Understanding Fractions, Improper Fractions, and Mixed Numbers

We've learned about division, and any time we perform division, there must be a dividend, a divisor, and a quotient.

At first, we looked at examples where the dividend is larger than the divisor.

This can sometimes result in a quotient that is a whole number, like fifteen divided by five, which gives us three.

But sometimes the divisor will be larger than the dividend, in which case we will get a fraction that has a value somewhere between zero and one.

It's a piece of one, or a fraction of one, hence the name fraction.

To see how these work, let's look at a pizza.

You'll typically see a pizza cut up into eight equal slices.

To represent this, we can take the number one to represent the pizza and divide by the number eight to show that it has been divided into eight sections.

That means that each slice is one-eighth of the pizza.

If you eat two slices, you have eaten two-eighths.

You can have three or four all the way up to eight-eighths if you are very hungry, in which case you have eaten the entire pizza.

Let's say the pizza was instead cut into fourths.

Then each slice would be one-fourth of the pizza.

We can see that eating one-fourth of the pizza is the same as eating two-eighths, and that's because these fractions are equivalent.

Two divided by eight is the same as one divided by four, and two-eighths can therefore reduce to one-fourth.

When we have a fraction like this, we can manipulate it any way we want by simply doing the same thing to both the top, or numerator, and the bottom, or denominator.

If we divide both the two and the eight by two, we get one-fourth, which again means that these fractions must be equivalent.

We've learned about division, and any time we perform division, there must be a dividend, a divisor, and a quotient.

At first, we looked at examples where the dividend is larger than the divisor.

This can sometimes result in a quotient that is a whole number, like fifteen divided by five, which gives us three.

But sometimes the divisor will be larger than the dividend, in which case we will get a fraction that has a value somewhere between zero and one.

It's a piece of one, or a fraction of one, hence the name fraction.

To see how these work, let's look at a pizza.

You'll typically see a pizza cut up into eight equal slices.

To represent this, we can take the number one to represent the pizza and divide by the number eight to show that it has been divided into eight sections.

That means that each slice is one-eighth of the pizza.

If you eat two slices, you have eaten two-eighths.

You can have three or four all the way up to eight-eighths if you are very hungry, in which case you have eaten the entire pizza.

Let's say the pizza was instead cut into fourths.

Then each slice would be one-fourth of the pizza.

We can see that eating one-fourth of the pizza is the same as eating two-eighths, and that's because these fractions are equivalent.

Two divided by eight is the same as one divided by four, and two-eighths can therefore reduce to one-fourth.

When we have a fraction like this, we can manipulate it any way we want by simply doing the same thing to both the top, or numerator, and the bottom, or denominator.

If we divide both the two and the eight by two, we get one-fourth, which again means that these fractions must be equivalent.