How To Add And Subtract Fractions

Last Updated on February 15, 2024.

If we want to add & subtract fractions, we first need to ensure that we’ll have the same denominators.

When they are not the same, such as in \(\frac{5}{9}\) and \(\frac{1}{6}\), we must find the LCD (least common denominator).

Let’s begin with adding fractions. The key element when we add fractions is that they must be “like” fractions.

This lesson is provided by Onsego GED Prep.

Video Transcription

That is meaning that you must ensure that both of the addends (the numbers that are added) include common denominators. So \(\frac{5}{6}\) plus \(\frac{6}{7}\) may initially look a bit difficult, but you’ll see that if they’ll have a common denominator of \(42\), this will be an easy addition question. Let’s begin.

Adding Fractions with the Same Denominators

Let’s get started with an easy one. When we have addends that have the same denominators, the only thing we need to is adding up both numerators (the numbers on the top) and simplify the answer. Let’s go and try a few.

\(\frac{1}{13}\) plus \(\frac{6}{13}\) equals?

• First, create common denominators. Both addends already have the same \((13)\), so we don’t need to do anything.

• Then, add up the two numerators: \(1\)  plus \(\,6 = 7\)

• Then, write the sum of the two numerators on the top (above the denominator): \(\frac{7}{13}\)

• Now simplify the fraction. Well, there’s no way to simplify \(\frac{7}{13}\). You’re done.

The answer is: \(\frac{1}{13}\) plus \(\frac{6}{13}\) is \(\frac{7}{13}\),

\(\frac{5}{9}\) plus \(\frac{1}{9}\) equals?

• First, create the common denominator. The denominators are already the same, so we do nothing.

• Then, add the numerators: \(5\) plus \(\,1 = 6\)

• Then, write the sum of those numerators above their common denominator: \(\frac{6}{9}\).

• Then simplify: well, \(6\) and \(9\) are having a common factor: \(3\). Now, when you divide both the numerator and the denominator by \(3\), we get \(\frac{2}{3}\).

The answer is: \(\frac{5}{9} \) plus\(\frac{1}{9} = \frac{6}{9} \) which equals \(\frac{2}{3} \)

How to create Common Denominators.

This is a little more advanced. How is it when we add “unlike” fractions? Then we don’t have a common denominator. In earlier lessons, we’ve been looking at creating equivalent fractions. Here, we’ll use that same process.

\(\frac{1}{7} \) plus \(\frac{1}{3} \) equals?

• First, create a common denominator. We have \(a\,7\) and \(a\, 3\). These numbers don’t have common factors. So let’s just do some multiplication so we can create two equivalent fractions. Do you remember how we earlier multiplied by equivalents of one \((1)\)? It went just like this

 

\(\frac{1}{7} = \frac{1}{7} * 1\) \(= \frac{1}{7} * \frac{3}{3}\) \(= \frac{(1*3)}{(7*3)} = \frac{3}{21}\)

 

\(\frac{1}{3} = \frac{1}{3} * 1\) \(= \frac{1}{3} * \frac{7}{7}\) \(= \frac{1}{7}*\frac{3}{7} = \frac{7}{21}\)

 

So now we have our common denominator, \(21\). So now, we can rewrite our problem as \(\frac{3}{21}\) plus \(\frac{7}{21} \) equals?

• Then, add the numerators: \(3\,\) plus \(\, 7 = 10\)

• Then, write the sum of the two numerators above their common denominator: \(\frac{10}{21}\)

• Then, simplify. Well, there is no way we can simplify \(\frac{10}{21}\), so we’re done.

 

The answer is: \(\frac{1}{7}\) plus \(\,\frac{1}{3} = \frac{10}{21}\)

Adding Mixed Numbers.
So now, we’ve got common denominators as well as unlike fractions covered and under control. Now, let’s take a look at some example that has mixed numbers before we continue. The first example covers mixed numbers that come with common denominators.

\(2\, \frac{2}{9}\) plus \(4\, \frac{3}{9}\) equals?

• First, check for a common denominator. The two are like fractions that have a denominator of nine \((9)\). So we do nothing.

• First, add the numerators of the two fraction: \(2\) plus \(\, 3 = 5\)

• Then, write the sum of the two numerators above our common denominator: \( \frac{5}{9}\)

• Then, add the two whole numbers as well: \(2\) plus \(\,4 = 6\)

• Then, write the mixed number. It is \( 6\,\frac{5}{9}\)

• Then, simplify. Well, \(\frac{5}{9}\) cannot be simplified. We’re done.

The answer is: \(2\,\frac{2}{9}\) plus \(\, 4\,\frac{3}{9} = 6\,\frac{5}{9}\)

What should we do if we would be winding up with an answer that has an improper fraction? Well, then we must simplify that improper fraction, after which we add the whole numbers. Here, I’ll use an example that’s like the last one. We only made the first addend somewhat bigger.

\( 2\,\frac{7}{9}\) plus \(4\,\frac{3}{9}\) equals?

• First, check for the common denominators. Well, these are like fractions that both have a denominator of nine \((9)\), so we do nothing.

• Then, add both numerators from the fraction: \(7\)plus \(\,3 = 10\)

• Then, write the sum of the two numerators above the common denominator, which gives us \(\frac{10}{9}\)

• Then, add the two whole numbers: \(2 \) plus \( \,4 = 6\)

• Then, write out our new mixed number, which is \(6\,\frac{10}{9}\)

• Then, simplify. This example includes an improper fraction that we need to simplify. We must use division to come up with a new fraction. And \(10 \div 9 = 1r1\). So our new mixed number is \(1\,\frac{1}{9}\). Now we’ll have to add our new whole number to our original one of \(6\). This entire process will go like this:

\(6 \frac{10}{9} = 6 + \frac{10}{9}\) \(= 6 + 1\,\frac{1}{9}\) \(= 6 + 1 + \frac{1}{9} = 7\, \frac{1}{9}\)

The answer is:

\(2\, \frac{7}{9}\) plus \(4\,\frac{3}{9}\) \(= 7\, \frac{1}{9}\)

Now, we can put this all together with the following example that comes with unlike fractions. Here, we’ll have to make common denominators and simplify the improper fraction.

\(2\, \frac{5}{8}\) plus \(5\, \frac{3}{4} =?\)

First, common denominators, and we begin with the fractions. You’ve got \(4\) and \(8\). These have the common factor of \(4\) (four), so we only have to deal with the \(\frac{3}{4}\) fraction.

\(\frac{3}{4} =\frac{3}{4} * 1\) \(= \frac{3}{4} *\frac{2}{2}\) \(= \frac{(3*2)}{(4*2)} = \frac{6}{8}\)

Then, add the two numerators: \(5\)plus \(\, 6 = 11\)

Then, rewrite the fraction and place the sum of the numerators on top, above the common denominator: \(\frac{11}{8}\)

Then, add the whole numbers: \(2\) plus \( \,5 = 7\)

Our new mixed number is \(7\, \frac{11}{8}\)

Then simplify. We have an improper fraction. Begin with dividing: \(11 \div 8 = 1r3\). So our new mixed number is \(1\, \frac{3}{8},\) and we need to add this new mixed number to our original of \(7\).

\(7 \,\frac{11}{8}= 7 + \frac{11}{8}\) \(= 7 + 1\, \frac{3}{8}\) \(= 7 + 1 + \frac{3}{8}\) \(= 8\, \frac{3}{8}\)

When you got it and want to keep going, move on to subtracting fractions. This is pretty close to addition, so you’ll be doing fine.