Probabilities are often stated as fractions. Sometimes we are interested in probabilities involving several events. The methods used to calculate these probabilities will require addition, subtraction, multiplication and division. This section will remind you how to do these operations on fractions. As you work through the rest of the course, you can return this section as needed for a quick reminder of operations on fractions.

Add Fractions

When you need to add or subtract fractions, you will need to first make sure that the fractions have the same denominator. The denominator tells you how many pieces the whole has been broken into, and the numerator tells you how many of those pieces you are using.

Suppose a pizza has been cut into 4 equal slices, and one slice of a pizza has been eaten leaving 3 slices. The fraction [latex]\frac{3}{4}[/latex] represents how much of a whole pizza we have left.

Suppose you have another pizza that had been cut into 8 equal parts and 3 of those parts were gone, leaving [latex]\frac{5}{8}[/latex]. To describe the total amount of pizza remaining as a single fraction, we need a common denominator. The lowest common denominator is the least common multiple (LCM) of the two original denominators. Finding the least common multiple of a list of numbers using prime factorizations was described in a previous section. See the example below for a demonstration of our pizza problem.

To add fractions with unlike denominators, first rewrite them with like denominators. Then add or subtract the numerators over the common denominator.

In the next example you are shown how to add two fractions with different denominators, then simplify the answer.

While any common multiple of the two denominators can be used, the least common multiple is the easiest.

In the following video you will see an example of how to add two fractions with different denominators.

Subtract Fractions

Subtracting fractions follows the same technique as adding them. First, determine whether or not the denominators are alike. If not, rewrite each fraction as an equivalent fraction, all having the same denominator. Below are some examples of subtracting fractions whose denominators are not alike.

We can perform additions and subtractions on three or more fractions, as seen in the example below.

In the following video you will see an example of how to subtract fractions with unlike denominators.

Multiply Fractions

When you multiply a fraction by a fraction, you are finding a “fraction of a fraction.” Suppose you have [latex]\Large\frac{3}{4}[/latex] of a candy bar and you want to find [latex]\Large\frac{1}{2}[/latex] of the [latex]\Large\frac{3}{4}[/latex]:

By dividing each fourth in half, you can divide the candy bar into eighths.

Then, choose half of those to get [latex]\Large\frac{3}{8}[/latex].

In both of the above cases, to find the answer, you can multiply the numerators together and the denominators together.

If a fraction has common factors in the numerator and denominator, we can reduce the fraction to its simplified form by removing the common factors.

You can simplify first, before you multiply two fractions, to make your work easier. This allows you to work with smaller numbers when you multiply.

In the following video you will see an example of how to multiply two fractions, then simplify the answer.

Divide Fractions

There are times when you need to use division to solve a problem. For example, if painting one coat of paint on the walls of a room requires [latex]3[/latex] quarts of paint and you have a bucket that contains [latex]6[/latex] quarts of paint, how many coats of paint can you paint on the walls? You divide [latex]6[/latex] by [latex]3[/latex] for an answer of [latex]2[/latex] coats. There will also be times when you need to divide by a fraction. Suppose painting a closet with one coat only required [latex]\Large\frac{1}{2}[/latex] quart of paint. How many coats could be painted with the 6 quarts of paint? To find the answer, you need to divide [latex]2[/latex] by the fraction, [latex]\Large\frac{1}{2}[/latex].

Before we begin dividing fractions, let’s cover some important terminology.

• reciprocal: two fractions are reciprocals if their product is [latex]1[/latex] (Don’t worry; we will show you examples of what this means.)

Note that for [latex]a \neq 0[/latex] and [latex]b \neq 0[/latex], [latex]\frac{a}{b} \cdot \frac{b}{a} = \frac{a \cdot b}{b \cdot a} = \frac{a \cdot b}{a \cdot b} = 1[/latex]. The reciprocal of a fraction can be found by interchanging the numerator and denominator.

The reciprocal of [latex]\frac{a}{b}[/latex] is [latex]\frac{b}{a}[/latex].

Dividing fractions requires using the reciprocal of a number or fraction. Here are some examples of reciprocals:

Caution! Division by zero is undefined and so is the reciprocal of any fraction that has a zero in the numerator. For any real number a, [latex]\Large\frac{a}{0}[/latex] is undefined. Additionally, the reciprocal of [latex]\Large\frac{0}{a}[/latex] will always be undefined.

Look at the diagram of two pizzas below. How can you divide what is left (the red shaded region) among [latex]6[/latex] people fairly?

Each person gets one piece, so each person gets [latex]\Large\frac{1}{4}[/latex] of a pizza.

Dividing a fraction by a whole number is the same as multiplying by the reciprocal, so you can always use multiplication of fractions to solve division problems.

Divide a Fraction by a Fraction

Sometimes you need to solve a problem that requires dividing by a fraction. Suppose you have a pizza that is already cut into [latex]4[/latex] slices. How many [latex]\Large\frac{1}{2}[/latex] slices are there?

There are [latex]8[/latex] slices. You can see that dividing [latex]4[/latex] by [latex]\Large\frac{1}{2}[/latex] gives the same result as multiplying [latex]4[/latex] by [latex]2[/latex].

What would happen if you needed to divide each slice into thirds?

You would have [latex]12[/latex] slices, which is the same as multiplying [latex]4[/latex] by [latex]3[/latex].

Any easy way to remember how to divide fractions is the phrase “keep, change, flip.” This means to KEEP the first number, CHANGE the division sign to multiplication, and then FLIP (use the reciprocal) of the second number.

When solving a division problem by multiplying by the reciprocal, remember to write all whole numbers and mixed numbers as improper fractions. The final answer should be simplified and written as a mixed number.

In the following video you will see an example of how to divide an integer by a fraction, as well as an example of how to divide a fraction by another fraction.