Adding and Subtracting Fractions with Unlike Denominators — Process

Adding and Subtracting Fractions

Fractions are a pain in the butt to add or subtract. They're superficial and picky. They only want to combine with other fractions that have the same denominator. Drama queens.

Same Denominator

When fractions have the same denominator, it is very easy to add and subtract. Just leave the denominator alone and add or subtract the numerators.

Let's try this with the two fractions in the background of the photo. What is: $$\frac32-\frac12$$

Since they have the same denominator, $2$, we can subtract the numerators. $$\frac32-\frac12=\frac{3-1}{2}=\frac22=1$$

Different Denominators

Now what about fractions with a different denominators? We need to alter one or more of the fractions so that they have a common denominator. Once they share a common denominator, we can add or subtract like we did above, leaving the denominator alone and focusing on the numerator.

Let's use the fractions in the photo. $$\frac35+\frac32$$ They have different denominators, $5$ and $2$. A quick way to find a common denominator is to multiply the denominators together. So, a common denominator for $2$ and $5$ is $2\times5=10$. This $10$ is also a common multiple of $2$ and $5$.

Now, we multiply or divide each fraction's top and bottom by a number that turns their denominator into the common denominator, $10$. We will produce equivalent fractions. Since the denominator changes, the numerator must also. Read on for more info.

Solution

Here is one way to solve this problem. There are faster methods, though!

First, combine $-\frac56$ and $\frac12:$ $$\begin{align} \frac85-\frac56+\frac12 &= \frac85- \frac22 \cdot\frac56+\frac66 \cdot \frac12\\ &= \frac85- \frac{2\cdot 5}{2 \cdot 6}+\frac{6\cdot 1}{6\cdot 2}\\ &= \frac85- \frac{10}{12}+\frac{6}{12}\\ &= \frac85 +\frac{6}{12} - \frac{10}{12}\\ &= \frac85 -\frac{4}{12}\\ &= \frac85 -\frac{1}{3}\\ \end{align}$$

Last, combine $\frac85$ and $-\frac13:$ $$\begin{align} &= \frac33 \cdot \frac85- \frac55 \cdot \frac{1}{3}\\ &= \frac{3\cdot 8}{3\cdot 5}- \frac{5\cdot 1}{5\cdot 3}\\ &= \frac{24}{15}- \frac{5}{15}\\ &= \frac{19}{15}\\ \end{align}$$

(Video) Adding and Subtracting Fractions / Mixed Numbers

by PatrickJMT

This video by PatrickJMT covers adding and subtracting fractions with unlike denominators.

$\large{\text{Summary}}$

The first example is: $$\frac79-\frac2{15}$$

The first step is to find the LCD. $$\begin{array}{lr} 9:&{\color{#29A603}{3}}\cdot3\\ 15:&{\color{#29A603}{3}}\cdot5 \end{array}$$ $${\color{#29A603}{3}}\cdot3\cdot5=45$$ The LCD is $45$.

We want to multiply the denominator of each fraction by some number that will make it equal the LCD. We also multiply the numerator by that same number. This will make sure the new fraction is equivalent to the previous. $$\begin{gather} \frac79{\color{#29A603}{{{\times\,5}\atop{\times\,5}}}}=\frac{35}{45}\\ \frac{2}{15}{\color{#29A603}{{{\times\,3}\atop{\times\,3}}}}=\frac{6}{45} \end{gather}$$

Now that we have fractions with similar denominators, we can go ahead and subtract. $$\frac{35}{45}-\frac{6}{45}=\frac{35-6}{45}=\frac{29}{45}$$

As with any fraction problem, we want to make sure our final answer is in simplest form. Since $29$ is a prime number, its only factors are $1$ and $29$. Factoring out a $1$ doesn't do anything to help reduce fractions. We cannot cancel out a $29$ either, because $45$ is not a multiple of $29$. Therefore our answer is already fully simplified.

To subtract with unlike denominators, equalize the denominators, then subtract.

Saying that two fractions have unlike denominators is the same as saying that they have different denominators. While subtracting fractions with like denominators is relatively simple, subtracting fractions with unlike denominators is really hard. So, we make sure that our fractions have the same denominator before we try to subtract. In other words, in order to subtract fractions with unlike denominators, follow these steps:

1. Make both denominators the same.
2. Subtract the numerators.

Step 1: Make both denominators the same

You can make the denominator of two fractions the same by multiplying each fraction (the numerator and the denominator) by the denominator of the other, or by finding their lowest common denominator.

For now, just multiply each by the other. This may make it necessary to reduce the final answer. However, that's easier than finding the lowest common denominator.

Step 2: Subtract the numerators

Now that your denominators are the same, you can just subtract one numerator from each other. Just use the strategy for subtracting fractions with like denominators, since now your fractions actually do have like denominators!

During this step we do not change the denominator. We took care of that in Step 1.

An example

$\frac{3}{5} - \frac{2}{6}$ is $\frac{8}{30}$ (which is also equal to $\frac{4}{15}$). To solve this problem, you should have multiplied each fraction by the denominator of the other fraction. Here's what that looks like:

$$\begin{align} \frac{3}{5} - \frac{2}{6} &= \frac{3\times6}{5\times6} - \frac{2\times5}{6\times5} \\\ &= \frac{18}{30} + \frac{8}{30} \\\ &= \frac{8}{30}\\\ &= \frac{4}{15} \end{align}$$

Why do we get the same denominator for the two fractions in the first step? Well, because multiplication is commutative. That means for any two numbers $a$ and $b$,

$$a \times b = b \times a$$

Above, we are multiplying the two numbers one way in the first fraction, and the other way in the second.