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1.2: Fractions

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Learning Objectives

By the end of this section, you will be able to:

  • Simplify fractions
  • Multiply and divide fractions
  • Add and subtract fractions
  • Use the order of operations to simplify fractions
Warm-up 1.2.1

Compute the following

  1. 16÷2
  2. 16÷8
  3. 6÷6
Answer
  1. 8
  2. 2
  3. -1

What is a Fraction?

Now try 1÷2. Is this an integer? No, it's just 1÷2

Conceptually this is asking how many friends you can feed with 1 slice of pizza if each friend eats 2 slices. Well, you can feed half of a friend, that is 12. If we expand our numbers once again we can accommodate these new numbers we will call fractions. Fractions represent smaller parts of whole numbers. If we now allow fractions, we get the rationals, Q. Rationals are any number that can be written as an integer over an integer.

FRACTION

A fraction is written ab, where b0 and

a is the numerator and b is the denominator.

A fraction represents parts of a whole. The denominator b is the number of equal parts the whole has been divided into, and the numerator a indicates how many parts are included.

Let's create a fraction for example, take a random integer.

7? Good choice! Take another integer.

3? Another good choice!Taking 7 over 3 we get 73. Really this is expressing the number we get when we divide 7 by 3.

Where does the negative sign go in a fraction? Usually the negative sign is in front of the fraction, but you will sometimes see a fraction with a negative numerator, or sometimes with a negative denominator. Remember that fractions represent division. When the numerator and denominator have different signs, the quotient is negative.

13=13negativepositive=negative

13=13positivenegative=negative

PLACEMENT OF NEGATIVE SIGN IN A FRACTION

For any positive numbers a and b,

ab=ab=ab

Example 1.2.1

If we have -7 or -3 but not both then we would get 73 This is the same as 73 or 73.

73=73=73

Any integer can be turned into a fraction with 1 as the denominator e.g 3=31.

Let's take a look at when we will get a nice integer from division.

Factor Trees

Definition: Divisible

A number x is "divisible" by another number y if x÷y is an integer

Example 1.2.2

9 is divisible by 3 since 9÷3=3
9 is not divisible by 2 since 9÷2 is not an integer.

Definition: Prime

A "prime number" is an integer that is only divisible by itself and 1 as well as negative itself and -1.

Example 1.2.3

7 is prime since it is not divisible by any integer except for ±7 and ±1

9 is not prime since 9 is divisible by 3 which is not ±9 or ±1.

We can "decompose" numbers into the prime numbers that were multiplied together to create them. This is called factoring.

Example 1.2.4

Factor

  1. 24
  2. 30
  3. 90
Solution
  1. 24=2223
  2. 30=235
  3. 90=2335

A popular trick to use for factoring numbers is called a factor tree. Let's take a look at a simple example on how to do this. Let's decompose 6 into its prime factors

decomp of 6.png

This involves dividing numbers until you get numbers that are prime. Here we note 6 is divisible by 3 and 6÷3=2 so we broke 6 up into 32. We can't break this up anymore since 3 and 2 are prime.

Example 1.2.5

Decompose 24 using a factor tree.

Answer

decomp of 24.png

You can divide 24 by 4 to get 6. You can decompose 6 into 3 and 2. While you can decompose 4 into 2 and 2. So you can decompose 24 into 2322 we see that 3 and 2 are prime so we can't decompose anymore. Notice that the factors produced are the same as in the previous example.

Exercise 1.2.6

Decompose 30 using a factor tree.

Answer

decomp of 30.png

30 is divisible by 5 and 30÷5=6 which we can decompose into 3 and 2.

So 30=532. This agrees with the answer in the previous example.

We see this isn't the only way to decompose 30 but you will still get the same result. For example you could have used 3 and 10 and decomposed 10 into 5 and 2 but the result is still 30=532

Exercise 1.2.7

Decompose 90 using a factor tree.

Answer

decomp of 90.png

We can break 90 up into 3 and 30 and decompose 30.

So 90=3352. This agrees with the example in the previous example.

