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# Equivalent Fractions

Some things are so obvious in mathematics that they don't need to be directly stated. For example, few would argue with the premise that 2 = 2 because it's the same value represented by the same number. However, fractions can have multiple numbers that mean the same value.

For example, consider the following diagrams illustrating the value of three fractions: $\frac{1}{2}$ , $\frac{2}{4}$ , and $\frac{4}{8}$ :

Exactly half of the circle is filled in in all three instances, so we would call these three fractions equivalent fractions. Fractions are essentially a different way of expressing division, meaning that infinite combinations of numbers yield the same value.

## How do you determine equivalent fractions without diagrams?

Generally speaking, you'll get equivalent fractions if you multiply both the numerator and denominator of a fraction by the same non-zero number. In the example above, you get $\frac{2}{4}$ if you multiply the numerator and denominator of $\frac{1}{2}$ by 2, and you get $\frac{4}{8}$ if you multiply the numerator and denominator of $\frac{2}{4}$ by 2.

The reason this works is that multiplying a fraction by a figure such as $\frac{2}{2}$ or $\frac{7}{7}$ is the same thing as multiplying by 1. Multiplying by 1 always gives you the same value you started with, though it might look a little different than it did before if you're working with fractions. If you multiply $\frac{2}{5}$ by $\frac{7}{7}$ , you get $\frac{14}{35}$ . It looks different, but it retains the same value since you effectively multiplied by one.

The reason zero doesn't work is that anything multiplied by zero becomes zero. The value is still equivalent, but we no longer call them equivalent fractions because zero isn't a fraction at all!

## Can I choose between equivalent fractions when giving answers on a test?

Typically you cannot. It would be a nightmare if students, teachers, and mathematicians had to work with all equivalent fractions interchangeably, so the standard convention is to convert everything to the simplest form for the "right answer". Some textbooks use language such as "written in lowest terms" to mean the same thing, so don't let that confuse you.

For instance, let's say that your family ordered a pizza and your brother ate four out of the eight slices. The most obvious fraction to use would be 4/8 because he ate four of the eight slices, and that language accurately captures how much pizza he ate. However, the simplest form of $\frac{4}{8}$ is $\frac{1}{2}$ , so you would say that your brother ate $\frac{1}{2}$ of the pizza if asked on a math test.

Putting equivalent fractions into practical terms like this can help you visualize how the value remains unchanged even if the numerators and denominators you're working with are getting larger or smaller.

## Equivalent fractions practice questions

a. Write three possible equivalent fractions for $\frac{1}{3}$ .

$\frac{2}{6},\frac{3}{9},\frac{4}{12},\frac{5}{15},$ etc.

b. Write three possible equivalent fractions for $\frac{25}{100}$ .

$\frac{1}{4},\frac{2}{8},\frac{3}{12},\frac{4}{16},$ etc.

c. In your own words, explain what it means if fractions are equivalent.

Fractions are equivalent if they share the same value

d. Express $\frac{10}{90}$ in simplest form

$\frac{1}{9}$

e. What is $\frac{30}{90}$ written in the lowest terms?

$\frac{1}{3}$

f. If you ate two slices of a birthday cake cut into a total of 12 pieces, how much of the cake did you eat? Express your answer in the simplest form.

$\frac{1}{6}$

## Get help with equivalent fractions now

Equivalent fractions can seem counterintuitive at times, but a solid understanding of how they work is essential as students climb the educational ladder. If your student cannot wrap their head around how ⅕ and 20/100 could possibly mean the same thing or why multiplying the numerator and denominator by the same number produces the same value, a private tutor could help them develop their quantitative problem-solving skills and feel more confident in math class. Contact the Educational Directors at Varsity Tutors for more information.

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