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.

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

product:the an outcome of multiplicationfactor: something gift multiplied – because that 3 cdot 2 = 6 , both 3 and 2 are determinants of 6numerator: the top part of a fraction – the molecule in the fractionLargefrac23 is 2denominator: the bottom component of a fraction – the denominator in the fractionLargefrac23 is 3

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.

InstructionInterpretation
FindPerform 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).
Simplify1) 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

Evaluate1) 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

ReduceWrite 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 4 pieces, and someone takes 1 piece. Now, Largefrac14of the pizza is gone and also Largefrac34 remains. Keep in mind that both of this fractions have a denominator of 4, which refers to the variety of slices the whole pizza has actually been reduced into. What if friend have an additional pizza that had been reduced into 8 equal parts and 3 of those parts were gone, leave Largefrac58?



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 1, 2, 3, 4, etc. Because that example, discover the least typical multiple of 2 and 5.

First, list all the multiples of 2:Then list all the multiples of 5:
2cdot 1 = 25cdot 1 = 5
2cdot 2 = 45cdot 2 = 10
2cdot 3 = 65cdot 3 = 15
2cdot 4 = 85cdot 4 = 20
2cdot 5 = 105cdot 5 = 25

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 dfrac34 and dfrac58as fractions with a least usual denominator.

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

Multiples that 4: 4, extbf8,12,16, extbf24

Multiples that 8: extbf8,16, extbf24

The least usual denominator is 8—the smallest multiple they have in common.

Rewrite dfrac34 through a denominator of 8. You need to multiply both the top and also bottom by 2 so you don’t adjust the relationship between them.

dfrac34cdotdfrac22=dfrac68

We don’t need to rewrite dfrac58 since it currently has the usual denominator.

Answer

Both dfrac68anddfrac58 have the very same denominator, and also you can explain how much pizza is leftwith typical terms. Include the numerators and put them end the usual denominator.

We have actually dfrac68 the the first pizza anddfrac58 that the 2nd pizza left. That’sdfrac118 of an identically sized pizza, or 1 anddfrac38, pizza quiet on the table.


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 practicesimplifying orreducing 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 a, dfracaa=1.

For example, to leveling dfrac69 you have the right to rewrite 6and 9using the smallest factors possible as follows:

dfrac69=dfrac2cdot33cdot3

Since over there is a 3 in both the numerator and also denominator, and also fractions deserve to be taken into consideration division, we deserve to divide the 3 in the optimal by the 3 in the bottom to minimize to 1.

dfrac69=dfrac2cdotcancel33cdotcancel3=dfrac2cdot13=dfrac23

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.

3cdot5=15

Rewrite each portion with a denominator of 15.

eginarraycLargefrac23cdotLargefrac55=Largefrac1015\\Largefrac15cdotLargefrac33=Largefrac315endarray

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

Largefrac1015+Largefrac315=Largefrac1315

Answer

Largefrac23+Largefrac15=Largefrac1315


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 7: 7, 14, extbf21

Multiples of 21: extbf21

Rewrite each fraction with a denominator the 21.

eginarraycLargefrac37cdotLargefrac33=Largefrac921\\Largefrac221endarray

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

Largefrac921+Largefrac221=Largefrac1121

Answer

Largefrac37+Largefrac221=Largefrac1121


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 Largefrac34+Largefrac16+Largefrac58. Leveling the answer and also write as a mixed number.

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 4, 6, and 8.

4=2cdot2\6=3cdot2\8=2cdot2cdot2\ extLCM:,,2cdot2cdot2cdot3=24

Rewrite each fraction with a denominator the 24.

eginarraycLargefrac34cdotLargefrac66=Largefrac1824\\Largefrac16cdotLargefrac44=Largefrac424\\Largefrac58cdotLargefrac33=Largefrac1524endarray

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

Largefrac1824+Largefrac424+Largefrac1524=Largefrac3724

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

Largefrac3724= ormalsize 1,,Largefrac1324

Answer

Largefrac34+Largefrac16+Largefrac58= ormalsize 1Largefrac1324


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.

5cdot6=30

Rewrite each fraction as one equivalent portion with a denominator the 30.

eginarraycLargefrac15cdotLargefrac66=Largefrac630\\Largefrac16cdotLargefrac55=Largefrac530endarray

Subtract the numerators. Leveling the price if needed.

Largefrac630-Largefrac530=Largefrac130

Answer

Largefrac15-Largefrac16=Largefrac130


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 6: 6, extbf12, 18, 24

Multiples the 4: 4, 8, extbf12,16, 20

12 is the least usual multiple the 6 and also 4.

Rewrite each fraction with a denominator of 12.

eginarraycLargefrac56cdotLargefrac22=Largefrac1012\\Largefrac14cdotLargefrac33=Largefrac312endarray

Subtract the fractions. Leveling the answer if needed.

