Learning More About Fractions
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Learning More About Fractions |
Adding Fractions
It is very easy to add two fractions that have the same denominator. You simply add the two numerators together to get the new numerator and keep the same denominator. Here are a few examples:
2/6 + 3/6 = 5/6
3/8 + 1/8 = 4/8 = 1/2In the second example above, we reduced the fraction 4/8 to the equivalent fraction 1/2 by dividing both numbers by 4. You can add more than two fractions together as shown in the following examples:
1/8 + 2/8 + 3/8 + 1/8 = 7/8
1/3 + 2/3 + 5/3 + 8/3 = 16/3 = 5 1/3The answer 16/3 reduces to 5 1/3 as follows:
16/3 = 3/3 + 3/3 + 3/3 + 3/3 + 3/3 + 1/3 = 1 + 1 + 1 + 1 + 1 + 1/3 = 5 1/3
What happens if the fractions to be added have denominators that are not the same? You have to make the denominators the same. Here is a simple example:
1/2 + 1/6
If you take the fraction 1/2 and multiply both numbers by 3 to get the equivalent fraction 3/6 you end up with a fraction that has the same denominator as the fraction 1/6. Now this becomes an easy problem to solve.
1/2 + 1/6 = 3/6 + 1/6 = 4/6 = 2/3
In this case, after the two fractions have been added we can simplify the answer 4/6 to 2/3 by dividing both numbers by 2.
Using the Least Common Multiple - Let's try another example where the denominators are not the same:
1/3 + 1/5
In this case the fraction 1/3 cannot be converted to an equivalent fraction with a denominator of 5. There is a special way to convert both fractions to have the same denominator and it is called finding the Least Common Multiple or LCM. Here is the process to solve the equation 1/3 + 1/5 using the LCM method.
- Find the multiples of both denominators:
- Choose the same multiple for both denominators (common multiple); in this case it is 15. If you have several multiples that are the same, you should always choose the smallest (least common multiple).
- Convert both fractions to use a denominator of 15. To convert 1/3, if you look at the table, we need to multiply both numbers by 5, which gives us 5/15. To convert 1/5 we need to multiply both numbers by 3 which gives us 3/15.
- Now we can add the two converted fractions
1/3 + 1/5 = 5/15 + 3/15 = 8/15So the final answer is 8/15.
You can add more than two fractions with different denominators by finding the Least Common Multiple (LCM) for all the denominators. Here is an example:
1/2 + 2/3 + 3/5
First, let's find the LCM:
16 32 48 80 As you can see in the table, the smallest common multiple for all three denominators is 30. We now use the multiples in the first row to convert each of the three fractions so they all have the same denominator of 30. We use 15 for the denominator 2, 10 for the denominator 3, and 6 for the denominator 5.
1/2 = 15/30
2/3 = 20/30
3/5 = 18/301/2 + 2/3 + 3/5 = 15/30 + 20/30 + 18/30 = 53/30
The improper fraction 53/30 can be written as the mixed fraction 1 23/30.
Subtracting Fractions
Just as in addition, it is easy to subtract two fraction that have the same denominator but when the denominators are different we must use the LCM method. Here is an example where the denominators are the same:
5/7 - 2/7 = 3/7
We keep the same denominator, 7, and then subtract the second numerator, 2, from the first numerator, 5, to get the answer. Our next example requires the LCM method because the two denominators are different:
5/7 - 2/3
First we need to find the LCM:
The LCM is 21. Now we convert each fraction so that the denominator is 21.
5/7 x 3/3 = 15/21
2/3 x 7/7 = 14/215/7 - 2/3 = 15/21 - 14/21 = 1/21
Once the fractions are converted it is very simple to get the answer of 1/21.
Adding and Subtracting Mixed Fractions
We explained mixed fractions in the previous page. Now we will add and subtract mixed fractions. Here is an example:
1 1/3 + 3 2/3
If you remember, the value 1 1/3 can be written as 1 + 1/3, so the above equation can be written as:
1 + 1/3 + 3 + 2/3
By using the associative rule for addition, this equation can be rewritten as:
1 + 3 + 1/3 + 2/3
which can be simplified into:
4 + 3/3 = 4 + 1 = 5
Let's try using the same method when subtracting mixed fractions:
7 1/2 - 5 3/4
As we did before, this equation can be written as:
7 + 1/2 - 5 - 3/4
which can be rewritten as:
7 - 5 + 1/2 - 3/4
We can convert 1/2 to 2/4 and the equation becomes:
7 - 5 + 2/4 - 3/4
You will notice that the fraction 3/4 that we are subtracting is larger than the fraction, 2/4. This makes it complicated so we will use a different method to solve this equation. We start with the same problem and first convert each mixed fraction into its equivalent improper fraction.
7 1/2 - 5 3/4
First convert the 7 and the 5 into the corresponding fractions:
14/2 + 1/2 - (20/4 + 3/4)
Then combine the halves and combine the quarters:
15/2 - 23/4
Next convert the 15/2 to 30/4 by multiplying by 2/2 to get a common denominator:
30/4 - 23/4 = 7/4 = 1 3/4
Finally simplify the answer 7/4 to the mixed fractions 1 3/4.
Another method of solving the same subtraction problem is to convert the 7 1/2 into an equivalent fraction:
7 1/2 = 6 + 1 +1/2 = 6 + 2/2 + 1/2 = 6 3/2
The resulting fraction 6 3/2 is equal to 6 6/4 which makes it possible to subtract 3/4 from 6/4. The equation can now be written as:
6 3/2 - 5 3/4 = 6 6/4 - 5 3/4 = (6 + 6/4) - (5 + 3/4)
Next, combine the integers and the fractions as follows:
6 - 5 + 6/4 - 3/4 = 1 + 3/4 = 1 3/4
Fraction Complements
Sometimes it is useful to know the complement of a fraction. For example, if there are 22 students in a class and 5 students have finished their reading assignment, how much of the class still needs to work on their reading assignment? For this example the unit is the whole class which is 22 students. So the fraction of students who have finished is 5/22. How do we determine the fraction of the class which has not yet finished? Fraction complements can be used to solve this problem.
complement -- The complement of a fraction is the value that needs to be added to it to get the value of 1. It can be calculated by subtracting the fraction from 1. Here is the equation:
1 - fraction = complement
To solve the previous problem we calculate the fraction complement of 5/22.
1 - 5/22 = 22/22 - 5/22 = 17/22
Here are a few more examples of calculating complements:
Here is a picture using integer bars to show the first example, the complement of 4/7:
The black bar is the size 7 bar which represents 1 unit. We then use 4 white bars in the top part of the picture to represent 4/7. Then, in the bottom part of the picture, we add as many white bars as needed to equal 1 unit in order to find that the complement of 4/7, which is 3/7.
Exercises -- The following set of exercises includes problems which use fraction addition, fraction subtraction, and finding the complement. Simplify the answers when possible. You can use the integer bars to help solve these problems.
<- Click on this image to start the applet
Once you have finished all of the problems, check your answers.
Learning the basics about fractions
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Learning more about fractions
Last Updated: Tuesday, 15-Jul-2003 23:45:55 GMT
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