Simplify Fractions

Try to input 48 and 12 into your calculator. What do you get? Why do you think the both inputs give you .5? It is because both of them express the same number. That is, 4 is half of 8 and 1 is half of 2. We can simplify 48 by decomposing 4 and 8 into a product of its primes, that is 48=22222. Now using the fact that we can break up a fraction, we can break up 22222=221222=222212=1112=12. This is why it is ok to cancel terms that are multiplied from the numerator and denominator. It is also common to just cross things out as in 22222=22222=12.

There are many ways to write a fraction, some may say there is an infinite way to write the same fraction, but we usually write it in simplest form, that is when we can no longer cancel anything from the top and bottom.

Fractions that have the same value are equivalent fractions.

The Equivalent Fractions Property allows us to find equivalent fractions and also simplify fractions.

If a, b, and c are numbers where b0,c0,

then ab=a·cb·c and a·cb·c=ab.

A fraction is considered simplified if there are no common factors, other than 1, in its numerator and denominator.

For example,

23 is simplified because there are no common factors of 2 and 3.

1015 is not simplified because 5 is a common factor of 10 and 15.

We simplify, or reduce, a fraction by removing the common factors of the numerator and denominator. A fraction is not simplified until all common factors have been removed. If an expression has fractions, it is not completely simplified until the fractions are simplified.

Sometimes it may not be easy to find common factors of the numerator and denominator. When this happens, it is a good idea to factor the numerator and the denominator into prime numbers using the factor tree, then cancel out the common factors using the Equivalent Fractions Property.

Example 1.2.8

Simplify 120252

Answer
Steps to solve the example.
Step 1: Write out the factor trees 252 over 120.png
Step 2: Rewrite the fraction as a product (multiplication) of primes on the top and bottom. 120252=3225222373
Step 3: Cancel like terms from top and bottom =3225222373=5273
Step 4: Multiply to get 1 term on top and bottom, that is, in the numerator and denominator. 5273=1021

Simplify 315770.

Answer
Solution to the example.
Step 1: Write out the factor trees 315 over 770.png
Step 2: Rewrite the fraction as a product (multiplication) of primes on the top and bottom. 315770=537371152
Step 3: Cancel like terms from top and bottom =537371152=922
Step 4: Multiply to get 1 term on top and bottom, that is, in the numerator and denominator. =922

Simplify 69120.

Answer

2340

EXAMPLE 1.2.9

Simplify 120192.

Answer

58

Multiplying Fractions

Multiplying fractions is a simple operation. All you do is multiple the numerator with the numerator and the denominator with the denominator. Let's first get an intuitive idea on why that works. First say you have 13 of a pizza and you multiplied it by 3, that is you put 3 13 pizzas together. You would get 1 whole pizza! In math, this would look like 313=3113=3113=33=1.

We can also go backwards as in we can rewrite 23 as 213.

What about the denominators? Why do we multiply those? Say you have a whole pizza and you divide it into 3 equal slices. You then cut a slice in half to get two slices. We see that the small slices from the 2nd cut make up 16 of a pizza. This makes sense because if you cut each slice in half, you would have 6 equally sized slices and 1 slice from that would be 16 a pizza. So we divided a pizza by 3 and then each slice in 2. We can express this using an equation as 1312=13÷2=(1÷3)÷2=16.

Now let's put all that intuition together.

Example 1.2.10

Compute

2375

Answer

2375=2735=1415

We that was way easier than all that intuition.

Let's do some more examples,

Exercise 1.2.11

Compute the following:

  1. 1414
  2. 1323
  3. 2345
Answer
  1. 1144=116
  2. 1233=29
  3. 2435=815

Now lets combine these strategies to compute and simplify a product of fractions.

Example 1.2.12

Compute 23274

Answer
Solution to previous example.
Step 1: Rewrite every number as a product of primes. =2333322
Step 2: Multiply =2333322
Step 3: Cancel like terms =332
Step 4: Simplify if possible =92

Compound Fractions and Dividing Fractions

The numerators or denominators of some fractions contain fractions themselves. A fraction in which the numerator or the denominator is a fraction is called a compound fraction. Some people may use complex fraction but we will see later that this is actually terrible and horrible terminology. Still just know that complex fraction is the same as a compound fraction.

