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## Equivalent Fractions

#### Introduction

Equivalent fractions are fractions that are equal. In order to determine if one fraction is equal or equivalent to another, both fractions must first be simplified if needed.

#### Terms

- Fractions are used when using numbers to express parts of whole.
- a fraction expressed in its lowest terms.

## Lesson

To determine if a fraction is equivalent to another, we need to compare the two fractions in their simplest form. So if we are given $rac{1}{4}$ and $rac{2}{8}$, we can see that $rac{1}{4}$ is in its simplest form, but $rac{2}{8}$ can be further reduced to $rac{1}{4}$ by dividing the top and bottom by 2.

$dfrac{2}{8} div dfrac{2}{2} = dfrac{1}{4}$

Therefore $rac{1}{4}$ and $rac{2}{8}$ are equivalent fractions.

Similarly, if we are given $rac{5}{10}$ and $rac{21}{30}$, we first simplify both fractions.

$dfrac{5}{10} div dfrac{5}{5} = dfrac{1}{2}$

$dfrac{21}{30} div dfrac{3}{3} = dfrac{7}{10}$

We can now see that $rac{5}{10}$ is NOT equivalent to $rac{21}{30}$, because $rac{1}{2}$ is not equivalent to $rac{7}{10}$

To find the equivalent of a fraction when we are given either the numerator or denominator is similar to expressing a fraction in higher terms. If we are told to solve the following problem:

$dfrac{3}{10} = dfrac{15}{?}$

we can determine that in order for the new numerator to equal 15, the original numerator must be multiplied by 5. So we multiply the denominator by 5 as well, and get the answer

$dfrac{3}{10} = dfrac{15}{50}$

## Examples

#### Equivalent Fractions (Example #1)

$\dfrac{8}{5}$ and $\dfrac{16}{10}$
First, write each fraction in lowest terms
The greatest common divisor of $8$ and $5$ is 1, so $\dfrac{8}{5}$ is already in lowest terms
$\dfrac{16}{10}$ can be reduced, since $2$ is a factor of both $16$ and $10$:
$\dfrac{16}{10} \div \dfrac{2}{2} = \dfrac{8}{5}$
The fraction is now in lowest terms
$\dfrac{8}{5}$ is equal to $\dfrac{16}{10}$

#### Make Equivalent Fraction (Example #1)

$\dfrac{\text{6}}{\text{5}} = \dfrac{\text{3}}{\text{x}}$
We have to multiply the first numerator (6) by 0 to get the second numerator (3)
To make an equivalent fraction, we need to multiply the denominator by 0 as well

#### Equivalent Fractions (Example #2)

$\dfrac{3}{4}$ and $\dfrac{5}{8}$
First, write each fraction in lowest terms
The greatest common divisor of $3$ and $4$ is 1, so $\dfrac{3}{4}$ is already in lowest terms
The greatest common divisor of $5$ and $8$ is 1, so $\dfrac{5}{8}$ is already in lowest terms
$\dfrac{3}{4}$ is not equal to $\dfrac{5}{8}$

#### Make Equivalent Fraction (Example #2)

$\dfrac{\text{3}}{\text{4}} = \dfrac{\text{3}}{\text{x}}$
We have to multiply the first numerator (3) by 1 to get the second numerator (3)
To make an equivalent fraction, we need to multiply the denominator by 1 as well