Subtracting Fractions

How to Subtract Fractions?

Subtracting fractions involves three simple steps –
  1. If the fractions have different denominators (bottom numbers), rename them so the denominators become equal.
  2. Subtract the numerators (top numbers) to get the numerator part of the answer. Copy the denominator.
  3. Simplify the resulting fraction, if possible.

Now let’s do some subtractions using these steps. I have divided the examples into different categories so it’s easier to understand.

Subtracting Fractions with the Same Denominator

This is the easiest. When the denominators are the same, the first step mentioned above (making the denominators equal) is already taken care of. That makes the subtraction really easy.

Example
Simplify :
79−29\frac{7}{9} - \frac{2}{9}

Solution

Step 1.  The denominators are already the same. So we move to step 2.

Step 2.  Subtract the top numbers and copy the denominator.

79−29=7−29=59\begin{align*} \frac{7}{9} - \frac{2}{9} \hspace{0.2em} &= \hspace{0.2em} \frac{7 - 2}{9} \\[1.5em] \hspace{0.2em} &= \hspace{0.2em} \frac{5}{9} \end{align*}

Step 3.  We can’t simplify the difference any further. It’s already in its lowest terms. So, that’s our answer.

And here’s the solution in pictures.

Subtracting fractions is simplest when the denominators are equal. In the image - taking away 2 slices from 7 slices of pizza.
Same denominator. Equal slices. Simple subtraction.
Example
Simplify :
512−112\frac{5}{12} - \frac{1}{12}

Solution

512−112=5−112=412=13\begin{align*} \frac{5}{12} - \frac{1}{12} \hspace{0.2em} &= \hspace{0.2em} \frac{5 - 1}{12} \\[1.5em] \hspace{0.2em} &= \hspace{0.2em} \frac{4}{12} \hspace{0.25em} = \hspace{0.25em} \frac{1}{3} \end{align*}

The only thing different in this example was that in the last step, we had to simplify the fraction – dividing the top and bottom by their largest common factor, 4.

Subtracting Fractions with Different Denominators

Very often, the fractions we are working with do not have the same bottom number. So, let’s see how to subtract fractions with different denominators.

Example
Simplify :
12−13\frac{1}{2} - \frac{1}{3}

Solution

Step 1.  When the fractions have different denominators, we must first make them equal.

For that, we need a common multiple of the denominators (preferably the Least Common Multiple or LCM). The LCM of 2\hspace{0.2em} 2 \hspace{0.2em} and 3\hspace{0.2em} 3 \hspace{0.2em} is 6\hspace{0.2em} 6 \hspace{0.2em}.

Now, we can make the two denominators equal to six by multiplying the first by 3\hspace{0.2em} 3 \hspace{0.2em} and the second by 2\hspace{0.2em} 2 \hspace{0.2em}.

But remember, when you multiply the denominator by any number, you must also multiply the numerator by the same number. Otherwise, the value of the fraction will change.

So, here's what we do.

12−13=1×32×3−1×23×2=36−26\begin{align*} {\color{Red} \frac{1}{2}} - \frac{1}{3} \hspace{0.2em} &= \hspace{0.2em} {\color{Red} \frac{1 \times 3}{2 \times 3}} - \frac{1 \times 2}{3 \times 2} \\[1.5em] \hspace{0.2em} &= \hspace{0.2em} {\color{Red} \frac{3}{6}} - \frac{2}{6} \end{align*}

Step 2.  Now that the denominators are equal, we can subtract the top numbers.

36−26=3−26=16\begin{align*} \frac{3}{6} - \frac{2}{6} \hspace{0.2em} &= \hspace{0.2em} \frac{3-2}{6} \\[1em] \hspace{0.2em} &= \hspace{0.2em} \frac{1}{6} \end{align*}

Step 3.  We can simplify it any further. So, that’s our answer.

Why Make the Denominators Equal?

Denominators in a fraction tell us how many parts make up the whole.

So when fractions have different denominators, they are referring to parts unequal in size. See how each slice in the first pizza is bigger than that in the other?

When denominators are different, it implies the slices are of differently-sized. Before we can do the subtraction, we must make them equal.
Different denominators – Unequal Slices

By making the denominators equal, we resize the slices so they become equal (without changing the quantities of pizza). And then, we can go ahead and subtract them easily.

By making the denominators equal, we resize the slices to make them equal.
Resizing the slices so they are all equal
Example
Simplify :
13+16\frac{1}{3} + \frac{1}{6}

Solution

Again, we make the denominators equal before subtracting the numerators.

The LCM of 3\hspace{0.2em} 3 \hspace{0.2em} and 6\hspace{0.2em} 6 \hspace{0.2em} is 6\hspace{0.2em} 6 \hspace{0.2em}. So we multiply the first fraction by 2\hspace{0.2em} 2 \hspace{0.2em} (top and bottom) and leave the other as it is (its denominator is already 6\hspace{0.2em} 6 \hspace{0.2em}).

