When two fractions have the same denominator, adding them is simple. We simply add the numerators of each fraction together with the same denominator.
Example
Add frac 2 3 to frac 5 3 to get frac 2 + 5 3 to get frac 7 3.
Fractions with Unlike Denominators Added
Let’s say we want to add frac1 to frac1 to get frac1+frac1
Because the denominators of the two fractions are not equal, we cannot simply add the numerators and denominators to obtain the answer.
Finding a common (same) denominator for both fractions that will allow us to add them correctly is the trick.
What number can we pick where 2 and 4 will split equally?
Since we always want to choose the smallest such number, our best option would be 4.
Any number that can be divided by both 2 and 4 is acceptable, such as 8, 12, or 16, and it will work. We simply save time by selecting the smallest one, as we will see in a moment.
Our task is to convert frac12 into an equivalent fraction with a 4 in the denominator because frac14 already has a 4 in the denominator.
With a 4 in the denominator, frac12 becomes frac24, which is the equivalent fraction.
We can now combine them by adding them together.
\frac{1}{4}+\frac{1}{2}=\frac{1}{4}+\frac{2}{4}=\frac{1+2}{4}=\frac{3}{4}
Multiplication by 1 and Common Denominators equal a/a
A common denominator for both fractions—preferably the Least Common Denominator—and equivalent fraction conversion are required to add two fractions correctly.
The last example's common denominator was relatively simple to find because 4 is a multiple of 2.
Here, we'll go over how we can actually find a common denominator.
For instance, add "frac 2 3 " and "frac 3 "
To solve this problem, multiply each fraction by a form of 1 that has an equivalent value and will give all the fractions a common denominator.
What is the first integer into which the numerators 3 and 4 divide equally?
In this instance, it is simply 3 x 4 = 12.
How can we convert each fraction to its equivalent form with a 12 as the denominator?
So to speak, we multiply each fraction by 1.
\frac{4}{4}\times\frac{2}{3}=\frac{8}{12} \sand
\frac{3}{3}\times\frac{3}{4}=\frac{9}{12}
Take note that frac 3 3 and frac 4 4 both equal 1.
Now that we've prepared ourselves, we can add these fractions.
\frac{2}{3}+\frac{3}{4}=(\frac{4}{4})
(\frac{2}{3})
+(\frac{3}{4})
(\frac{3}{3})
=\frac{8}{12}+\frac{9}{12}=\frac{8+9}{12}=\frac{17}{12}
In order to find a common denominator, we multiplied both fractions by a form of 1: 1=fracaa for the first and 1=fracbb for the second.
Example \sAdd
\frac{5}{6}+\frac{3}{8}
The first number that 6 and 8 divide evenly into is: \frac{24}{6}=4 and frac 24 8 = 3.
Divide each fraction by the equivalent form of 1 to get the denominator, which is 24. For frac5-6, 1=frac4-4, and for frac3-8, 1=frac3-3.
\frac{5}{6}+\frac{3}{8}=(\frac{4}{4})(\frac{5}{6})+(\frac{3}{8})(\frac{3}{3})
=\frac{20}{24}+\frac{9}{24}=\frac{20+9}{4}=\frac{29}{24}
Example
Add
\frac{2}{5}+\frac{1}{6} \s\frac{2}{5}+\frac{1}{6}=(\frac{6}{6})
(\frac{2}{5})+(\frac{1}{6})(\frac{5}{5})=\frac{12}{30}+\frac{5}{30}=\frac{17}{30}
3 or more fractions added
The same procedure is used to add three or more fractions, but there are now three or more denominators to take into consideration when determining a common denominator.
Example \sAdd
\frac{7}{6}+\frac{4}{3}+\frac{5}{4}+\frac{3}{2}
2, 3, 4, and 6 all have a common denominator. It is 12.
\frac{7}{6}+\frac{4}{3}+\frac{5}{4}+\frac{3}{2}= \s(\frac{2}{2})
(\frac{7}{6})+(\frac{4}{4})
(\frac{4}{3})+(\frac{5}{4})
(\frac{3}{3})+(\frac{3}{2})
(\frac{6}{6})
= \s\frac{14}{12}+\frac{16}{12}+\frac{15}{12}+\frac{18}{12}=\frac{63}{12}=\frac{21}{4}
Quick and simple Formula \s\frac{a}{b}+\frac{c}{d}=\frac{ad+bc}{bd}
