We will look at addition and subtraction of fractions using pie slices.

## Visualizing fractions

In this lesson, we will use partially filled circles to represent fractions. For example, the circle to the left is divided into 5 slices, and three of those slices are filled in with green. So it represents the fraction \$\$3/5\$\$. (If they were filled in with pink, it would represent the fraction \$\$-3/5\$\$.)

What fractions are represented by each of the pie diagrams to the left? Do the problems in the left-hand table below first, then continue to the right-hand table. Do not reduce these answers to lowest terms.

If we want to represent a fraction that’s greater than 1 or less than \$-1\$, we’ll put several pies next to each other to get enough total slices. For instance, there are two pairs of pies to the left, each representing \$\$-7/4\$\$. In this course, we write improper fractions (like \$\$3/2\$\$), instead of “mixed” fractions (like \$\$1{1/2}\$\$).

Click the first button below, and then look at the top pie diagram to the left. Type the fraction that it represents into the table below. Your answer will be illustrated in the bottom pie diagram. Do this for each row of the tables below.

## Adding and subtracting fractions with matching denominators

When two fractions have the same denominator (bottom part), their pie slices are the same size. This means the fractions can be added or subtracted by adding or subtracting their numerators (top parts).

How many pie slices of the given size are shown?
What is the total fractional quantity represented by the pie slices shown?

## Equivalent fractions

Two fractions which represent the same number (cover the same amount of pie) are called equivalent. For example, \$\$1/2\$\$ and \$\$3/6\$\$ are equivalent fractions.

For each fraction in the table to the right, find the equivalent fraction with the given denominator.

Look at the first row of the table from fracEqQ. Notice that you have to multiply the first denominator (\$2\$) by \$3\$ in order to get the second denominator (\$6\$). Similarly, you have to multiply the first numerator (\$1\$) by \$3\$ in order to get the second numerator (\$3\$).

Now look at the second row of the table from fracEqQ.

 What number do you have to multiply the first denominator (\$3\$) by to get the second denominator (\$9\$)? In the same row, what do you have to multiply the first numerator (\$2\$) by to get the second numerator (\$6\$)?

Now look at the last row of the table from fracEqQ.

 What number do you have to divide the first denominator (\$12\$) by to get the second denominator (\$3\$)? In the same row, what do you have to divide the first numerator (\$20\$) by to get the second numerator (\$5\$)?

Notice that any time you multiply or divide both the numerator and the denominator of a fraction by the same number, you get an equivalent fraction.

## Lowest terms

Writing a fraction in lowest terms means finding the equivalent fraction with the smallest possible denominator.

Type in the denominator to rewrite each of these fractions in lowest terms. If there is an equivalent fraction to the given one that has that denominator, its numerator will be filled in automatically. (You must click or tap that row’s button first.)
Write each fraction in lowest terms. The fraction you type will be drawn on the bottom pie.

From now on, unless a question asks you to give a version of a fraction with a specific denominator, you should always write the fraction in lowest terms.

## Adding and subtracting fractions with different denominators

You already know how to add and subtract fractions with the same denominator. You can combine this with the idea of equivalence to add and subtract any fractions you want.

Each row of the table below has two fractions in it. Find fractions that are equivalent to each of these two fractions, and have the same denominator as each other.

For each of the arithmetic problems listed below, make it so the two fractions have the same denominator by replacing one or both of them with an equivalent fraction. Then solve the arithmetic problem.

You can add or subtract any two fractions by following this process:

1. Replace one or both fractions by another equivalent fraction, to make the two fractions have the same denominator.
2. Add or subtract the two new fractions.

For each of the arithmetic problems listed below, turn it into a problem where the fractions have the same denominator. Then perform the addition, and write the result in lowest terms.