conversion factor.

We're also told "3 weeks," but this value isn't equated to any other value, so we'll write it down as a standalone piece of information.
We're also told "3 weeks," but this value isn't equated to any other value, so we'll write it down as a standalone piece of information.

The medicine is only sold in 225 milligram tablets.
The medicine is only sold in 225 milligram tablets.

So 225 milligrams equals 1 tablet; this is another conversion factor.
So 225 milligrams equals 1 tablet; this is another conversion factor.

A bottle of 18 tablets costs $10.95 including tax.
A bottle of 18 tablets costs $10.95 including tax.

So 1 bottle equals 18 tablets which equals $10.95.
So 1 bottle equals 18 tablets which equals $10.95.

How many bottles of medicine should you buy to save your kitten from the grip of death--no pressure--and how much will it cost you?
How many bottles of medicine should you buy to save your kitten from the grip of death--no pressure--and how much will it cost you?

Next let's come up with a plan for our unit conversion.
Next let's come up with a plan for our unit conversion.

Our only standalone piece of information in this problem is 3 weeks.
Our only standalone piece of information in this problem is 3 weeks.

In general it can be helpful to start your dimensional analysis with standalone information,
In general it can be helpful to start your dimensional analysis with standalone information,

so we'll start with 3 weeks and convert it eventually into bottles and then dollars.
so we'll start with 3 weeks and convert it eventually into bottles and then dollars.

None of our conversion factors have weeks in them but one does have hours,
None of our conversion factors have weeks in them but one does have hours,

So we can convert our weeks into days and then days into hours.
So we can convert our weeks into days and then days into hours.

Then we can convert from hours to milligrams of medicine using 550 milligrams equals 8 hours.
Then we can convert from hours to milligrams of medicine using 550 milligrams equals 8 hours.

Our other conversion factor with milligrams will let us convert from milligrams to tablets.
Our other conversion factor with milligrams will let us convert from milligrams to tablets.

And our less conversion factor will allow us to convert from tablets to bottles and bottles to dollars.
And our less conversion factor will allow us to convert from tablets to bottles and bottles to dollars.

Next let's add numbers.
Next let's add numbers.

We'll start with 3 weeks.
We'll start with 3 weeks.

Since weeks are on top we'll multiply by 7 days divided by 1 week.
Since weeks are on top we'll multiply by 7 days divided by 1 week.

Putting weeks in the denominator of our conversion factor allows weeks to cancel and we're left with days.
Putting weeks in the denominator of our conversion factor allows weeks to cancel and we're left with days.

Then we can convert days to hours by multiplying by 24 hours divided by 1 day.
Then we can convert days to hours by multiplying by 24 hours divided by 1 day.

Now that were in hours, we can multiply by 550 milligrams of medicine per 8 hours and we'll be left with milligrams.
Now that were in hours, we can multiply by 550 milligrams of medicine per 8 hours and we'll be left with milligrams.

Next we'll multiply by 1 tablet per 225 milligrams of medicine
Next we'll multiply by 1 tablet per 225 milligrams of medicine

and finally 1 bottle divided by 18 tablets.
and finally 1 bottle divided by 18 tablets.

If you put this in your calculator you would enter it as 3 times 7 times 24 times 550
If you put this in your calculator you would enter it as 3 times 7 times 24 times 550

divided by parentheses 8 times 225 times 18, end parentheses.
divided by parentheses 8 times 225 times 18, end parentheses.

You should get 8.56 bottles.
You should get 8.56 bottles.

Since it's not likely that we can buy 0.56 bottles, we'll round up to 9 bottles.
Since it's not likely that we can buy 0.56 bottles, we'll round up to 9 bottles.

To determine the cost of 9 bottles, we'll multiply by $10.95 per bottle.
To determine the cost of 9 bottles, we'll multiply by $10.95 per bottle.

So it should cost us $98.55 to save our kitten from the grip of death.
So it should cost us $98.55 to save our kitten from the grip of death.