Fractions
FRACTIONS-GETTING STARTED WITH BASICS, MIXED FRACTIONS, IMPROPER FRACTIONS, & REDUCING FRACTIONS TO LOWEST TERMS

Getting started-fraction basics
Changing an improper fraction into a mixed fraction
Changing a mixed fraction into an improper fraction
Changing a fraction to Lowest terms
Changing a fraction's denominator
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Note: Click on any of the word(s) in red type to see a definition.



A recipe calls for 1/8 teaspoon salt
You order 1 1/2 lbs. of bologna at a deli
Your favorite teams has a record of 25 wins out of 32 gams giving them a record of 25/32
The odds of winning a lottery are 1 in 1000 or 1/1000
These are four examples of how fractions are used in everyday life.


GETTING STARTED-FRACTION BASICS

As you probably know a fraction is considered to be a part of a whole. For example in the illustration below I have a circle cut into 8 equal pieces. One wedge is 1 part of 8 or 1/8.
The entire circle could be expressed by the fraction 8/8.

Similarly, if I have the rectangle shown below cut up into 6 equal pieces, two of the pieces would represent 2/6 of the total rectangle. The entire rectangle would be represented as the fraction 6/6.


Your could also read fractions as an extended division problem. 1/8 could be read as 1 divided by 8. 2/6 would be 2 divided by 6. Similarly 6/6 would be 6 divided by 6 (which is equal to 1), thus we could write 6/6=1.
8/8 is 8 divided by 8, which is also equal to 1 (in fact any number divided by itself equals 1), so 8/8=1.

The top of a fraction is called the numerator. The bottom is called the denominator.

Examples

In the fraction 2/3 2 is the numerator, 3 is the denominator.
In the fraction 5/7 5 is the numerator, 7 is the denominator.
In the fraction 11/23 11 is the numerator, 23 is the denominator.


CHANGING AN IMPROPER FRACTION INTO A MIXED FRACTION

A mixed fraction has a whole number part and a fraction part
Examples

In 3 1/2(read as three and one-half), 3 is the whole number part, and 1/2 is the fraction part.
In 5 7/8 (read as five and seven-eighths), 5 is the whole number part, and 7/8 is the fraction part.
In 8 1/9 (read as eight and one-ninth), 8 is the whole number part, and 1/9 is the fraction part.
An improper fraction is one in which the numerator is higher than the denominator.
Examples of mixed fractions include 35/18, 3/2, and 7/3
In order to change an improper fraction into a mixed fraction, first divide the denominator into the numerator. The quotient you receive will be the whole number part. Any remainder you receive will be your new numerator.
I'll work out a few examples of this in order to make it clear, then it's your turn.
Example 1
35/18. 18 divides 35 one time with a remainder of 17. 1 will be the whole number part, and 17 will be our new numerator. Thus 35/18=1 17/18.

7/3. 3 divides 7 two times with a remainder of 1. 2 will be the whole number part, and 1 will be our new numerator. Thus 7/3=2 1/3.


Now it's your turn. In each of these exercises you'll be given a problem and asked to place your answer in the text box. Then, click on the submit button, and you'll be told if your answer was correct. If your answer was not correct, you'll be given instructions on whether you would like to try another time, or have the answer displayed on the screen along with a brief explanation.
There are a total of 5 exercises. After the first four exercises you will be given the option of going on to the next section of this unit, going back to the top of this page, or continuing on to the next problem. I strongly suggest working out at least 2 exercises.
Exercise 1
Change 23/4 to a mixed fraction
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Exercise 2
Change 3/2 to a mixed fraction
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Exercise 3
Change 24/10 to a mixed fraction
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Exercise 4
Change 18/5 to a mixed fraction
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Exercise 5
Change 37/7 to a mixed fraction
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CHANGING A MIXED FRACTION INTO AN IMPROPER FRACTION

Now we're going to do the same thing as above in reverse (no. we're not going to actually do the same problems while walking backwards).
To create an improper fraction from a mixed fraction, multiply the denominator by the whole number, and add this product to the Numerator. This will be the new numerator. This will become more clear after doing a few examples. Again, I'll show you the first two, then you're on your own.
Example 1
8 1/9
Okay 8*9=72 and 72+1=73 so 8 1/9=73/9

Example 2
5 7/8
Okay 5*8=40 and 40+7=47 so 5 7/8=47/8
Now, you try it.
Exercise 1
Change 1 4/7 to an improper fraction
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Exercise 2
Change 2 3/8 to an improper fraction
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Exercise 3
Change 3 1/3 to an improper fraction
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Exercise 4
Change 1 2/13 to an improper fraction
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Exercise 5
Change 4 3/5 to an improper fraction
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REDUCING FRACTIONS TO LOWEST TERMS

In the whole numbers unit, we discussed factors and factoring. You will need to know that information in this section. If you would like to go back and review that unit click here.

A fraction is in lowest terms if there is no factor that divides both the numerator and the denominator.

