Modeling and Finding Equivalent Fractions

Learning Outcomes

Model equivalent fractions
Find equivalent fractions

Let’s think about Andy and Bobby and their favorite food again. If Andy eats ${\Large\frac{1}{2}}$ of a pizza and Bobby eats ${\Large\frac{2}{4}}$ of the pizza, have they eaten the same amount of pizza? In other words, does ${\Large\frac{1}{2}}={\Large\frac{2}{4}}?$ We can use fraction tiles to find out whether Andy and Bobby have eaten equivalent parts of the pizza.

Equivalent Fractions

Equivalent fractions are fractions that have the same value.

Fraction tiles serve as a useful model of equivalent fractions. You may want to use fraction tiles to do the following activity. Or you might make a copy of the fraction tiles shown earlier and extend it to include eighths, tenths, and twelfths.

Start with a ${\Large\frac{1}{2}}$ tile. How many fourths equal one-half? How many of the ${\Large\frac{1}{4}}$ tiles exactly cover the ${\Large\frac{1}{2}}$ tile?

One long, undivided rectangle is shown. Below it is a rectangle divided vertically into two pieces, each labeled as one half. Below that is a rectangle divided vertically into four pieces, each labeled as one fourth.
Since two ${\Large\frac{1}{4}}$ tiles cover the ${\Large\frac{1}{2}}$ tile, we see that ${\Large\frac{2}{4}}$ is the same as ${\Large\frac{1}{2}}$ , or ${\Large\frac{2}{4}}={\Large\frac{1}{2}}$ .

How many of the ${\Large\frac{1}{6}}$ tiles cover the ${\Large\frac{1}{2}}$ tile?

One long, undivided rectangle is shown. Below it is a rectangle divided vertically into two pieces, each labeled as one half. Below that is a rectangle divided vertically into six pieces, each labeled as one sixth.
Since three ${\Large\frac{1}{6}}$ tiles cover the ${\Large\frac{1}{2}}$ tile, we see that ${\Large\frac{3}{6}}$ is the same as ${\Large\frac{1}{2}}$ .

So, ${\Large\frac{3}{6}}={\Large\frac{1}{2}}$ . The fractions are equivalent fractions.

Example

Use fraction tiles to find equivalent fractions. Show your result with a figure.

How many eighths ( ${\Large\frac{1}{8}}$ ) equal one-half ( ${\Large\frac{1}{2}}$ )?
How many tenths ( ${\Large\frac{1}{10}}$ ) equal one-half ( ${\Large\frac{1}{2}}$ )?
How many twelfths ( ${\Large\frac{1}{12}}$ ) equal one-half ( ${\Large\frac{1}{2}}$ )?

Solution
1. It takes four ${\Large\frac{1}{8}}$ tiles to exactly cover the ${\Large\frac{1}{2}}$ tile, so ${\Large\frac{4}{8}}={\Large\frac{1}{2}}$ .

One long, undivided rectangle is shown, labeled 1. Below it is an identical rectangle divided vertically into two pieces, each labeled 1 half. Below that is an identical rectangle divided vertically into eight pieces, each labeled 1 eighth.
2. It takes five ${\Large\frac{1}{10}}$ tiles to exactly cover the ${\Large\frac{1}{2}}$ tile, so ${\Large\frac{5}{10}}={\Large\frac{1}{2}}$ .

One long, undivided rectangle is shown. Below it is a rectangle divided vertically into two pieces, each labeled as one half. Below that is a rectangle divided vertically into ten pieces, each labeled as one tenth.
3. It takes six ${\Large\frac{1}{12}}$ tiles to exactly cover the ${\Large\frac{1}{2}}$ tile, so ${\Large\frac{6}{12}}={\Large\frac{1}{2}}$ .

One long, undivided rectangle is shown. Below it is a rectangle divided vertically into two pieces, each labeled as one half. Below that is a rectangle divided vertically into twelve pieces, each labeled as one twelfth.

Suppose you had tiles marked ${\Large\frac{1}{20}}$ . How many of them would it take to equal ${\Large\frac{1}{2}}$ ? Are you thinking ten tiles? If you are, you’re right, because ${\Large\frac{10}{20}}={\Large\frac{1}{2}}$ .

We have shown that ${\Large\frac{1}{2},\frac{2}{4},\frac{3}{6},\frac{4}{8},\frac{5}{10},\frac{6}{12}}$ , and ${\Large\frac{10}{20}}$ are all equivalent fractions.

Try it

Find Equivalent Fractions

We used fraction tiles to show that there are many fractions equivalent to ${\Large\frac{1}{2}}$ . For example, ${\Large\frac{2}{4},\frac{3}{6}}$ , and ${\Large\frac{4}{8}}$ are all equivalent to ${\Large\frac{1}{2}}$ . When we lined up the fraction tiles, it took four of the ${\Large\frac{1}{8}}$ tiles to make the same length as a ${\Large\frac{1}{2}}$ tile. This showed that ${\Large\frac{4}{8}}={\Large\frac{1}{2}}$ . See the previous example.

