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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
License: CC BY-NC 3.0

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

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

In this concept, you will learn how to compare and order improper fractions and mixed numbers.

Comparing Improper Fractions and Mixed Numbers

An improper fraction is a fraction where the numerator is larger than the denominator.

A mixed number is composed of a whole number and a fraction.

To compare a mixed number and an improper fraction, first make sure that they are in the same form. Convert the improper fraction to a mixed number or the mixed number to an improper fraction, then compare.

\begin{align*}6 \frac{1}{2} \ \underline{\;\;\;\;\;\;\;\;\;} \ \frac{15}{4}\end{align*}612  154

Convert \begin{align*}\frac{15}{4}\end{align*}154 into a mixed number. Divide 15 by 4 and write the quotient as a whole number and a fraction.

\begin{align*}\frac{15}{4} = 3\frac{3}{4}\end{align*}154=334

Compare the numbers.

\begin{align*}6 \frac{1}{2} > 3 \frac{3}{4}\end{align*}612>334

\begin{align*}6 \frac{1}{2}\end{align*}612 is greater than \begin{align*}\frac{15}{4}\end{align*}154.

If the whole number is the same, compare the fractions. You may have to convert the fractions using the lowest common denominator.

You can order improper fractions and mixed numbers in the same way. Convert them all to the same form and then write them in order.

Order these fractions from least to greatest.

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

First, change the fractions so that they are all in the same form. Let’s change them all to mixed numbers. Simplify if you can.

\begin{align*}\frac{10}{6} = \frac{5}{3} = 1 \frac{2}{3}\end{align*}106=53=123

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

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

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

Examples

Example 1

Earlier, you were given a problem about Keith and the sandwiches.

Keith has \begin{align*}\frac{15}{2}\end{align*}152 tuna sandwiches and his sister has \begin{align*}6 \frac{3}{4}\end{align*}634 ham sandwiches. Compare the fractions to see who has more sandwiches.

First, convert the improper fraction to a mixed number.

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

Then, compare the two quantities.

\begin{align*}7 \frac{1}{2} > 6 \frac{3}{4}\end{align*}712>634

Keith has more sandwiches.

For the following examples, compare the fractions.

Example 2

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

First, convert the improper fraction to a mixed number.

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

Compare the numbers.

\begin{align*}9 \frac{2}{3} > 7 \frac{1}{3}\end{align*}

\begin{align*}\frac{29}{3}\end{align*} is greater than \begin{align*}7 \frac{1}{3}\end{align*}.

Example 3

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

First, change the improper fraction to a mixed number

\begin{align*}\frac{12}{5} = 2\frac{2}{5}\end{align*}

Then, compare the numbers.

\begin{align*}4\frac{1}{2} > 2\frac{2}{5}\end{align*}

\begin{align*}4\frac{1}{2}\end{align*} is greater than \begin{align*}\frac{12}{5}\end{align*}.

Example 4

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

Both fractions are improper. Let’s try comparing the fractions using the lowest common denominator of 3 and 5. The LCD is 15.

First, find the equivalent fraction for each with the denominator of 15.

\begin{align*}\frac{16}{3} = \frac{80}{15}\end{align*}

\begin{align*}\frac{22}{5} = \frac{66}{15}\end{align*}

Then, compare the fractions.

\begin{align*}\frac{80}{15} > \frac{66}{15}\end{align*}

\begin{align*}\frac{16}{3}\end{align*} is greater than \begin{align*}\frac{22}{5}\end{align*}.

Example 5

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

First, convert the mixed fraction to an improper fraction. Multiply the whole number by the denominator and add the numerator. Write it as a fraction over 4.

\begin{align*}4 \times 4 + 1 = 17\end{align*}

\begin{align*}4 \frac{1}{4} = \frac{17}{4}\end{align*}

Then, compare the fractions.

\begin{align*}\frac{17}{4} = \frac{17}{4}\end{align*}

\begin{align*}\frac{17}{4}\end{align*} is equal to \begin{align*}4 \frac{1}{4}\end{align*}.

Review

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

  1. \begin{align*}\frac{12}{5} \ \underline{\;\;\;\;\;\;\;\;\;} \ 2 \frac{1}{4}\end{align*} 
  2. \begin{align*}\frac{16}{5} \ \underline{\;\;\;\;\;\;\;\;\;} \ 3 \frac{1}{2}\end{align*}
  3. \begin{align*}\frac{44}{9} \ \underline{\;\;\;\;\;\;\;\;\;} \ 6 \frac{1}{3}\end{align*}
  4. \begin{align*}\frac{45}{7} \ \underline{\;\;\;\;\;\;\;\;\;} \ 6 \frac{1}{2}\end{align*}
  5. \begin{align*}\frac{19}{4} \ \underline{\;\;\;\;\;\;\;\;\;} \ 4 \frac{3}{4}\end{align*}
  6. \begin{align*}\frac{16}{8} \ \underline{\;\;\;\;\;\;\;\;\;} \ 2\end{align*}
  7. \begin{align*}\frac{49}{5} \ \underline{\;\;\;\;\;\;\;\;\;} \ 6 \frac{2}{3}\end{align*}
  8. \begin{align*}\frac{99}{10} \ \underline{\;\;\;\;\;\;\;\;\;} \ 10\end{align*}
  9. \begin{align*}\frac{69}{4} \ \underline{\;\;\;\;\;\;\;\;\;} \ 8 \frac{2}{5}\end{align*}
  10. \begin{align*}\frac{70}{3} \ \underline{\;\;\;\;\;\;\;\;\;} \ 10 \frac{4}{7}\end{align*}
  11. \begin{align*}\frac{80}{8} \ \underline{\;\;\;\;\;\;\;\;\;} \ \frac{40}{4}\end{align*}
  12. \begin{align*}\frac{75}{3} \ \underline{\;\;\;\;\;\;\;\;\;} \ 25\end{align*}
  13. \begin{align*}\frac{18}{3} \ \underline{\;\;\;\;\;\;\;\;\;} \ \frac{24}{6}\end{align*}
  14. \begin{align*}\frac{99}{3} \ \underline{\;\;\;\;\;\;\;\;\;} \ \frac{33}{11}\end{align*}
  15. \begin{align*}\frac{78}{4} \ \underline{\;\;\;\;\;\;\;\;\;} \ 10 \frac{8}{9}\end{align*} 

Review (Answers)

To see the Review answers, open this PDF file and look for section 5.16.

Resources

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Vocabulary

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.

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  1. [1]^ License: CC BY-NC 3.0

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