Definition: COMPOUND FRACTION

A compound fraction is a fraction in which the numerator or the denominator contains a fraction.

Some examples of compound fractions are:

6733458x256

To simplify a compound fraction, remember that the fraction bar means division. For example, the compound fraction 3458 means 34÷58.

Surprisingly, division is just as easy as multiplication! All we need to do is take the top fraction and multiply it by the "reciprocal" of the bottom fraction.

Definition: RECIPROCAL

The reciprocal of a number is the multiplicative inverse. That is, the number you need to multiply to it to get 1.

This is very easy for the rationals as we just need to switch the numerator with the denominator.

Example 1.2.13

The reciprocal of 2=12 is 12 since 212=1.

The reciprocal of 32 is 23 since 3223=1

Now we can look at an intuitive example of dividing by a fraction. Lets say a child eats half a slice of pizza and you have a pizza that contains 8 slices. How many children can you feed with 8 slices? Well each child eats half a slice so you would be able to feed 2 children with each slice and you have 8 slices, so you can feed 16 children. In an equation, this looks like 8112=8121=161=16

Now lets get a bit more complicated with...teenagers...that eat 34 of a slice each. That is, each teenager eat 3 quarter slices. Say you have 32 slices of pizza, that is 3 half slices. How many teenagers can you feed with that? Well let's rewrite 32 as 312 and save the multiplication with 12 for later. Let's cut up 3 slices into quarter slices so we have 34=12 quarter slices. each teenager eats 3 quarter slices so we can feed 123=4 teenagers. BUT WAIT we had 3 half slices so we can actually feed 412=42=2 teenagers. This is just
3234=3243=126=2

WOW that was way easier than the intuition. You don't need to remember the intuitive examples for the test but hopefully you now understand why you are doing what you are doing. Now lets take a look at a step by step problem. To help with the examples, remember "invert and multiply"; this means that to divide, we flip over the denominator and multiply it by the numerator.

Example 1.2.14

Compute 92274

Answer
Solution to previous example.
Step 1: Rewrite the quotient as a product. =92427
Step 2: Rewrite every number as a product of primes. =33222333
Step 3: Multiply =33222333
Step 4: Cancel like terms =23
Step 5: Simplify if possible =23

This is the same process as above just with an extra step in the beginning.

Example 1.2.15

Compute 718÷(1427).

Answer
Solution to previous example.
Step 1: Rewrite the quotient as a product. =7182714
Step 2: Rewrite every number as a product of primes. =733233372
Step 3: Multiply =733333272
Step 4: Cancel like terms =322
Step 5: Simplify if possible =34

Divide: 727÷(3536).

Answer

415

Divide: 514÷(1528).

Answer

23

Add and Subtract Fractions

Let's move on to addition. Say you have half a pizza and your friend has half a pizza, then between the two of you, you all have a whole pizza. That is, 12+12=22=1. Note that we added the numerators together and left the denominators unchanged.

When we multiplied fractions, we just multiplied the numerators and multiplied the denominators right straight across. To add or subtract fractions, they must have a common denominator.

FRACTION ADDITION AND SUBTRACTION

If a, b, and c are numbers where c0, then

ac+bc=a+bc and acbc=abc

To add or subtract fractions, add or subtract the numerators and place the result over the common denominator.

What if the denominators are different? Then make them the same! We can make them the same by using a fundamental trick of math

Theorem: 1st fundamental trick of math 1.2.16

A number multiplied by 1 is the same number. That is, if a is any number, then a1=a

That doesn't seem like much of a trick but there are many creative ways to write the number 1.

For example 22 and 6262 are 1.

Definition: Least Common Multiple (LCM)

The least common multiple (LCM) of two integers x,y is the smallest integer, c such that c is divisible by x and y.

Example 1.2.17

The LCM of 4 and 6 is 12 because 4 and 6 divide 12 and 12 is the smallest integer that both 6 and 4 divide evenly.

The LCM of 4 and 6 is not 24 because, even though 4 and 6 divide 24, it is not the smallest integer that both 6 and 4 divide evenly.

Let's do an example of how to find the LCM

Example 1.2.18

Find the lcm of 8 and 12.

Answer
Solution to previous example.
Step 1: factor the numbers lcm 8 and 12.png
Step 2: multiply all the primes but skip repeats 2223
Step 3: Compute the lcm by doing the multiplication =24

Note: Starting from left to right, we write down the three 2s we got from 8. We skip the two 2s from 12 since we already picked up three 2s from 8, and we pick up a 3 since we didn't already have it.

The least common denominator (LCD) of two fractions is the smallest number that can be used as a common denominator of the fractions. The LCD of the two fractions is the least common multiple (LCM) of their denominators.

Definition: LEAST COMMONE DENOMINATOR

The least common denominator (LCD) of two fractions is the least common multiple (LCM) of their denominators.

After we find the least common denominator of two fractions, we convert the fractions to equivalent fractions with the LCD. Putting these steps together allows us to add and subtract fractions because their denominators will be the same!

Let's first do an intuitive example.

Let's say you have half a pizza and a quarter of a pizza. How much pizza do you have all together and why is it 34th of a pizza? We broke up the half of a pizza into 2 quarters of a pizza and then added them to the quarter of a pizza to get 3 quarters of a pizza. In an equation this looks like
12+14=24+14=34


Did you notice that 4 is the LCM of 2 and 4? If we want to add fractions all we do is make them have the same denominator and the denominator that is easiest to work with is the LCM!

Example 1.2.19

Compute 310+215

Answer
Solution to previous example.
Step 1: Compute lcm of denominator if they are different, if same then skip to step 5.
Solution to previous example.
Substep 1: factor the denominators lcm 10 and 15.png
Substep 2: multiply all the primes but skip repeats 523
Substep 3: Compute the lcm =30
Step 2: divide the lcm by the denominators 3010=3,3015=2

Step 3: multiply the 1st numerator and denominator by the 1st number above.

33103=930

This is where we multiply by a 1 since 310=3101=31033=33103. Same thing goes for the next step.

Step 4: multiply the 2nd numerator and denominator by the 2nd number above 22152=430
Step 5: add the two numbers together 930+430=1330
Step 6: Simplify if possible =1330

This is the same procedure for subtraction too.

Example 1.2.20

Compute 2345

Answer
Solution to previous example.
Step 1: Compute lcm of denominator
Solution to previous example.
Substep 1: factor the denominators 3 and 5 are both prime.
Substep 2: multiply all the primes but skip repeats 35
Substep 3: Compute the lcm =15
Step 2: divide the lcm by the denominators

153=5

155=3

Step 3: multiply the 1st numerator and denominator by the 1st number above 2535=1015
Step 4: multiply the 2nd numerator and denominator by the 2nd number above 4353=1215
Step 5: subtract the two numbers 10151215=215
Step 6: Simplify if possible =215

Lets take a look at some more examples.

EXAMPLE 1.2.21

Add: 712+1115.

Answer

7960

EXAMPLE 1.2.22

Add: 1315+1720.

Answer

10360

We now have all four operations for fractions. The Table below summarizes fraction operations.

Four operations for fractions.
Fraction Multiplication Fraction Division
abcd=acbd ab÷cd=abdc
Multiply the numerators and multiply the denominators Multiply the first fraction by the reciprocal of the second.
Fraction Addition Fraction Subtraction
ac+bc=a+bc acbc=abc
Add the numerators and place the sum over the common denominator. Subtract the numerators and place the difference over the common denominator.

To multiply or divide fractions, an LCD is NOT needed.

To add or subtract fractions, an LCD is needed.

Use the Order of Operations to Simplify Fractions

The fraction bar in a fraction acts as grouping symbol. The order of operations then tells us to simplify the numerator and then the denominator. Then we divide.

SIMPLIFY AN EXPRESSION WITH A FRACTION BAR.
  1. Simplify the expression in the numerator. Simplify the expression in the denominator.
  2. Simplify the fraction.
EXAMPLE 1.2.23

Simplify: 4(3)+6(2)3(2)2.

Answer

The fraction bar acts like a grouping symbol. So completely simplify the numerator and the denominator separately.

