Interpreting units in formulas: novel units | Mathematics I | High School Math | Khan Academy

So, we're told to consider the formula ( Y ) is equal to ( \frac{2C}{P} ) where ( y ) represents the carrot yield.

So, ( y ) represents the carrot yield, ( c ) represents the number of expected carrots, and ( P ) represents the number of plants.

So, ( P ) represents a number of plants. Select an appropriate measurement unit for carrot yield.

So, the key here is to realize that we can treat units like we would treat algebraic variables.

And so, for example, over here, if we're saying our yield is going to be two times the number of expected carrots divided by the number of plants, well, our units are going to be—well, actually, we could put some numbers in here actually just to make something more interesting.

Let's just say that ( c ) is equal to 10 carrots, and I'm just making these numbers up, just hopefully it makes sense what I'm about to do.

And let's say that ( P ) is equal to, I don't know, 30 plants. ( P ) is equal to 30 plants.

And so, using this formula, ( y ) would be equal to ( \frac{2 \times 10 \text{ carrots}}{30 \text{ plants}} ).

So, ( 2 \times 10 ) carrots, that's going to be 20 carrots.

So, we can write the numerator as this is 20 carrots divided by 30 plants.

And so that is going to be equal to—you could divide the numbers, so it would be ( \frac{20}{30} ), which would be ( \frac{2}{3} ).

( \frac{2}{3} ) and then the units would be carrots per plant: ( \frac{2}{3} ) carrots per plant, carrots per plant, carrots per plant.

And once again, the whole point of what I just did is to see what the units would be for our carrot yield, and we can see the units would be carrots per plant.

And I put the numbers there just so hopefully it makes a little bit of intuitive sense of what we just did, that we're algebraically manipulating or mathematically manipulating the numbers; we do the same thing with the units.

And so when we try to find the appropriate measurement unit for carrot yield, which I had never heard of before this video, we see it's carrots per plant, not carrots per plant squared.

That would have been the case if we were squaring this ( P ) over here, or plants times the square root of carrots—well, that would be more so if we were taking the square root up here and we weren't dividing by ( P ), but we were multiplying by ( P ).

But the general theme that you could see is this constant, this two, didn't affect what happens to the units, but what did matter is how these variables relate to each other.

We're taking the variable ( C ), dividing it by ( P ), so whatever the units were for ( C ), we divide those units by the units for ( P ), and we get the units for yield: carrots per plant.