13−16=1×23×2−16=26−16=2−16=16\begin{align*} {\color{Red} \frac{1}{3}} - \frac{1}{6} \hspace{0.2em} &= \hspace{0.2em} {\color{Red} \frac{1 \times 2}{3 \times 2}} - \frac{1}{6} \\[1.5em] \hspace{0.2em} &= \hspace{0.2em} {\color{Red} \frac{2}{6}} - \frac{1}{6} \\[1.5em] \hspace{0.2em} &= \hspace{0.2em} \frac{ {\color{Red} 2} - 1}{6} \hspace{0.25em} = \hspace{0.25em} \frac{1}{6} \end{align*}

Subtractions with Whole and Mixed Numbers

When subtracting fractions, if you come across whole numbers or mixed numbers, generally the simplest way is to write them as simple fractions and then proceed with the subtraction as we saw earlier.

Example
Simplify :
5−135 - \frac{1}{3}

Solution

We begin by writing the whole number, 5, as a fraction with one as its denominator.

5−13=51−13 {\color{Red} 5} - \frac{1}{3} = {\color{Red} \frac{5}{1}} - \frac{1}{3}
Now it’s two fractions with different denominators. We've already learned how to deal with them.

51−13=5×31×3−13=153−13=15−13=143=423\begin{align*} {\color{Red} \frac{5}{1}} - \frac{1}{3} \hspace{0.2em} &= \hspace{0.2em} {\color{Red} \frac{5 \times 3}{1 \times 3}} - \frac{1}{3} \\[1.5em] \hspace{0.2em} &= \hspace{0.2em} {\color{Red} \frac{15}{3}} - \frac{1}{3} \\[1.5em] \hspace{0.2em} &= \hspace{0.2em} \frac{ {\color{Red} 15} - 1}{3} \\[1.5em] \hspace{0.2em} &= \hspace{0.2em} \frac{14}{3} \hspace{0.25em}=\hspace{0.25em} 4 \frac{2}{3} \end{align*}

See how I rewrote the top-heavy (improper) fraction as a mixed number in the last step?

It wasn’t necessary, but unless the question uses improper fractions, it’s preferred that you don’t have them in your final answer either. So, convert them into mixed numbers (mixed numbers are easier to visualize).

Example
Simplify :
115−341 \frac{1}{5} - \frac{3}{4}

Solution

Here we have a mixed number. No problem. Just as we did in the previous example, we start by converting the mixed number into an improper fraction.

115−34=65−34 {\color{Red} 1\frac{1}{5}} - \frac{3}{4} = {\color{Red} \frac{6}{5}} - \frac{3}{4}

From here, it's business as usual.

65−34=6×45×4−3×54×5=2420−1520=24−1520=920\begin{align*} \frac{6}{5} - \frac{3}{4} \hspace{0.15em} \hspace{0.2em} &= \hspace{0.2em} \hspace{0.15em} \frac{6 \times 4}{5 \times 4} - \frac{3 \times 5}{4 \times 5} \\[1.5em] \hspace{0.2em} &= \hspace{0.2em} \hspace{0.15em} \frac{24}{20} - \frac{15}{20} \\[1.5em] \hspace{0.2em} &= \hspace{0.2em} \hspace{0.15em} \frac{24 - 15}{20} \\[1.5em] \hspace{0.2em} &= \hspace{0.2em} \hspace{0.15em} \frac{9}{20} \end{align*}
Example
Simplify :
4−2124 - 2 \frac{1}{2}

Solution

This example brings together a whole number and a mixed number. No worries. We just convert them both into simple fractions.

4−212=41−2124 - {\color{Red} 2 \frac{1}{2}} = \frac{4}{1} - {\color{Red} 2 \frac{1}{2}}

And again, there are two simple fractions to deal with.

41−52=6×25×2−34=82−52=8−52=32=112\begin{align*} \frac{4}{1} - \frac{5}{2} \hspace{0.15em} \hspace{0.2em} &= \hspace{0.2em} \hspace{0.15em} \frac{6 \times 2}{5 \times 2} - \frac{3}{4} \\[1.5em] \hspace{0.2em} &= \hspace{0.2em} \hspace{0.15em} \frac{8}{2} - \frac{5}{2} \\[1.5em] \hspace{0.2em} &= \hspace{0.2em} \hspace{0.15em} \frac{8 - 5}{2} \\[1.5em] \hspace{0.2em} &= \hspace{0.2em} \hspace{0.15em} \frac{3}{2} \hspace{0.15em} = \hspace{0.15em} 1\frac{1}{2} \end{align*}

So that was all about subtracting fractions. Give yourself some practice, and you will master it in no time. Until next time.