Simplifying Fractions

Learning Objectives

Introduction

Fractions are used to represent a part of a whole. Fractions that represent the same part of a whole are called equivalent fractionsTwo or more fractions that name the same part of the whole. . Factoring, multiplication, and division are all helpful tools for working with equivalent fractions.

Equivalent Fractions

We use equivalent fractions every day. Fifty cents can be `2` quarters, and we have `2/4` of a dollar, because there are `4` quarters in a dollar. Fifty cents is also `50` pennies out of `100` pennies, or `50/100` of a dollar. Both of these fractions are the same amount of money, but written with a different numerator and denominator.

Think about a box of crackers that contains `3` packets of crackers. Two of these packets are `2/3` of the box. Suppose each packet has `30` crackers in it. Two packets are also `60` (`30*2`) crackers out of `90` (`30*3`) crackers. This is `60/90` of the box. The fractions `2/3` and `60/90` both represent two packets of crackers, so they are equivalent fractions.

Equivalent fractions represent the same part of a whole, even if the numerator and denominator are different. For example, `1/4=5/20`. In these diagrams, both fractions represent one of four rows in the rectangle.

The image shows a grid with 1 column and 4 rows where blocks are divided equally horizontally. The top block is shaded. The image is labeled with the fraction 1 over 4, or one-fourth.The image shows a grid with 5 columns and 4 rows where squares are divided into 20 equal parts. The top row of five square is shaded. It is labeled with the fraction 5 over 20, or five twentieths.

Since `1/4` and `5/20` are naming the same part of a whole, they are equivalent.

There are many ways to name the same part of a whole using equivalent fractions.

Let’s look at an example where you need to find an equivalent fraction.

Example

Problem

 

 

 

 

John is making cookies for a bake sale. He made `20` large cookies, but he wants to give away only `3/4` of them for the bake sale. What fraction of the cookies does he give away, using `20` as the denominator?

 

The image shows 20 circles representing cookies that are randomly organized.

Start with `20` cookies.

 

The image shows 20 circles representing cookies that are grouped into four sets of five cookies each. The image is labeled “4 groups of 5 cookies.”

Because the denominator of `3/4` is `4`, make `4` groups of cookies, `5` in each group.

 

The image shows 20 circles representing cookies that are grouped into four sets of five cookies each. Three of the sets of cookies are colored pink. One set of cookies is colored green. The image is labeled “3 groups of 5 cookies equals 15 cookies.”

`3/4=(3*5)/(4*5)` because there are `5` cookies in each group.

 

`3/4=(3*5)/(4*5)=15/20`

 

Answer

 

He gives away `15/20` of the cookies.

When you regroup and reconsider the parts and whole, you are multiplying the numerator and denominator by the same number. In the above example, you multiply `4` by `5` to get the needed denominator of `20`, so you also need to multiply the numerator `3` by `5`, giving the new numerator of `15`.

Finding Equivalent Fractions

 

To find equivalent fractions, multiply or divide both the numerator and the denominator by the same number.

 

Examples:

`20/25=(20-:5)/(25-:5)=4/5`

 

`2/7=(2*6)/(7*6)=12/42`

 

 

Write an equivalent fraction to `2/3` that has a denominator of `27`.

 

A) `26/27`

 

B) `11/27`

 

C) `18/27`

 

D) `12/18`

 

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Simplifying Fractions

A fraction is in its simplest formA fraction is in simplest form if the numerator and denominator have no common factors other than `1`., or lowest termsA fraction is in lowest terms if the numerator and denominator have no common factors other than `1`. , when it has the least numerator and the least denominator possible for naming this part of a whole. The numerator and denominator have no common factor other than `1`.

Here are `10` blocks, `4` of which are green. So, the fraction that is green is `4/10`. To simplify, you find a common factor and then regroup the blocks by that factor.

Example

Problem

Simplify `4/10`.

