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# Fraction and Mixed Number Comparison

## Use <, > and/or = to compare fractions and mixed numbers.

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Fraction and Mixed Number Comparison

Have you ever had to clean up after a party?

Keith and his sister were assigned the task of cleaning up after a party. Keith took all of the leftover tuna sandwiches and his sister took all of the left over ham sandwiches.

Keith has 152\begin{align*}\frac{15}{2}\end{align*} of tuna sandwiches.

His sister has 634\begin{align*}6 \frac{3}{4}\end{align*} of ham sandwiches.

Who has more sandwiches?

This is a situation where you will need to know how to compare and order mixed numbers and improper fractions. By the end of the Concept you will know how to figure out this question.

### Guidance

Now that you have learned how to write mixed numbers and improper fractions and how to convert them back and forth, we can look at how to compare them.

How do we compare a mixed number and an improper fraction?

We compare them by first making sure that they are in the same form. They both need to be mixed numbers otherwise it is difficult to determine which one is greater and which one is less than.

612  154

The easiest thing to do here is to convert fifteen-fourths into a mixed number.

154=334

Now we know that six and one-half is greater than fifteen-fourths.

Our answer is 612>154\begin{align*}6 \frac{1}{2} > \frac{15}{4}\end{align*}.

We can order improper fractions and mixed numbers in the same way. We have to be sure to convert them all to the same form and then we write them in order.

412,106,43,719\begin{align*}4 \frac{1}{2},\frac{10}{6},\frac{4}{3},7 \frac{1}{9}\end{align*}

To order these improper fractions from least to greatest, we must first change them all so that they are the same unit. Let's change them all to mixed numbers.

106=146=123\begin{align*}\frac{10}{6} = 1 \frac{4}{6} = 1 \frac{2}{3}\end{align*}

43=113\begin{align*}\frac{4}{3} =1 \frac{1}{3}\end{align*}

Now we can write them in order from least to greatest.

43,106,412,719\begin{align*}\frac{4}{3}, \frac{10}{6}, 4 \frac{1}{2},7 \frac{1}{9}\end{align*}

Try a few of these on your own. Compare the following mixed numbers and improper fractions.

#### Example A

412  125\begin{align*}4 \frac{1}{2} \ {\underline{\;\;\;\;\;\;\;}} \ \frac{12}{5}\end{align*}

Solution: >

#### Example B

163  225\begin{align*}\frac{16}{3} \ {\underline{\;\;\;\;\;\;\;}} \ \frac{22}{5}\end{align*}

Solution: <

#### Example C

174  414\begin{align*}\frac{17}{4} \ {\underline{\;\;\;\;\;\;\;\;}} \ 4 \frac{1}{4}\end{align*}

Solution: =

Now let's go back to the party clean up. Here is the original dilemma once again.

Keith and his sister were assigned the task of cleaning up after a party. Keith took all of the leftover tuna sandwiches and his sister took all of the left over ham sandwiches.

Keith has 152\begin{align*}\frac{15}{2}\end{align*} of tuna sandwiches.

His sister has 634\begin{align*}6 \frac{3}{4}\end{align*} of ham sandwiches.

Who has more sandwiches?

To answer this question, we will need to convert the improper fraction to a mixed number. Once this is done, it will be easier to compare the two quantities.

152=712\begin{align*}\frac{15}{2} = 7 \frac{1}{2}\end{align*}

Based on this mixed number, Keith has more sandwiches.

There are more tuna sandwiches left over.

### Vocabulary

Mixed Number
a number made up of a whole number and a fraction
Improper Fractions
a fraction where the numerator is greater than the denominator
Equivalent
means equal

### Guided Practice

Here is one for you to try on your own.

293  713\begin{align*}\frac{29}{3} \ {\underline{\;\;\;\;\;\;\;\;}} \ 7 \frac{1}{3}\end{align*}

To compare these two values, we first need to convert the improper fraction to a mixed number.

293=923\begin{align*}\frac{29}{3} = 9 \frac{2}{3}\end{align*}

Now it is easy to compare them.

293>713\begin{align*}\frac{29}{3}>7 \frac{1}{3}\end{align*}

### Practice

Directions:Compare each set of values using <, > or =.

1. 125  214\begin{align*}\frac{12}{5} \ {\underline{\;\;\;\;\;\;\;\;}} \ 2 \frac{1}{4}\end{align*}

2. 165  312\begin{align*}\frac{16}{5} \ {\underline{\;\;\;\;\;\;\;\;}} \ 3 \frac{1}{2}\end{align*}

3. 449  613\begin{align*}\frac{44}{9} \ {\underline{\;\;\;\;\;\;\;\;}} \ 6 \frac{1}{3}\end{align*}

4. 457  612\begin{align*}\frac{45}{7} \ {\underline{\;\;\;\;\;\;\;\;}} \ 6 \frac{1}{2}\end{align*}

5. 194  434\begin{align*}\frac{19}{4} \ {\underline{\;\;\;\;\;\;\;\;}} \ 4 \frac{3}{4}\end{align*}

6. 168  2\begin{align*}\frac{16}{8} \ {\underline{\;\;\;\;\;\;\;\;}} \ 2 \end{align*}

7. 495  623\begin{align*}\frac{49}{5} \ {\underline{\;\;\;\;\;\;\;\;}} \ 6 \frac{2}{3}\end{align*}

8. 9910  10\begin{align*}\frac{99}{10} \ {\underline{\;\;\;\;\;\;\;\;}} \ 10 \end{align*}

9. 694  825\begin{align*}\frac{69}{4} \ {\underline{\;\;\;\;\;\;\;\;}} \ 8 \frac{2}{5}\end{align*}

10. 703  1047\begin{align*}\frac{70}{3} \ {\underline{\;\;\;\;\;\;\;\;}} \ 10 \frac{4}{7}\end{align*}

11. 808  404\begin{align*}\frac{80}{8} \ {\underline{\;\;\;\;\;\;\;\;}} \ \frac{40}{4}\end{align*}

12. 753  25\begin{align*}\frac{75}{3} \ {\underline{\;\;\;\;\;\;\;\;}} \ 25\end{align*}

13. 183  246\begin{align*}\frac{18}{3} \ {\underline{\;\;\;\;\;\;\;\;}} \ \frac{24}{6}\end{align*}

14. 993  3311\begin{align*}\frac{99}{3} \ {\underline{\;\;\;\;\;\;\;\;}} \ \frac{33}{11}\end{align*}

15. 784  1089\begin{align*}\frac{78}{4} \ {\underline{\;\;\;\;\;\;\;\;}} \ 10 \frac{8}{9}\end{align*}

### Vocabulary Language: English

Equivalent

Equivalent

Equivalent means equal in value or meaning.
improper fraction

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.