Math students and working adults often uncover their understanding of just how to add, subtract, multiply, and divide fractions has grown rusty through disuse. We tend to depend on calculators to carry out a most the occupational of fractions because that us. University Algebra, though, builds up some necessary techniques for managing expressions and also equations that are based upon operations top top fractions. So that is necessary to refamiliarize yourself v these skills. This ar will repeat you how to do operations on fractions. Together you work through the rest of the course, you have the right to return this section as needed for a rapid reminder of operations on fractions.

You are watching: Two equivalent fractions for 1 4

Before we gain started, let’s define some terminology.

**product:**the an outcome of multiplication

**factor:**something gift multiplied – because that

**numerator:**the top part of a fraction – the molecule in the fraction

**denominator:**the bottom component of a fraction – the denominator in the fraction

## A Note around Instructions

Certain indigenous are supplied in math textbooks and by teacher to administer students with instructions ~ above what to do with a given problem. Because that example, you may see indict such together *find* or *simplify.*It is crucial to know what these words typical so you can successfully work through the difficulties in this course. Right here is a short list of some problem instructions together with their descriptions together they will certainly be supplied in this module.

Find | Perform the suggested mathematical operations i m sorry may encompass addition, subtraction, multiplication, division (later, use of the wordfind will certainly be broadened to solving equations, as infind the worth of variable). |

Simplify | 1) do the indicated mathematical operations consisting of addition, subtraction, multiplication, division 2) compose a mathematical statement in smallest terms so there space no various other mathematical to work that deserve to be performed—often found in problems related come fractions and the stimulate of operations |

Evaluate | 1) perform the indicated mathematical operations consisting of addition, subtraction, multiplication, division 2) substitute a offered value for a variable in an expression and also then perform the shown mathematical operations |

Reduce | Write a mathematical declare in smallest or lowest terms so there are no various other mathematical operations that can be performed—often uncovered in problems related to fountain or division |

## Add Fractions

When you require to add or subtract fractions, girlfriend will require to an initial make sure that the fractions have the very same denominator. The denominator tells you how many pieces the whole has been damaged into, and also the molecule tells girlfriend how many of those pieces you are using.The “parts the a whole” ide can be modeled v pizzas and also pizza slices. Because that example, imagine a pizza is reduced into

How have the right to you explain the full amount that pizza that is left v one number rather than two various fractions? You need a common denominator,technicallycalledthe **least typical multiple. **Remember thatif a number is a multiple of another, you can divide them and also have no remainder.

One means to uncover the least usual multiple of 2 or much more numbers is to an initial multiply eachby

First, list all the multiples of | Then list all the multiples of 5: |

The smallest multiple they have in common will be the usual denominator to usage to rewrite each fraction into identical fractions. See the example below for a demonstration of our pizza problem.

### Example

One pizza, reduced into 4 slices, has one missing. An additional pizza the the same size has actually been reduced into eight pieces, of i m sorry three have actually been removed. Define the full amount that pizza left in the two pizzas using usual terms.

Rewrite the fractions

Find the least usual multiple that the denominators. This is the least common denominator.

Multiples that

Multiples that

The least usual denominator is

Rewrite

We don’t need to rewrite

Both

We have actually

To include fractions v unlike denominators, an initial rewrite castle with choose denominators. Then include or subtract the numerators over the common denominator.

### Adding Fractions v Unlike Denominators

Find a usual denominator.Rewrite each fraction as an equivalent fraction using the typical denominator.Now that the fractions have actually a common denominator, girlfriend can add the numerators.Simplify through canceling out all usual factors in the numerator and also denominator.## Simplify a Fraction

It is a typical convention in mathematics to current fractions in lowest terms. We speak to this practice*simplifying or**reducing *the fraction, and it deserve to be accomplished by canceling (dividing) the common factors in a fraction’s numerator and also denominator. We can do this due to the fact that a fraction represents division, and also for any type of number

For example, to leveling

Since over there is a

In the next example you are presented how to add twofractions with different denominators, then simplify the answer.

Since the denominators are not alike, discover a usual denominator by multiplying the denominators.

Rewrite each portion with a denominator of

Add the fountain by adding the numerators and also keeping the denominator the same. Make certain the fraction cannot be simplified.

You can uncover a usual denominator by finding the typical multiples the the denominators. The least typical multiple is the simplest to use.

Since the denominators room not alike, uncover the least common denominator by detect the least typical multiple (LCM) that 7 and 21.