 

 

The image shows 10 blocks. 4 of the blocks are shaded green. A label shows words set up as a fraction: green blocks over blocks, equals 4 over 10.

We start with `4` green blocks out of `10` total blocks.

 

The image shows 10 blocks. 4 of the blocks are shaded green. A label shows words set up as a fraction: green blocks over blocks, equals 2 times 2 over 5 times 2.

Group the blocks in twos, since `2` is a common factor. You have `2` groups of green blocks and a total of `5` groups, each group containing `2` blocks.

 

The image shows 10 blocks. 4 of the blocks are shaded green. A label shows words set up as a fraction: green blocks over blocks, equals 2 times 2 over 5 times 2 equals 2 over 5.

Now, consider the groups as the part and you have `2` green groups out of `5` total groups.

Answer

`4/10=2/5` The simplified fraction is `2/5`.

Once you have determined a common factor, you can divide the blocks into the groups by dividing both the numerator and denominator to determine the number of groups that you have.

For example, to simplify `6/9` you find a common factor of `3`, which will divide evenly into both `6` and `9`. So, you divide `6` and `9` into groups of `3` to determine how many groups of `3` they contain. This gives `(6-:3)/(9-:3)=2/3`, which means `2` out of `3` groups, and `2/3` is equivalent to `6/9`.

It may be necessary to group more than one time. Each time, determine a common factor for the numerator and denominator using the tests of divisibility, when possible. If both numbers are even numbers, start with `2`. For example:

Example

Problem

Simplify `32/48`.

 

 

`32/48=(32-:2)/(48-:2)=16/24`

`32` and `48` have a common factor of `2`. Divide each by `2`.

 

`16/24=(16-:2)/(24-:2)=8/12`

`16` and `24` have a common factor of `2`. Divide each by `2`.

 

`8/12=(8-:4)/(12-:4)=2/3`

`8` and `12` have a common factor of `4`. Divide each by `4`.

 

 

 

Answer

`32/48=2/3`

`2/3` is the simplified fraction equivalent to `32/48`.

In the example above, `16` is a factor of both `32` and `48`, so you could have shortened the solution.

`32/48=(2*16)/(3*16)=2/3`

You can also use prime factorizationA number written as the product of its prime factors. to help regroup the numerator and denominator.

Example

Problem

Simplify `54/72`.

 

 

`54/72=(2*3*3*3)/(2*2*2*3*3`

The prime factorization of `54` is `2*3*3*3`.

The prime factorization of `72` is `2*2*2*3*3`.

 

`(3*(2*3*3))/(2*2*(2*3*3))`

Rewrite, finding common factors.

 

`3/(2*2)*1`

`(2*3*3)/(2*3*3)=1`

 

`3/4`

Multiply: `2*2`.

Answer

`54/72=3/4`

`3/4` is the simplified fraction equivalent to `54/72`.

Notice that when you simplify a fraction, you divide the numerator and denominator by the same number, in the same way you multiply by the same number to find an equivalent fraction with a greater denominator. In the example above, you could have divided the numerator and denominator by `9`, a common factor of `54` and `72`.

`(54-:9)/(72-:9)=6/8`

Since the numerator (`6`) and the denominator (`8`) still have a common factor, the fraction is not yet in lowest terms. So, again divide by the common factor `2`.

`(6-:2)/(8-:2)=3/4`

Repeat this process of dividing by a common factor until the only common factor is `1`.

Simplifying Fractions to Lowest Terms

 

To simplify a fraction to lowest terms, divide both the numerator and the denominator by their common factors. Repeat as needed until the only common factor is `1`.

 

 

Simplify `36/72`.

 

A) `3/6`

 

B) `9/18`

 

C) `18/38`

 

D) `1/2`

 

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Summary

Equivalent fractions do not always have the same numerator and denominator, but they have the same value. A fraction is in lowest terms when the numerator and denominator of the fraction share no common factors other than `1`. A fraction written in lowest terms is called a simplified fraction.