Have you ever had to combine pieces of something to make a whole?

Travis is doing exactly that. He has three pieces of pipe that has to be clamped together. It will be connected by a professional, but Travis needs to combine the pieces of pipe and figure out the total length that he has.

The first piece of pipe measures \begin{align*}5 \frac{1}{3}\end{align*} feet.

The second piece of pipe measures \begin{align*}6 \frac{1}{2}\end{align*} feet.

The third piece of pipe measures \begin{align*}2 \frac{1}{3}\end{align*} feet.

If Travis is going to combine these together, then he has to add mixed numbers.

**This Concept will teach you how to evaluate numerical expressions involving mixed numbers. Then we will return to this original problem once again.**

### Guidance

Sometimes, we can have ** numerical expressions** that have both addition and subtraction in them. When this happens, we need to add or subtract the mixed numbers in order from left to right.

\begin{align*}4\frac{1}{6}+3\frac{4}{6}-1\frac{4}{6}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}

Here is a problem with two ** operations** in it. These operations are addition and subtraction. All of these fractions have the same common denominator, so we can begin right away. We start by performing the first operation. To do this, we are going to add the first two mixed numbers.

\begin{align*}4\frac{1}{6}+3\frac{4}{6}=7\frac{5}{6}\end{align*}

Now we can perform the final operation, subtraction. We are going to take the sum of the first two mixed numbers and subtract the final mixed number from this sum.

\begin{align*}7\frac{5}{6}-1\frac{4}{6}=6\frac{1}{6}\end{align*}

**Our final answer is** \begin{align*}6\frac{1}{6}\end{align*}.

**What about when the fractions do not have a common denominator?**

**When this happens, you must rename as necessary to be sure that all of the mixed numbers have one common denominator before performing any operations.**

After this is done, then you can add/subtract the mixed numbers in order from left to right.

\begin{align*}2\frac{4}{6}+1\frac{1}{6}-1\frac{1}{2}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}

The fraction parts of these mixed numbers do not have a common denominator. We must change this before performing any operations. The lowest common denominator between 6, 6 and 2 is 6. Two of the fractions are already named in sixths. We must rename the last one in sixths.

\begin{align*}1\frac{1}{2}=1\frac{3}{6}\end{align*}

Next we can rewrite the problem.

\begin{align*}2\frac{4}{6}+1\frac{1}{6}-1\frac{3}{6}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}

Add the first two mixed numbers.

\begin{align*}2\frac{4}{6}+1\frac{1}{6}=3\frac{5}{6}\end{align*}

Now we can take that sum and subtract the last mixed number.

\begin{align*}3\frac{5}{6}-1\frac{3}{6}=2\frac{2}{6}\end{align*}

**Don’t forget to simplify.**

\begin{align*}2\frac{2}{6}=2\frac{1}{3}\end{align*}

**This is our final answer.**

Now it's time for you to try a few on your own. Be sure your answer is in simplest form.

#### Example A

\begin{align*}6\frac{4}{8}+2\frac{2}{8}-1\frac{1}{8}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}

**Solution: \begin{align*}7 \frac{5}{8}\end{align*}**

#### Example B

\begin{align*}4\frac{3}{9}+2\frac{1}{3}-1\frac{2}{9}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}

**Solution: \begin{align*}5 \frac{4}{9}\end{align*}**

#### Example C

\begin{align*}2\frac{1}{3}+ 5\frac{1}{3}-6\frac{1}{4}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}

**Solution: \begin{align*}1 \frac{5}{12}\end{align*}**

Now back to Travis and the pipe. Here is the original problem once again.

Travis is doing exactly that. He has three pieces of pipe that has to be clamped together. It will be connected by a professional, but Travis needs to combine the pieces of pipe and figure out the total length that he has.

The first piece of pipe measures \begin{align*}5 \frac{1}{3}\end{align*} feet.

The second piece of pipe measures \begin{align*}6 \frac{1}{2}\end{align*} feet.

The third piece of pipe measures \begin{align*}2 \frac{1}{3}\end{align*} feet.

If Travis is going to combine these together, then he has to add mixed numbers.

To solve this, we can begin by writing an expression that shows all three mixed numbers being added together.

\begin{align*}5\frac{1}{3}+ 6\frac{1}{2} + 2\frac{1}{3}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}

Now we can convert all of the mixed numbers to improper fractions.

\begin{align*} \frac{16}{3} + \frac{13}{2} + \frac{7}{3}\end{align*}

Next, we rename each fraction using the lowest common denominator. The LCD of 3 and 2 is 6.

\begin{align*} \frac{32}{6} + \frac{39}{6} + \frac{14}{6}\end{align*}

Next, we add the numerators.

\begin{align*} \frac{85}{6} = 14 \frac{1}{6}\end{align*} feet.

**This is our answer.**

### Vocabulary

- Mixed Number
- a number that has a whole number and a fraction.

- Numerical Expression
- a number expression that has more than one operation in it.

- Operation
- addition, subtraction, multiplication and division

### Guided Practice

Here is one for you to try on your own.

\begin{align*}2\frac{1}{8}+ 3\frac{1}{4}-2\frac{1}{2}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}

**Answer**

To start, we need to convert all of the mixed numbers to improper fractions.

\begin{align*} \frac{17}{8} + \frac{13}{4} - \frac{5}{2}\end{align*}

Now we rename each fraction using the lowest common denominator. The LCD of 8, 4 and 2 is 8.

\begin{align*} \frac{17}{8} + \frac{26}{8} - \frac{20}{8}\end{align*}

Now we can combine and simplify.

\begin{align*} \frac{23}{8} = 2 \frac{7}{8}\end{align*}

**This is our answer.**

### Video Review

Khan Academy Subtracting Mixed Numbers

James Sousa Subtracting Mixed Numbers

James Sousa Example of Subtracting Mixed Numbers

### Practice

Directions: Evaluate each numerical expression. Be sure your answer is in simplest form.

1. \begin{align*}2\frac{1}{3}+4\frac{1}{3}-1\frac{1}{3}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}

2. \begin{align*}6\frac{2}{5}+6\frac{2}{5}-1\frac{1}{5}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}

3. \begin{align*}7\frac{3}{9}+8\frac{1}{9}-1\frac{2}{9}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}

4. \begin{align*}8\frac{3}{10}+2\frac{5}{10}-6\frac{4}{10}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}

5. \begin{align*}6\frac{1}{5}+2\frac{3}{5}-1\frac{1}{5}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}

6. \begin{align*}9\frac{4}{9}+2\frac{4}{9}-3\frac{5}{9}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}

7. \begin{align*}6\frac{9}{12}+3\frac{2}{12}-8\frac{4}{12}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}

8. \begin{align*}7\frac{8}{9}-1\frac{1}{9}+1\frac{3}{9}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}

9. \begin{align*}6\frac{4}{8}+3\frac{4}{8}-6\frac{6}{8}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}

10. \begin{align*}14\frac{2}{3}-2\frac{1}{3}+1\frac{1}{3}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}

11. \begin{align*}12\frac{6}{9}+12\frac{8}{9}-10\frac{7}{9}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}

12. \begin{align*}9\frac{1}{7}+12\frac{3}{7}+1\frac{2}{7}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}

13. \begin{align*}14\frac{3}{4}+2\frac{1}{4}-1\frac{3}{4}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}

14. \begin{align*}18\frac{6}{15}+2\frac{3}{15}-4\frac{2}{15}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}

15. \begin{align*}12\frac{1}{9}+2\frac{1}{3}-1\frac{1}{6}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}