Remember Travis and his Uncle from the Differences of Fractions with Different Denominators Concept? Well, after lunch, the team on the job site all enjoyed some brownies. Of course this presented an interesting math problem. Take a look.

There were two pans of unfinished brownies left for dessert.

One pan had one - fourth of a pan left.

One pan had one - third of a pan left.

Travis combined the two pans of brownies and then ate two - sixths of the pan.

After Travis had eaten his brownies, how much of the pan was left?

**This problem has two different operations in it. This Concept will teach you how to evaluate numerical expressions like this one.**

### Guidance

Previously we worked on evaluating numerical expressions that had multiple operations and multiple fractions in them. This Concept is going to build on that Concept, except this time, our fractions are going to have different denominators. We are going to need to rename them with a lowest common denominator before evaluating the expression.

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

Right away, you can see that we have THREE different denominators. We need to find the LCM for all three denominators. Begin by naming the multiples of each number.

2, 4, 6, 8, 10, 12

3, 6, 9, 12

6, 12

**All three have the number six in common. This is our lowest common denominator.** Next, we rename all three fractions in terms of sixths, by creating an equivalent fraction for each one.

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

Notice that one-sixth is already written in terms of sixths, so it can remain the same. Now we rewrite the problem.

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

We can add/subtract in order from left to right.

3 + 2 = 5 - 1 = 4

**Our answer is** \begin{align*}\frac{4}{6}\end{align*}.

\begin{align*}\frac{4}{6}\end{align*} can be simplified by dividing by the greatest common factor of 2.

**Our final answer is** \begin{align*}\frac{2}{3}\end{align*}.

Now it is time for you to try a few. Be sure that your answer is in simplest form.

#### Example A

\begin{align*} \frac{4}{5} + \frac{2}{10} - \frac{1}{2}\end{align*}

**Solution: \begin{align*} \frac{5}{10} = \frac{1}{2}\end{align*}**

#### Example B

\begin{align*} \frac{4}{8} - \frac{1}{2} + \frac{1}{4}\end{align*}

**Solution: \begin{align*} \frac{1}{4}\end{align*}**

#### Example C

\begin{align*} \frac{3}{4} - \frac{5}{8} + \frac{1}{4}\end{align*}

**Solution: \begin{align*} \frac{3}{8}\end{align*}**

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

There were two pans of unfinished brownies left for dessert.

One pan had one - fourth of a pan left.

One pan had one - third of a pan left.

Travis combined the two pans of brownies and then ate two - sixths of the pan.

After Travis had eaten his brownies, how much of the pan was left?

First, we can write an expression to explain the problem.

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

Next, we rename the fractions using the lowest common denominator which is 12.

\begin{align*} \frac{3}{12} + \frac{4}{12} - \frac{4}{12}\end{align*}

\begin{align*} \frac{3}{12} = \frac{1}{4}\end{align*}

**There was \begin{align*} \frac{1}{4}\end{align*} of a pan of brownies left.**

### Vocabulary

- Renaming fractions
- renaming fractions means rewriting them with a different denominator, but not changing the value of the fraction.

- Least Common Multiple
- the lowest multiple that two or more numbers have in common.

- Lowest Common Denominator
- the least common multiple becomes the lowest common denominator when adding or subtracting fractions with different denominators.

- Equivalent Fractions
- equal fractions. Created by multiplying the numerator and the denominator of a fraction by the same number.

### Guided Practice

Here is one for you to try on your own.

\begin{align*} \frac{6}{9} + \frac{1}{3} - \frac{4}{5}\end{align*}

**Answer**

First, we have to rename the fractions in terms of the lowest common denominator. The LCD of 9, 3 and 5 is 45.

\begin{align*} \frac{30}{45} + \frac{15}{45} - \frac{36}{45}\end{align*}

Next we perform the order of operations from left to right.

\begin{align*} \frac{45}{45} - \frac{36}{45} = \frac{9}{45}\end{align*}

Now we can simplify.

\begin{align*} \frac{9}{45} = \frac{1}{5}\end{align*}

**This is our answer.**

### Video Review

Khan Academy Adding and Subtracting Fractions

James Sousa Example of Adding Fractions with Different Denominators

James Sousa Another Example of Adding Fractions with Different Denominators

Jame Sousa Subtracting Fractions

Jame Sousa Example of Subtracting Fractions with Unlike Denominators

### Practice

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

1. \begin{align*}\frac{1}{2} + \frac{1}{3} + \frac{2}{4}\end{align*}

2. \begin{align*}\frac{6}{9} + \frac{1}{3} - \frac{2}{3}\end{align*}

3. \begin{align*}\frac{4}{5} + \frac{1}{3} - \frac{1}{5}\end{align*}

4. \begin{align*}\frac{8}{9} + \frac{1}{2} - \frac{1}{3}\end{align*}

5. \begin{align*}\frac{3}{4} + \frac{1}{3} - \frac{2}{10}\end{align*}

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

7. \begin{align*}\frac{1}{5} + \frac{2}{5} - \frac{2}{7}\end{align*}

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

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

10. \begin{align*}\frac{8}{11} + \frac{1}{3} - \frac{2}{3}\end{align*}

11. \begin{align*}\frac{6}{7} + \frac{1}{2} - \frac{2}{7}\end{align*}

12. \begin{align*}\frac{4}{9} + \frac{2}{9} - \frac{2}{3}\end{align*}

13. \begin{align*}\frac {11}{12} + \frac{1}{12} - \frac{6}{8}\end{align*}

14. \begin{align*}\frac{13}{14} + \frac{1}{28} - \frac{4}{7}\end{align*}

15. \begin{align*}\frac{17}{18} + \frac{2}{18} - \frac{5}{9}\end{align*}