Additional Example
Example 3
Let's look at an example highlighting each of these three steps.
Find the value of \(\dfrac{7}{6}+\dfrac{3}{10}\) and give your answer in its simplest form.
Solution
I want to point out that the first fraction we have is an improper fraction. Now this has no impact on how we add two fractions so we're just going to continue as we did in the previous examples.
The second thing we want to notice right away is that the denominators of our two fractions are different.
\(\begin{align*} \dfrac{7}{6} + \dfrac{3}{10} \end{align*}\)
Step 1: Find a common denominator
So we look at the number \(6\) and \(10\). The lowest common multiple of \(6\) and \(10\) is \(\class{hl2}{30}\). \(30\) is going to be our common denominator.
\(\begin{align*} \dfrac{7}{6} + \dfrac{3}{10}& = \dfrac{\boxed{\phantom{\square}}}{\class{hl2}{30}} + \dfrac{\boxed{\phantom{\square}}}{\class{hl2}{30}}\\ \end{align*}\)
Step 2: Create equivalent fractions
We need to create equivalent fractions of \(\dfrac{7}{6}\) and \(\dfrac{3}{10}\) using our common denominator.
\(\dfrac{7}{6} = \dfrac{35}{\class{hl2}{30}} \qquad \qquad \dfrac{3}{10}= \dfrac{9}{\class{hl2}{30}}\)
Putting this into our calculation, we get
\(\begin{align*} \dfrac{7}{6} + \dfrac{3}{10}& = \dfrac{\boxed{\phantom{\square}}}{30} + \dfrac{\boxed{\phantom{\square}}}{30}\\ & = \dfrac{35}{\class{hl2}{30}} + \dfrac{9}{\class{hl2}{30}} \\ \end{align*}\)
Step 3: Add the numerators
\(\dfrac{35}{30} + \dfrac{9}{30} = \dfrac{35+9}{30} = \dfrac{44}{30}\)
Again, notice that we have kept the denominator and are adding the numerators.
We then want to pay special attention to make sure that our answer is in its simplest form. We see that \(44\) and \(30\) have a common factor of \(2\). Therefore, we get that
\(\begin{align*} \dfrac{7}{6} + \dfrac{3}{10}& = \dfrac{\boxed{\phantom{\square}}}{30} + \dfrac{\boxed{\phantom{\square}}}{30}\\ & = \dfrac{35}{30} + \dfrac{9}{30} \\ & = \dfrac{35+9}{30} \\ & \class {timed add1-cover remove15-cover add16-toGrey}{= \dfrac{44}{30}} \\ & = \dfrac{22}{15} \end{align*}\)
Our answer is not in its simplest form. We could rewrite this improper fraction as a mixed number, if necessary, but for this example, I'm just going to keep it as \(\dfrac{22}{15}\).
Alternate Solution
I want you to consider the same example of asking you to find the value of \(\dfrac{7}{6}+\dfrac{3}{10}\).
How would our calculation change if we found a common denominator that was not the lowest common multiple (LCM)? For example, consider \(60\), which is the product of \(6\) and \(10\), as the common denominator.
Take a moment and try this problem on your own.
Our steps for finding this sum are exactly the same.
Step 1: Find a common denominator
Suppose that we chose \(6\times 10=60\) as the common denominator.
\(\begin{align*} \dfrac{7}{6} + \dfrac{3}{10} & = \dfrac{\boxed{\phantom{\square}}}{60} + \dfrac{\boxed{\phantom{\square}}}{60}\\ \end{align*}\)
So let's go back to the suggestion I gave. What happens if we chose the common denominator of \(60\)?
Well, we're still going to create the equivalent fractions.
Step 2: Create equivalent fractions
\(\dfrac{7}{6} = \dfrac{70}{60} \qquad \qquad \dfrac{3}{10}= \dfrac{18}{60}\)
Putting this into our calculation, we get
\(\begin{align*} \dfrac{7}{6} + \dfrac{3}{10} & = \dfrac{\boxed{\phantom{\square}}}{60} + \dfrac{\boxed{\phantom{\square}}}{60}\\ & = \dfrac{70}{60} + \dfrac{18}{60} \end{align*}\)
Step 3: Add the numerators
\(\dfrac{70}{60} +\dfrac{18}{60} = \dfrac{70+18}{60} =\dfrac{88}{60}\)
Again, the denominator has stayed the same, and we've added the numerators.
We notice that \(88\) and \(60\) have a common factor of \(4\). Therefore, we get
\(\begin{align*} \dfrac{7}{6} + \dfrac{3}{10} & = \dfrac{\boxed{\phantom{\square}}}{60} + \dfrac{\boxed{\phantom{\square}}}{60}\\ & = \dfrac{70}{60} + \dfrac{18}{60} \\ & = \dfrac{70+18}{60} \\ & = \dfrac{88}{60} \\ & = \dfrac{22}{15} \end{align*}\)
Example 3 Summary
Consider the two solutions side by side.
Solution 1: Original
Choosing a common denominator of \(30\):
\(\begin{align*} \dfrac{7}{6} + \dfrac{3}{10} & = \dfrac{\boxed{\phantom{\square}}}{30} + \dfrac{\boxed{\phantom{\square}}}{30}\\ & = \dfrac{35}{30} + \dfrac{9}{30} \\ & = \dfrac{35+9}{30}\\ & = \dfrac{44}{30} \\ &= \dfrac{22}{15} \end{align*}\)
Solution 2: Alternate
Choosing a common denominator of \(60\):
\(\begin{align*} \dfrac{7}{6} + \dfrac{3}{10} & = \dfrac{\boxed{\phantom{\square}}}{60} + \dfrac{\boxed{\phantom{\square}}}{60}\\ & = \dfrac{70}{60} + \dfrac{18}{60} \\ & = \dfrac{70+18}{60} \\ & = \dfrac{88}{60} \\ & = \dfrac{22}{15} \end{align*}\)
Notice that the answer to both solutions is the same, and this makes sense because we're adding the same fractions in both cases.
What's different is the work in the middle. In our second solution, where we used a common denominator of \(60\), the numbers that we're working with are much larger overall.
Using the smallest common denominator keeps the numerators as small as possible, which is sometimes easier, when we're performing mental arithmetic.
What I want you to get out of performing this calculation a second time, is that choosing the smallest common denominator isn't necessary — any common multiple of the original denominators will work. The lowest common denominator keeps the numbers as small as possible.
Check Your Understanding 2
Question
Evaluate \(\dfrac{2}{3} + \dfrac{15}{16}\)
Answer
\(\dfrac{77}{48}\) or \(1\dfrac{29}{48}\)
Feedback
To evaluate the expression we complete the following steps.
\(\begin{align*} \dfrac{2}{3} + \dfrac{15}{16} &= \dfrac{\boxed{\phantom \square}}{48} + \dfrac{\boxed{\phantom \square}}{48} & 1. \text{ Find a common denominator} \\ & = \dfrac{32}{48} + \dfrac{45}{48} & 2. \text{ Create equivalent fractions} \\ & = \dfrac{77}{48} & 3.\text{ Add the numerators}\end{align*}\)
The answer can also be represented as a mixed number \(1\dfrac{29}{48}\).