10/23 is in lowest terms since there is no factor that divides both 10 and 23.
However, 10/24 is not in lowest terms since both 10 and 24 can both be divided by 2.
In order to reduce a fraction into lowest terms, first we need to find the GCF (Greatest Common Factor) (also known as the Greatest Common Divisor or GCD).
The GCF is the largest factor that divides both the numerator and the denominator.
To find the GCF we must first know the factors of the numerator and the factors of the denominator. By comparing these two sets of factors, we'll determine the GCF.
For example let's examine the fraction 28/40.
Using what we learned in the whole number factor section, the factors of 28 are {1,2,4,7,14,28}.
The factors of 40 are {1,2,4,5,8,10,20,40}.
Now let's look at the numbers the two sets have in common {1,2,4} the largest of which is 4.
Therefore the GCF of 28 and 40 is 4.
Okay, now we know the GCF, what now?
Well let's rewrite the fraction as follows:
28/40=(4*7)/(4*10)=(4/4)*(7/10).
Now 4/4=1, so they cancel each other out.
Therefore 28/40=7/10.

For our second example let's examine the fraction 2 12/36.
For the moment let's forget about the whole number part, 2.
Using what we learned in the whole number factor section, the factors of 12 are {1,2,3,4,6,12}.
The factors of 36 are {1,2,3,4,6,9,12,13,36}.
Now let's look at the numbers the two sets have in common (1,2,3,4,6,12} the largest of which is 12.
Therefore the GCF of 12 and 36 is 12.
let's rewrite the fraction as follows:
12/36=(12*1)/(12*3)=(12/12)*(1/3).
Now 12/12=1, so they cancel each other out.
Therefore 12/36=1/3 and it follows that 2 12/36=2 1/3.
Okay now it's your turn at bat
Exercise 1
Reduce the fraction 12/16
Step 1: Enter what you think is the GCF
Step 2: Now, using the GCF reduce 12/16 to lowest terms.

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Exercise 2
Reduce the fraction 44/46 to lowest terms
Step 1: Enter what you think is the GCF
Step 2: Now, using the GCF reduce 44/46 to lowest terms.

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Exercise 3
Reduce the fraction 1 12/60 to lowest terms
Step 1: Enter what you think is the GCF
Step 2: Now, using the GCF reduce 1 12/60 to lowest terms.

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Exercise 4
Reduce the fraction 3 12/28 to lowest terms
Step 1: Enter what you think is the GCF
Step 2: Now, using the GCF reduce 3 12/28 to lowest terms.

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Exercise 5
Reduce the fraction 1 12/23 to lowest terms
Step 1: Enter what you think is the GCF
Step 2: Now, using the GCF reduce 1 12/23 to lowest terms.

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CHANGING A FRACTION'S DENOMINATOR

You have a recipe that calls for 3/4 c. of sugar, but you can't seem to find your 1/4 measuring cup. All you have on hand is the 1/8 measuring cup. What should you do? Note: Going to the store to get new measuring cups is not the answer.
We'll return to this question at the end of this section and give it an answer.

Okay now we're about to go backwards again (getting dizzy?), then we'll return to our cooking problem.

You have a fraction and want to change the denominator. into one of it's multiples.
Well first divide the new denominator (the one you want to get to) by the current denominator and then multiply both the numerator and denominator by this factor.

For example let's say I wanted to go from 3/7 to a fraction with a denominator of 28.
First I would take the denominator I wanted to get to (28) and divide it by my current denominator (7). 28/7=4.
Now let's write the fraction as follows:
3/7=(3/7)*(4/4) (note:4/4=1 so we're really just multiplying by 1 and thus not really changing our original fraction.) Continuing we have 3/7=(3/7)*(4/4)=(3*4)/(7*4)=12/28.
3/7=12/28


For our second example let's say I wanted to go from 1 2/5 to a fraction with a denominator of 25.
Until we get to the end, let's forget the whole number part.
First take the denominator we want to get to (25) and divide it by the current denominator (5). 25/5=5.
Now write the fraction as follows:
2/5=(2/5)*(5/5) (note:5/5=1 so we're really just multiplying by 1 and thus not really changing our original fraction)
continuing we have 2/5=(2/5)*(5/5)=(2*5)/(5*5)=10/25.
2/5=10/25 and it follows that 1 2/5=1 10/25
Now here's your chance to show off
Exercise1
Change the fraction 3/8 to one with a denominator of 32
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Exercise2
Change the fraction 4/9 to one with a denominator of 72
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Exercise3
Change the fraction 7/12 to one with a denominator of 36
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Exercise4
Change the fraction 2 3/4 to one with a denominator of 44
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Exercise5
Change the fraction 3 1/3 to one with a denominator of 90
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Now we'll end this section by going back to the question posed at the beginning of this section.
You have a recipe that calls for 3/4 c. of sugar, but you can't seem to find your 1/4 measuring cup. All you have on hand is the 1/8 measuring cup. What should you do?
Well 8 (the denominator of the 1/8 measure) is a multiple of 4 (the denominator of the missing 1/4 c. measure).
8/4=2
Thus (3/4)*(2/2)=6/8.
3/4=6/8
Use 6 "1/8 measures" to get the same amount as 3 "1/4 measures"

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