Multiples the

Multiples of

Rewrite each fraction with a denominator the

Add the fountain by including the numerators and keeping the denominator the same. Make sure the portion cannot it is in simplified.

In the following video clip you will see an instance of exactly how to add two fractions with various denominators.

You can likewise add an ext than two fractions as long as you very first find a usual denominator for all of them. An instance of a amount of three fractions is shown below. In this example, you will usage the element factorization an approach to find the LCM.

### Think around It

Add

What renders this example different than the ahead ones? usage the box listed below to compose down a few thoughts about how you would include three fountain with different denominators together.

Since the denominators are not alike, discover the least usual denominator by finding the least typical multiple (LCM) the

Rewrite each fraction with a denominator the

Add the fractions by adding the numerators and keeping the denominator the same.

Write the improper fraction as a mixed number and also simplify the fraction.

## Subtract Fractions

Subtracting fractions follows the same technique as including them. First, recognize whether or no the denominators room alike. If not, rewrite each portion as an tantamount fraction, all having the same denominator. Listed below are some examples of subtracting fractions who denominators are not alike.

The fractions have actually unlike denominators, so you require to discover a common denominator. Recall that a common denominator deserve to be found by multiplying the 2 denominators together.

Rewrite each fraction as one equivalent portion with a denominator the

Subtract the numerators. Leveling the price if needed.

The example listed below shows how to usemultiples to find the least typical multiple, which will certainly be the least common denominator.

Find the least common multiple of the denominators—this is the least typical denominator.

Multiples the

Multiples the

Rewrite each fraction with a denominator of

Subtract the fractions. Leveling the answer if needed.

In the following video clip you will certainly see an example of just how to subtract fractions through unlike denominators.

## Multiply Fractions

Just as you add, subtract, multiply, and divide when working with entirety numbers, you likewise use these operations as soon as working v fractions. There are plenty of times once it is vital to multiply fractions. A design may assist you understand multiplication the fractions.

When you multiply a portion by a fraction, you space finding a “fraction that a fraction.” expect you have

To review: if a fraction hascommon components in the numerator and also denominator, we deserve to **reduce** the portion to that is simplified form by removed the common factors.

For example,

GivenYou cansimplify first, prior to you multiply two fractions, to make your work-related easier. This allows you to occupational with smaller numbers once you multiply.

In the following video you will see an instance of how to multiply 2 fractions, then leveling the answer.

### Think about It

Multiply

What renders this example various than the previous ones? usage the box below to compose down a couple of thoughts around how you would certainly multiply three fractions together.

Multiply the numerators and multiply the denominators.

Simplify very first by canceling (dividing) thecommon determinants of

## Divide Fractions

There space times as soon as you should use department to fix a problem. Because that example, if paint one coat of paint on the wall surfaces of a room calls for

Before we start dividing fractions, let’s covering some important terminology.

**reciprocal:**two fractions room reciprocals if your product is

**quotient:**the resultof division

Dividing fractions calls for using the mutual of a number or fraction. If you multiply two numbers together and get

Sometimes us callthe reciprocalthe “flip” that the other number: flip

## Division through Zero

You know what it method to divide by

We deserve to read it as, “zero divided by eight.” since multiplication is the station of division, we can rewrite this as a multiplication problem.

We have the right to infer that the unknown have to be

Now let’s take into consideration the mutual of

This doesn’t make any type of sense. There room no numbers that you have the right to multiply by zero to obtain a an outcome of 8. The reciprocal of

Caution! department by zero is undefined and also so is the mutual of any fraction that has actually a zero in the numerator. For any kind of real number a,

## Divide a fraction by a totality Number

When you division by a entirety number, you room multiplying by the reciprocal. In the painting example where you need

### Dividing is multiplying by the Reciprocal

For every division, you can turn the operationinto multiplication by utilizing the reciprocal. Splitting is the very same as multiplying by the reciprocal.

The very same idea will job-related when the divisor (the thing being divided) is a fraction. If you have

Each human gets

If you have actually a recipe that requirements to be split in half, you deserve to divide each ingredient through

For example, dividing by

Each human being gets one piece, for this reason each human gets

Dividing a fraction by a totality number is the same as multiply by the reciprocal, so friend can always use multiplication of fractions to solve division problems.

Write your answer in shortest terms.

Dividing by

Multiply numerators and also multiply denominators.

Simplify to lowest terms by splitting numerator and also denominator by the usual factor

Write her answer in shortest terms.

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Dividing by