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Fraction and Mixed Number Comparison

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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 \frac{15}{2} of tuna sandwiches.

His sister has 6 \frac{3}{4} 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.

6 \frac{1}{2} \ {\underline{\;\;\;\;\;\;\;}} \ \frac{15}{4}

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

\frac{15}{4} = 3 \frac{3}{4}

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

Our answer is 6 \frac{1}{2} > \frac{15}{4} .

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.

4 \frac{1}{2},\frac{10}{6},\frac{4}{3},7 \frac{1}{9}

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.

\frac{10}{6} = 1 \frac{4}{6} = 1 \frac{2}{3}

\frac{4}{3} =1 \frac{1}{3}

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

\frac{4}{3}, \frac{10}{6}, 4 \frac{1}{2},7 \frac{1}{9}

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

Example A

4 \frac{1}{2} \ {\underline{\;\;\;\;\;\;\;}} \ \frac{12}{5}

Solution: >

Example B

\frac{16}{3} \ {\underline{\;\;\;\;\;\;\;}} \ \frac{22}{5}

Solution: <

Example C

\frac{17}{4} \ {\underline{\;\;\;\;\;\;\;\;}} \ 4 \frac{1}{4}

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 \frac{15}{2} of tuna sandwiches.

His sister has 6 \frac{3}{4} 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.

\frac{15}{2} = 7 \frac{1}{2}

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.

\frac{29}{3} \ {\underline{\;\;\;\;\;\;\;\;}} \ 7 \frac{1}{3}

Answer

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

\frac{29}{3} = 9 \frac{2}{3}

Now it is easy to compare them.

\frac{29}{3}>7 \frac{1}{3}

This is our answer.

Video Review

Khan Academy Mixed Numbers and Improper Fractions

James Sousa Mixed Numbers and Improper Fractions

Practice

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

1. \frac{12}{5} \ {\underline{\;\;\;\;\;\;\;\;}} \ 2 \frac{1}{4}

2. \frac{16}{5} \ {\underline{\;\;\;\;\;\;\;\;}} \ 3 \frac{1}{2}

3. \frac{44}{9} \ {\underline{\;\;\;\;\;\;\;\;}} \ 6 \frac{1}{3}

4. \frac{45}{7} \ {\underline{\;\;\;\;\;\;\;\;}} \ 6 \frac{1}{2}

5. \frac{19}{4} \ {\underline{\;\;\;\;\;\;\;\;}} \ 4 \frac{3}{4}

6. \frac{16}{8} \ {\underline{\;\;\;\;\;\;\;\;}} \ 2

7. \frac{49}{5} \ {\underline{\;\;\;\;\;\;\;\;}} \ 6 \frac{2}{3}

8. \frac{99}{10} \ {\underline{\;\;\;\;\;\;\;\;}} \ 10

9. \frac{69}{4} \ {\underline{\;\;\;\;\;\;\;\;}} \ 8 \frac{2}{5}

10. \frac{70}{3} \ {\underline{\;\;\;\;\;\;\;\;}} \ 10 \frac{4}{7}

11. \frac{80}{8} \ {\underline{\;\;\;\;\;\;\;\;}} \ \frac{40}{4}

12. \frac{75}{3} \ {\underline{\;\;\;\;\;\;\;\;}} \ 25

13. \frac{18}{3} \ {\underline{\;\;\;\;\;\;\;\;}} \ \frac{24}{6}

14. \frac{99}{3} \ {\underline{\;\;\;\;\;\;\;\;}} \ \frac{33}{11}

15. \frac{78}{4} \ {\underline{\;\;\;\;\;\;\;\;}} \ 10 \frac{8}{9}

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