Fractions

Numerator and Denominator

A fraction has two numbers organized vertically with a line between them. The upper number is called the numerator and the lower number the denominator.

Meaning of the Denominator

The denominator of a fraction refers to the number of equal parts that a whole has been divided into.

For example, if we have a pie and we divide it into two pieces, our associated fraction would contain 2 in the denominator.

If we divide our pie into three pieces, our associated fraction would contain 3 in the denominator.

If we divide our pie into four pieces, our associated fraction would contain 4 in the denominator.

Meaning of the Numerator

We have learned that the denominator of a fraction refers to the number of pieces that a whole quantity has been divided into. The numerator of a fraction refers to the number of these pieces that we have.

For example, if we divide a pie into four pieces, our associated fraction would contain 4 in the denominator to indicate that we have divided the pie into four pieces. If we have three of these pieces, our numerator would contain 3.

If we divide a pie into three pieces, our associated fraction would contain 3 in the denominator to indicate that we have divided the pie into three pieces. If we have two of these pieces, our numerator would contain 2.

Click below to practice visualizing the relationship between numerators, denominators and the associated fraction.

Click below to practice visually obtaining the correct fraction of a whole.

Equivalent Fractions

Given a pie, we can use the meanings of the numerator and the denominator to obtain the quantity associated with the following fractions.

We can see that these fractions refer to the same quantity.

Given a pie, we can use the meanings of the numerator and the denominator to obtain the quantity associated with the following fractions.

Once again, we can see that these fractions refer to the same quantity.

As refers to two wholes or 2, it is worth noting that whole numbers have fractional forms.

Fractions in Simplest Form

We have seen that there are numerous ways of expressing the same quantity using fractions. Given all the possible representations for a quantity, if we select the fraction where the the numerator and denominator have the smallest possible values, the fraction is said to be in simplest form.

For example: Given

We can express this quantity with the following fractions.

Thus, in simplest form would be .

Practice:

Summing Fractions

There is no problem summing pieces of the same size. That is to say two fractions with the same denominator.
E.g.

E.g.

If pieces are of different sizes, however, they are more difficult to add.

For example,

We can see the quantity that reflects this sum very easily however expressing the sum is more difficult.

If we remember that fractions can be expressed in multiple forms, however, we can get around this problem.

Solution for :

Expand both fractions using the possible denominators

Select the smallest denominator common to both these expansions and express the fractions in terms of this "least common denominator''.

,

Now that the two denominators are the same, the two fractions can be summed

Practice:

Subtracting Fractions

There is no problem subtracting pieces of the same size. That is to say two fractios with the same denominator. E.g.

E.g.

If pieces are of different sizes, however, they are more difficult to subtract. For example

We can see the quantity that reflects the difference very easily however expressing it is more difficult.

Once again, if we remember that fractions can be expressed in multiple form, we can get around this problem.

Solution for :

Expand both fractions using the possible denominators

Select the smallest denominator common to both these expansions and express the fractions in terms of this "least common denominator''.

,

Now that the two denominators are the same, the two fractions can be subtracted.

Multiplying Fractions

If we multiply a quantity by a fraction , indicates that we are going to divide the quantity into parts and then multiply it by the numerator .

For example: means we first take one third of one half.

The result of this is . We then multiply the result by 2.

Hence .

For example: means we first take one half of two thirds.

The result of this is one third which we then multiply by 3.

Hence .

Observing these examples, we find that a convenient approach to this can be found by the expression .

Dividing Fractions

In the preceding section we saw that multiplying by is the same as dividing by 2. Also, that multiplying by is the same as dividing by 3. In general multiplying by is the same as dividing by . Hence, if we divide a quantity by a fraction , we need only interchange the and and multiply.

More simply stated, .

For example: .

Practice: