An Introduction to Fractions

Making Friends with

Fractions

What is a fraction?

A fraction is basically an unworked division problem.

¾ could be read “three divided by four.”The top number (numerator) tells how many

parts you are talking about.The bottom number (denominator) tells how

many parts it takes to make a whole unit. In ¾, you are talking about 3 pieces out of 4 pieces.

Equivalent Fractions

You can have two fractions that are worth the same amount.

This is just like having half a dollar by having 2 quarters or 5 dimes or 10 nickels or 50 pennies.

1/2= 5/10 = 10/20 = 50/100All of these fractions are worth the same

amount.

The Rule of Equivalent Fractions

To create equivalent fractions, you simply multiply the top and bottom by the same number.

It can be any number you need it to be, but you must do the same thing to the top and the bottom.

Creating Equivalent Fractions

Let’s use the rule to create some equivalent fractions.

1 = 2 = 3 = 4 = 5

3 6 9 12 15

x2/2 x3/3 x4/4 x5/5

In each case, the top and bottom of the original fraction were multiplied by the same thing.

Equivalent Fractions

Use the equivalent fraction rule to make three equivalent fractions for each of these fractions:

1/4 2/5 3/8 1/10

Equivalent Fractions

Use the equivalent fraction rule to make three equivalent fractions for each of these fractions:

1/4 = 2/8 = 3/12 = 4/16 2/5 = 4/10 = 6/15 = 10/25 3/8 = 6/16 = 12/32 = 15/40 = 21/56 1/10 = 2/20 = 5/50 = 9/90 = 10/100

Simplifying Fractions

When you work with fractions, it is considered proper to leave the fraction in its most reduced form.

That means that the top and the bottom don’t share any factors. They are “relatively prime.”

One fool-proof way is to look at all the factors of the top and the bottom and get rid of the ones they share.

Simplifying Fractions

For example, take the fraction 6/10

6 = 3 x 2 Since 2/2 = 1,

10 = 5 x 2 the 2s will divide out.

(We say they “cancel”

out.”)

We end up with 3

5

Simplifying Fractions

Here are the steps:List the prime factors of

both top and bottom.Cross out the ones

they share.Whatever is left is your

simplified fraction.

Simplifying Fractions

Another example:

30 = 2 x 3 x 5 = 2 x 3 = 6

35 = 5 x 7 = 7 7

And another example:

24 = 2 x 2 x 2 x 3 = 2 x 2 = 4

90 = 2 x 3 x 3 x 5 = 3 x 5 = 15

Another Way

Once you are used to this method, it may be faster for you to divide out common factors that are not prime.

For example, we can simplify 40/72 if werealize that 8 will go into both 40 and 72.The result is 5/9.

This way can be faster, but also is more open to mistakes, since people sometimes overlook common factors.

Simplifying Fractions

Can you put these fractions in their simplest forms?

5/20 = 8/10 = 18/27 = 12/14 =

Simplifying Fractions

Can you put these fractions in their simplest forms?

5/20 = 1/48/10 = 4/518/27 = 2/312/14 = 6/7

I’ve got it now!

Common Denominators

When you compare or add or subtract fractions, both fractions must have the same denominator.

This makes sense. If we are going to add pieces together, they need to be the same size, or we will not be able to add them or compare them.

Common Denominators

Of course, if we need to change a denominator, we will probably be changing OUT of the simplest form.

That’s OK!! We will change the fraction to a form we can use. If we need to change it back later, we can.

Common Denominators

Take two fractions like 3/4 and 2/3. Which is bigger?

We need to change the fourths and the thirds into pieces that are the same size.

We need the bottom numbers to be the same. No problem! We will just use the Equivalent Fraction Rule to make the denominators the same.

Common Denominators

3/4 and 2/3 Take a look at the bottom of each

fraction. We need the bottom numbers to be the same.

What number is a multiple of both 3 and 4? 12 is!

12

8

4

4

3

2

12

9

3

3

4

3 and

When the denominators are the same, we can tell that 3/4 is the bigger fraction!

Common Denominators

So we used the equivalent fraction rule to make the fraction have a denominator that was convenient for us.

Do not let the common denominator problem psych you out! Make it work for you! It’s like playing a game – you just have to know the rules!

Common Denominators

Here is another example: Compare 5/6 and 7/10 We need to change the 6 and the 10 to the

same number. 60 would work, but there is an easier way. Think about the multiples of 10: 10, 20, 30, 40. . . – 30 is the smallest number that is divisible by both 6 and 10. We want new denominator to be 30 for both fractions.

30

21

3

3

10

7

30

25

5

5

6

5 and

Common Denominator Questions

In the last problem, would I have been wrong to use 60? No, but it is always easier to use the smallest

number that will do the job. Why work with big numbers when you can use small ones?

Is there an easy way to find the smallest common denominator? Yes, I am glad you asked!!

Common Denominators This is one of those weird math tricks. It is almost like a

shortcut. We will use the last problem to demonstrate. (5/6 and 7/10)

Step 1: Make a fraction out of the two denominators. Step 2: Simplify that fraction & write it next to the first

one. Step 3: Multiply the top of one side by the bottom of the

other side. The answer you get is the smallest common denominator (sometimes called the least common multiple).

5

3

10

6

LCM = 30

Common Denominators

Another common denominator example: 3/4 and 5/8 Find the common denominator:

Get equivalent fractions so that both denominators are 8:

2

1

8

4

LCM = 8

8

5

8

5

8

6

4

3 and

This isn’t so bad

after all!

Common Denominators

Another common denominator example: 7/15 and 9/20 Find the common denominator:

Get equivalent fractions so that both denominators are 60:

4

3

20

15

LCM = 60

60

27

20

9

60

28

15

7 and

These fractions

are almost the same

size!

To Simplify or To Unsimplify?

Here’s the deal: If you need to add or subtract or compare fractions, you will need common denominators: This usually means Unsimplifying your fractions.

If you are finished working your problem, you always want to leave it in simplest form. That’s just good math manners!

Now What?

Now you are ready to begin operating with fractions:AddingSubtractingMultiplyingDividing

Now that I understand

them, I really am friends

with fractions!