4(3)+6(2)3(2)2Multiply.12+(12)62Simplify.248Divide.3

EXAMPLE 1.2.24

Simplify:8(2)+4(3)5(2)+3.

Answer

4

EXAMPLE 1.2.25

Simplify: 7(1)+9(3)5(3)2.

Answer

2

Now we’ll look at compound fractions where the numerator or denominator contains an expression that can be simplified. So we first must completely simplify the numerator and denominator separately using the order of operations. Then we divide the numerator by the denominator as the fraction bar means division.

EXAMPLE 1.2.26

Simplify 125235+15

Answer
Solution to the example.
Step 1: Simplify the numerator =4235+15
Step 2: Simplify the denominator =4245
Step 3: Compute the division =4254=52
Step 4: Simplify if possible =52\
EXAMPLE 1.2.27

Simplify: (13)2+2.

Answer

112

EXAMPLE 1.2.28

Simplify: 1+44(14).

Answer

68

SIMPLIFY COMPOUND FRACTIONS.
  1. Simplify the numerator.
  2. Simplify the denominator.
  3. Divide the numerator by the denominator.
  4. Simplify if possible.
EXAMPLE 1.2.29

Simplify: 12+233416.

Answer

It may help to put parentheses around the numerator and the denominator.

12+233416Simplify the numerator (LCD=6) and simplify the denominator (LCD=12).(36+46)(912212)Simplify.(76)(712)Divide the numerator by the denominator.76÷712Simplify.76127Divide out common factors.762671Simplify.2

EXAMPLE 1.2.30

Simplify: 13+123413.

Answer

2

EXAMPLE 1.2.31

Simplify: 231214+13.

Answer

27

Key Concepts

  • If a, b, and c are numbers where b0,c0, then

ab=a·cb·c and a·cb·c=ab.

  • How to simplify a fraction.
    1. Rewrite the numerator and denominator to show the common factors.
      If needed, factor the numerator and denominator into prime numbers first.
    2. Simplify using the Equivalent Fractions Property by dividing out common factors.
    3. Multiply any remaining factors.
  • If a, b, c, and d are numbers where b0, and d0, then

    ab·cd=acbd

    To multiply fractions, multiply the numerators and multiply the denominators.

  • If a, b, c, and d are numbers where b0,c0, and d0, then

    ab÷cd=abdc

    To divide fractions, we multiply the first fraction by the reciprocal of the second.

  • If a, b, and c are numbers where c0, then

    ac+bc=a+bc and acbc=abc

    To add or subtract fractions, add or subtract the numerators and place the result over the common denominator.

  • How to add or subtract fractions.
    1. Do they have a common denominator?
      • Yes—go to step 2.
      • No—rewrite each fraction with the LCD (least common denominator).
        • Find the LCD.
        • Change each fraction into an equivalent fraction with the LCD as its denominator.
    2. Add or subtract the fractions.
    3. Simplify, if possible.
  • How to simplify an expression with a fraction bar.
    1. Simplify the expression in the numerator. Simplify the expression in the denominator.
    2. Simplify the fraction.
  • For any positive numbers a and b,

    ab=ab=ab

  • How to simplify compound fractions.
    1. Simplify the numerator.
    2. Simplify the denominator.
    3. Divide the numerator by the denominator. Simplify if possible.

Glossary

compound fraction
A fraction in which the numerator or the denominator is a fraction is called a compound fraction.
denominator
In a fraction, written ab, where b0, the denominator b is the number of equal parts the whole has been divided into.
equivalent fractions
Equivalent fractions are fractions that have the same value.
fraction
A fraction is written ab, where b0, and a is the numerator and b is the denominator. A fraction represents parts of a whole.
least common denominator
The least common denominator (LCD) of two fractions is the least common multiple (LCM) of their denominators.
numerator
In a fraction, written ab, where b0, the numerator a indicates how many parts are included.
reciprocal
The reciprocal of a fraction is found by inverting the fraction, placing the numerator in the denominator and the denominator in the numerator.

This page titled 1.2: Fractions is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Stanislav A. Trunov and Elizabeth J. Hale via source content that was edited to the style and standards of the LibreTexts platform.

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