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# Mixed Numbers as Improper Fractions

## 2 and 3/4 = [(4 x 2)+3]/4=11/4

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Mixed Numbers as Improper Fractions

Casey ordered eight pizzas for the drama club to enjoy. Each pizza had ten slices. At the end of the pizza party, there were two whole pizzas and two slices left. How many slices weren’t eaten? How can you express this as an improper fraction of pizza slices?

In this concept, you will learn how to rewrite mixed numbers as an improper fraction.

### Writing Mixed Numbers as Improper Fractions

A mixed number is a number that has both wholes and parts in it. Here is a mixed number.

514\begin{align*}5 \frac{1}{4}\end{align*}

There are five whole items and one-fourth of a whole. The opposite of a mixed number is an improper fraction.

An improper fraction is a fraction that has a larger numerator than the denominator. Here is an improper fraction.

125\begin{align*}\frac{12}{5}\end{align*}

The denominator tells you how many parts the whole has been divided into. This whole has been divided into 5 parts. The numerator tells you the number of parts. In this case, there are twelve parts. There are more parts than there are in 1 whole.

To write a mixed number as an improper fraction, write the mixed number as a fraction in terms of parts instead of in terms of wholes and parts. Remember that a whole number can also be written as a fraction. The numerator is equal to the denominator.

1=55\begin{align*}1 = \frac{5}{5}\end{align*}

Change 213\begin{align*}2\frac{1}{3}\end{align*} to an improper fraction.

First, multiply the whole number by the denominator to convert the whole to a fraction and add the numerator. This will give you the new numerator.

2×3+1=7\begin{align*}2 \times 3 + 1 = 7\end{align*}

Then, put the sum over the denominator. The denominator is 3.

213=73\begin{align*}2 \frac{1}{3} = \frac{7}{3}\end{align*}

The mixed number 213\begin{align*}2 \frac{1}{3}\end{align*} is also written as 73\begin{align*}\frac{7}{3}\end{align*}.

Change the following mixed numbers to improper fractions.

### Examples

#### Example 1

Earlier, you were given a problem about Casey and the pizzas

Convert 2 pizzas and 2 slices to an improper fraction to find the number of uneaten slices of pizza.

Two whole pizzas and two slices =2210\begin{align*}= 2 \frac{2}{10}\end{align*}

First, multiply the whole number by the denominator and add the numerator.

2×10+2=22\begin{align*}2 \times 10 +2 = 22\end{align*}

Then, put the sum over the denominator. The denominator is 10.

2210=2210\begin{align*}2 \frac{2}{10} = \frac{22}{10}\end{align*}

There were a total of 22 slices of pizza left uneaten.

#### Example 2

Express 478\begin{align*}4 \frac{7}{8}\end{align*} as an improper fraction.

First, multiply the whole number by the denominator and add the numerator.

4×8+7=39\begin{align*}4 \times 8 + 7 = 39\end{align*}

Then, put the sum over the denominator. The denominator is 8.

478=398\begin{align*}4 \frac{7}{8} = \frac{39}{8}\end{align*}

The mixed number 478\begin{align*}4 \frac{7}{8}\end{align*} is expressed as 398\begin{align*}\frac{39}{8}\end{align*}.

#### Example 3

313\begin{align*}3\frac{1}{3}\end{align*}

First, multiply the whole number by the denominator and add the numerator.

3×3+1=10\begin{align*}3 \times 3 +1 = 10\end{align*}

Then, put the sum over the denominator. The denominator is 3.

313=103\begin{align*}3 \frac{1}{3} = \frac{10}{3}\end{align*}

The mixed number 313\begin{align*}3 \frac{1}{3}\end{align*} is expressed as 103\begin{align*}\frac{10}{3}\end{align*}.

#### Example 4

523\begin{align*}5\frac{2}{3}\end{align*}

First, multiply the whole number by the denominator and add the numerator.

3×5+2=17\begin{align*}3 \times 5 +2 = 17\end{align*}

Then, put the sum over the denominator. The denominator is 3.

313=103\begin{align*}3 \frac{1}{3} = \frac{10}{3}\end{align*}

The mixed number 523\begin{align*}5 \frac{2}{3}\end{align*} is expressed as 173\begin{align*}\frac{17}{3}\end{align*}.

#### Example 5

618\begin{align*}6 \frac{1}{8}\end{align*}

First, multiply the whole number by the denominator and add the numerator.

6×8+1=49\begin{align*}6 \times 8 +1 = 49\end{align*}

Then, put the sum over the denominator. The denominator is 3.

618=498\begin{align*}6 \frac{1}{8} = \frac{49}{8}\end{align*}

The mixed number 618\begin{align*}6 \frac{1}{8}\end{align*} is expressed as 498\begin{align*}\frac{49}{8}\end{align*}.

### Review

Write each mixed number as an improper fraction.

1. 212\begin{align*}2\frac{1}{2}\end{align*}
2. 314\begin{align*}3\frac{1}{4}\end{align*}
3. 513\begin{align*}5\frac{1}{3}\end{align*}
4. 423\begin{align*}4\frac{2}{3}\end{align*}
5. 614\begin{align*}6\frac{1}{4}\end{align*}
6. 625\begin{align*}6\frac{2}{5}\end{align*}
7. 713\begin{align*}7\frac{1}{3}\end{align*}
8. 825\begin{align*}8\frac{2}{5}\end{align*}
9. 745\begin{align*}7\frac{4}{5}\end{align*}
10. 827\begin{align*}8\frac{2}{7}\end{align*}
11. 834\begin{align*}8\frac{3}{4}\end{align*}
12. 956\begin{align*}9\frac{5}{6}\end{align*}
13. 658\begin{align*}6\frac{5}{8}\end{align*}
14. 923\begin{align*}9\frac{2}{3}\end{align*}
15. 512\begin{align*}5\frac{1}{2}\end{align*}
16. 1614\begin{align*}16\frac{1}{4}\end{align*}

To see the Review answers, open this PDF file and look for section 5.14.

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Color Highlighted Text Notes

### Vocabulary Language: English

Equivalent

Equivalent means equal in value or meaning.

improper fraction

An improper fraction is a fraction in which the absolute value of the numerator is greater than the absolute value of the denominator.

Mixed Number

A mixed number is a number made up of a whole number and a fraction, such as $4\frac{3}{5}$.