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(male narrator) In this video,
we will look at a process called "dimensional analysis"
to convert a single unit to another single unit.
The way we make this conversion is we multiply by 1,
because this does not change the value.
However, this 1 will usually be a fraction.
We can make 1 a fraction
by putting 1 over 1, or even 2 over 2.
As long as the numerator and denominator are the same,
the value is always equal to 1, such as 3x over 3x.
With dimensional analysis, we will make different units
in both the numerator and denominator,
but they will still have the same value.
For example, 60 minutes over 1 hour.
They both have the same value, so it still is equal to 1.
Another example is 5,280 feet is the same as 1 mile.
This fraction is equal to 1.
Our goal will be to set up the right fraction
to clear the unit we don't want
and make it into the unit we do want.
We do this by asking three questions.
First, we ask, "Where does the units go?
Where does it go?"
Once we know that, we ask, "What does it become?"
Finally, we ask ourselves, "What is the relationship...
between those two units?"
Let's take a look at some examples
where we can see this conversion take place
asking these questions.
In this first example,
we're going to convert 5 feet into meters.
Let's write those 5 feet as a fraction
by putting them over 1.
We're gonna convert the 5 feet
by multiplying by another rational expression
or a conversion factor.
Going through the questions we were asking,
to get rid of the feet that are already in the numerator,
they must go in the denominator, so they can divide out.
That's where it goes, and it will become meters.
Then, we ask ourselves
what the relationship is between feet and meters.
Using a conversion chart, we can see 1 meter is 3.28 feet.
We can now simplify by dividing out the feet
which are in common, and multiplying straight across.
In the numerator, 5 times 1 meter is 5 meters.
In the denominator, 1 times 3.28 is just 3.28.
Finally, we can divide those values--5 by 3.28--
to get 1.52 meters.
This is the same as 5 feet.
Let's take a look at another example
where we make this conversion.
Here, we're going to convert 3 miles into yards.
We'll start by making the 3 miles into a fraction
by putting them over 1
and start multiplying by a conversion factor.
To get rid of the miles, which are already on top,
we need to put them on the bottom.
We want them to become yards.
However, we might not know the number of miles in a yard
or yards in a mile.
So it might be more useful to convert to feet first,
because we can quickly look up that 1 mile is 5,280 feet.
That will allow the miles to divide out,
but we still have feet that we need to get rid of,
which we can do with another conversion factor.
To get rid of the feet,
we must put them in the denominator.
And we want those feet to become yards,
and we know that 1 yard is equal to 3 feet.
Now, the feet will divide out,
and we can multiply straight across:
3 times 5,280; times 1 yard;
is 15,840 yards; over 1; times 1; times 3; is 3;
and when we divide, we get 5,280 yards.
This process of dimensional analysis
is setting up conversion factors,
where the numerator and denominator
have the same value,
and the units we don't want will always divide out.