Largefrac1012-Largefrac312=Largefrac712

Answer

Largefrac56-Largefrac14=Largefrac712


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 Largefrac34of a liquid bar and also you desire to uncover Largefrac12of the Largefrac34:





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,

Given Largefrac815, the factors of 8 are: 1, 2, 4, 8 and the components of 15 are: 1, 3, 5, 15. Largefrac815 is simplified because there are no typical factors the 8 and 15.Given Largefrac1015, the factors of 10 are: 1, 2, 5, 10 and the determinants of 15 are: 1, 3, 5, 15. Largefrac1015 is not simplified because 5is a typical factor the 10and 15.

You 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 Largefrac23cdotLargefrac14cdotLargefrac35. Simplify the answer.

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.

Largefrac2cdot 1cdot 33cdot 4cdot 5

Simplify very first by canceling (dividing) thecommon determinants of 3 and also 2. 3 divided by 3 is 1, and 2 separated by 2 is 1.

eginarraycLargefrac2cdot 1cdot33cdot (2cdot 2)cdot 5\Largefraccancel2cdot 1cdotcancel3cancel3cdot (cancel2cdot 2)cdot 5\Largefrac110endarray

Answer

Largefrac23cdotLargefrac14cdotLargefrac35 = Largefrac110


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 3 quarts the paint and you havea bucket that consists of 6 quarts the paint, how plenty of coats the paint have the right to you paint on the walls? You division 6 through 3 for an answer of 2coats. Over there will likewise be times when you should divide through a fraction. Suppose painting a closet with one coat only compelled Largefrac12 quart the paint. How numerous coats could be painted with the 6 quarts that paint? To find the answer, you must divide 2by the fraction, Largefrac12.

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

reciprocal: two fractions room reciprocals if your product is 1(Don’t worry; we will present you examples of what this means.)quotient: the resultof division

Dividing fractions calls for using the mutual of a number or fraction. If you multiply two numbers together and get 1as a result, then the 2 numbers room reciprocals. Here are some instances of reciprocals:

Original numberReciprocalProduct
Largefrac34Largefrac43Largefrac34cdotLargefrac43=Largefrac3cdot 44cdot 3=Largefrac1212=1
Largefrac12Largefrac21Largefrac12cdotLargefrac21=Largefrac1cdot22cdot1=Largefrac22=1
3=Largefrac31Largefrac13Largefrac31cdotLargefrac13=Largefrac3cdot 11cdot 3=Largefrac33=1
2Largefrac13=Largefrac73Largefrac37Largefrac73cdotLargefrac37=Largefrac7cdot33cdot7=Largefrac2121= ormalsize 1

Sometimes us callthe reciprocalthe “flip” that the other number: flip Largefrac25 to gain the reciprocal Largefrac52.

Division through Zero

You know what it method to divide by 2 or divide by 10, however what does it average to divide a quantity by 0? Is this also possible? deserve to you divide 0 by a number? consider thefraction

Largefrac08

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.

ext?cdot8=0.

We have the right to infer that the unknown have to be 0 because that is the just number the will provide a an outcome of 0 when it is multiply by 8.

Now let’s take into consideration the mutual of Largefrac08 which would certainly be Largefrac80. If werewrite this as a multiplication problem, we will have

ext?cdot0=8.

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 ofLargefrac80 is undefined, and also in fact, all department by zero is undefined.


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, Largefraca0 is undefined. Additionally, the mutual ofLargefrac0a will constantly be undefined.

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 3 quarts of repaint for a coat and have 6 quarts of paint, girlfriend can find the total variety of coats that have the right to be painted by separating 6 through 3, 6div3=2. Friend can additionally multiply 6 by the reciprocal of 3, i m sorry is Largefrac13, therefore the multiplication problem becomes

Largefrac61cdotLargefrac13=Largefrac63= ormalsize2


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 Largefrac34 that a candy bar and also need to divide it amongst 5 people, each human gets Largefrac15 of the accessible candy:

Largefrac15 ormalsize ext of Largefrac34=Largefrac15cdotLargefrac34=Largefrac320

Each human gets Largefrac320of a entirety candy bar.

If you have actually a recipe that requirements to be split in half, you deserve to divide each ingredient through 2, or you deserve to multiply every ingredient by Largefrac12to uncover the brand-new amount.

For example, dividing by 6 is the same as multiplying by the reciprocal of 6, i beg your pardon is Largefrac16. Look in ~ the chart of 2 pizzas below. How have the right to you division what is left (the red shaded region) amongst 6 civilization fairly?



Each human being gets one piece, for this reason each human gets Largefrac14 that a pizza.

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 4 or Largefrac41 is the exact same as multiplying by the mutual of 4, i m sorry is Largefrac14.

Largefrac23 ormalsizediv 4=Largefrac23cdotLargefrac14

Multiply numerators and also multiply denominators.

Largefrac2cdot 13cdot 4=Largefrac212

Simplify to lowest terms by splitting numerator and also denominator by the usual factor 4.

Largefrac16

Answer

Largefrac23 ormalsizediv4=Largefrac16


Write her answer in shortest terms.

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Dividing by Largefrac12 is the very same as multiplying by the mutual of Largefrac12, which is Largefrac21.