Area with fraction division example

We're told a yoga mat is three-fifths of a meter wide and has an area of one and two twenty-fifths square meters. What is the length of the mat? Well, we know that length times width is going to give you area.

Or another way of thinking about it is if the product of two numbers gives you a third number, if you take that third number and divide it by one of these, you're going to get the other one. So another way of thinking about it is length would be the same thing as area divided by width.

So we're trying to figure out the length here. We have the area, we have the width, so our length is going to be one and two twenty-fifths divided by three-fifths. Now this is going to be the same thing as... let me write this as an improper fraction. It's going to be easier to do some arithmetic with it.

So one is the same thing as twenty-five twenty-fifths plus two twenty-fifths. This is twenty-seven twenty-fifths divided by three-fifths. We've already talked about how this is saying how many three-fifths can fit into twenty-seven twenty-fifths, and we've given the intuition why this is the same thing as just multiplying twenty-seven twenty-fifths times the reciprocal of three-fifths, which is five-thirds.

So this is going to be equal to... and actually, I'm going to factor this out a little bit to simplify things a bit. Twenty-seven is three times three times three, and twenty-five is five times five. So this is going to be equal to... in our numerator, we're going to have three times three times three times five, and then in our denominator, we're going to have five times five times three.

Then we can reduce this a little bit. We can divide both the numerator and the denominator by five; we can divide both the numerator and the denominator by three. So in the numerator, we're gonna have three times three, which is nine. Thus, we find that nine-fifths is going to be equal to... so the yoga mat is three-fifths of a meter wide and nine-fifths of a meter long.

Now let's make sure that this makes sense. So I'm gonna make a grid. This right over here is one-fifth of a meter in that dimension and one-fifth of a meter in that dimension.

And then we can see, well, if this is one-fifth of a meter, then the width right over here is three-fifths of a meter, and our length right over here we have one, two, three, four, five, six, seven, eight, nine fifths. It is nine-fifths.

Now each of these units, what is its area? Well, it is one twenty-fifth meter squared. And how many of these do we have? Well, we can see we have three rows of nine, which is twenty-seven of these twenty-fifths. So we're gonna have twenty-seven twenty-fifths square meters, which is the same thing as one and two-fifths square meters.