We can show this with pizzas, too. Image (a) shows a single pizza, cut into two equal pieces with ${\Large\frac{1}{2}}$ shaded. Image (b) shows a second pizza of the same size, cut into eight pieces with ${\Large\frac{4}{8}}$ shaded.

Two pizzas are shown. The pizza on the left is divided into 2 equal pieces. 1 piece is shaded. The pizza on the right is divided into 8 equal pieces. 4 pieces are shaded.
This is another way to show that ${\Large\frac{1}{2}}$ is equivalent to ${\Large\frac{4}{8}}$ .

How can we use mathematics to change ${\Large\frac{1}{2}}$ into ${\Large\frac{4}{8}}$ ? How could you take a pizza that is cut into two pieces and cut it into eight pieces? You could cut each of the two larger pieces into four smaller pieces! The whole pizza would then be cut into eight pieces instead of just two. Mathematically, what we’ve described could be written as:

${\Large\frac{1\cdot\color{blue}{4}}{2\cdot\color{blue}{4}}}={\Large\frac{4}{8}}$

These models lead to the Equivalent Fractions Property, which states that if we multiply the numerator and denominator of a fraction by the same number, the value of the fraction does not change.

Equivalent Fractions Property

If $a,b$ , and $c$ are numbers where $b\ne 0$ and $c\ne 0$ , then

${\Large\frac{a}{b}}={\Large\frac{a\cdot c}{b\cdot c}}$

When working with fractions, it is often necessary to express the same fraction in different forms. To find equivalent forms of a fraction, we can use the Equivalent Fractions Property. For example, consider the fraction one-half.

${\Large\frac{1\cdot\color{blue}{3}}{2\cdot\color{blue}{3}}}={\Large\frac{3}{6}}$ so ${\Large\frac{1}{2}}={\Large\frac{3}{6}}$

${\Large\frac{1\cdot\color{blue}{2}}{2\cdot\color{blue}{2}}}={\Large\frac{2}{4}}$ so ${\Large\frac{1}{2}}={\Large\frac{2}{4}}$

${\Large\frac{1\cdot\color{blue}{10}}{2\cdot\color{blue}{10}}}={\Large\frac{10}{20}}$ so ${\Large\frac{1}{2}}={\Large\frac{10}{20}}$

So, we say that ${\Large\frac{1}{2},\frac{2}{4},\frac{3}{6}}$ , and ${\Large\frac{10}{20}}$ are equivalent fractions.

Example

Find three fractions equivalent to ${\Large\frac{2}{5}}$ .

Show Solution

Solution
To find a fraction equivalent to ${\Large\frac{2}{5}}$ , we multiply the numerator and denominator by the same number (but not zero). Let us multiply them by $2,3$ , and $5$ .

${\Large\frac{2\cdot\color{blue}{2}}{5\cdot\color{blue}{2}}}={\Large\frac{4}{10}}$ ${\Large\frac{2\cdot\color{blue}{3}}{5\cdot\color{blue}{3}}}={\Large\frac{6}{15}}$ ${\Large\frac{2\cdot\color{blue}{5}}{5\cdot\color{blue}{5}}}={\Large\frac{10}{25}}$

So, ${\Large\frac{4}{10},\frac{6}{15}}$ , and ${\Large\frac{10}{25}}$ are equivalent to ${\Large\frac{2}{5}}$ .

Try it

Find three fractions equivalent to ${\Large\frac{3}{5}}$ .

Show Solution

Correct answers include ${\Large\frac{6}{10},\frac{9}{15}},\text{and }{\Large\frac{12}{20}}$ .

Find three fractions equivalent to ${\Large\frac{4}{5}}$ .

Show Solution

Correct answers include ${\Large\frac{8}{10},\frac{12}{15}},\text{and }{\Large\frac{16}{20}}$ .

Example

Find a fraction with a denominator of $21$ that is equivalent to ${\Large\frac{2}{7}}$ .

Show Solution

Solution
To find equivalent fractions, we multiply the numerator and denominator by the same number. In this case, we need to multiply the denominator by a number that will result in $21$ .

Since we can multiply $7$ by $3$ to get $21$ , we can find the equivalent fraction by multiplying both the numerator and denominator by $3$ .

${\Large\frac{2}{7}}={\Large\frac{2\cdot\color{blue}{3}}{7\cdot\color{blue}{3}}}={\Large\frac{6}{21}}$

Try it

In the following video we show more examples of how to find an equivalent fraction given a specific denominator.

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UNIT 4