Margaret’s dad is trying to cut back on the salt in his diet. He learned that many packaged foods contain a lot of salt, so he is trying to make some of his favorite foods himself. One of his favorite snacks is salsa, so he is making his own salsa. He found two different recipes. One calls for

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

### Comparing Fractions and Mixed Numbers

When two fractions have the same denominator, the fraction with the larger numerator will be the bigger fraction. When two fractions do not have the same denominator, comparison is not as easy.

One way to compare fractions is by using approximation and benchmarks. Approximate each fraction with one of the three benchmarks:

Here is an example.

Use approximation to order

First, approximate each fraction using the fraction benchmarks.

78:7 out of 8 is almost 8 out of 8 which would be a whole, so78 is approximately 1.25:2 is a little less than half of 5, so25 is approximately12 .358 is a mixed number greater than 1, so it will automatically be the greatest number in our list.129 the denominator of 29 is much larger than the numerator of 1, so129 is approximately 0.2930:29 out of 30 is almost 30 out of 30 which would be a whole, so2930 is approximately 1.

Now, write the fractions in a preliminary greatest to least order with the benchmarks in parentheses:

Notice that the approximation method helped to order most of the numbers, but there are two fractions that are close to 1. You will have to use another method to decide how

One way to determine which fraction is closest to 1 is to draw two number lines between 0 and 1, arranged so that one number line is above the other. Divide the top number line into eighths and find

Now you can see that

The answer is that the numbers ordered from greatest to least are

Another way to compare two fractions with different denominators is by rewriting one or both fractions so that they have the same denominator. To rewrite a fraction, find an equivalent fraction by multiplying both the numerator and the denominator by the same number. Your goal is to choose numbers to multiply by so that the denominators of the equivalent fractions will be the same.

Here is an example.

Compare

First, notice that the denominators of 3 and 7 are different. You will need to find an equivalent fraction for each given fraction so that their denominators are the same.

Next, find a common denominator. You are looking for a number that is a multiple of both 3 and 7. The product of 3 and 7 is 21 and in this case that is the least common multiple of 3 and 7.

Now, rewrite each fraction as an equivalent fraction with a denominator of 21. Remember to always multiply the numerator and denominator of the fraction by the same number.

Next, compare the rewritten fractions. Now that they have the same denominator, you can see that

The answer is

### Examples

#### Example 1

Earlier, you were given a problem about Margaret's dad, who is making salsa.

He found two recipes. One calls for

First, Margaret should notice that because both numbers are between 1 and 2, she can compare the fractional parts of the mixed numbers,

Next, she can find a common denominator. She is looking for a number that is a multiple of both 2 and 8. 8 is the least common multiple of 2 and 8.

Now, she can rewrite each fraction as an equivalent fraction with a denominator of 8. Note that because

Finally, Margaret can compare the rewritten fractions. Now that they have the same denominator, she can see that

The answer is that Margaret’s dad should use the recipe that calls for

#### Example 2

In the long jump contest, Peter jumped

First, notice that

Next, compare

Now, find a common denominator. You are looking for a number that is a multiple of both 5 and 7. 35 is the least common multiple of 5 and 7.

Next, rewrite each fraction as an equivalent fraction with a denominator of 35.

\begin{align*}\begin{matrix} \frac{3}{5} & = & \frac{3 \times 7}{5 \times 7} & = & \frac{21}{35} \\ \frac{2}{7} & = & \frac{2 \times 5}{7 \times 5} & = & \frac{10}{35} \end{matrix}\end{align*}

Finally, compare the rewritten fractions. Now that they have the same denominator, you can see that \begin{align*}\frac{21}{35}\end{align*} is greater than \begin{align*}\frac{10}{35}\end{align*}. This means \begin{align*}6 \frac{3}{5}\end{align*} is greater than \begin{align*}6 \frac{2}{7}\end{align*}.

The answer is that the distances ordered from greatest to least are \begin{align*}6 \frac{3}{5}, 6 \frac{2}{7}, 5 \frac{3}{8}\end{align*}.

#### Example 3

Compare \begin{align*}\frac{1}{8}\end{align*} and \begin{align*}\frac{5}{6}\end{align*}.

First, try the approximation method and approximate each fraction with a fraction benchmark.

- \begin{align*}\frac{1}{8}:\end{align*} The numerator of 1 is much less than the denominator of 8, so \begin{align*}\frac{1}{8}\end{align*} is approximately 0.
- \begin{align*}\frac{5}{6} : 5\end{align*} out of 6 is almost 6 out of 6 which would be a whole, so \begin{align*}\frac{5}{6}\end{align*} is approximately 1.

Next, compare the two numbers. \begin{align*}\frac{1}{8}\end{align*} is approximately 0 and \begin{align*}\frac{5}{6}\end{align*} is approximately 1. That means \begin{align*}\frac{1}{8}\end{align*} is definitely less than \begin{align*}\frac{5}{6}\end{align*}.

The answer is \begin{align*}\frac{1}{8} < \frac{5}{6}\end{align*}.

#### Example 4

Compare \begin{align*}\frac{4}{9}\end{align*} and \begin{align*}\frac{7}{15}\end{align*}.

First, notice each fraction is approximately \begin{align*}\frac{1}{2}\end{align*}, so the approximation method for comparison won’t work this time. Because the denominators are different, you will need to find an equivalent fraction for each given fraction so that their denominators are the same.

Next, find a common denominator. You are looking for a number that is a multiple of both 9 and 15. 45 is the least common multiple of 9 and 15, though any common multiple of 9 and 15 would work.

Now, rewrite each fraction as an equivalent fraction with a denominator of 45.

\begin{align*}\begin{matrix} \frac{4}{9} & = & \frac{4 \times 5}{9 \times 5} & = & \frac{20}{45} \\ \frac{7}{15} & = & \frac{7 \times 3}{15 \times 3} & = & \frac{21}{45} \end{matrix}\end{align*}

Next, compare the rewritten fractions. Now that they have the same denominator, you can see that \begin{align*}\frac{20}{45}\end{align*} is less than \begin{align*}\frac{21}{45}\end{align*}, so \begin{align*}\frac{4}{9}\end{align*} is less than \begin{align*}\frac{7}{15}\end{align*}.

The answer is \begin{align*}\frac{4}{9} < \frac{7}{15}\end{align*}.

#### Example 5

Compare \begin{align*}\frac{8}{9}\end{align*} and \begin{align*}\frac{3}{4}\end{align*}.

First, notice each fraction is approximately 1, so the approximation method for comparison won’t work this time. Because the denominators are different, you will need to find an equivalent fraction for each given fraction so that their denominators are the same.

Next, find a common denominator. You are looking for a number that is a multiple of both 9 and 4. 36 is the least common multiple of 9 and 4.

Now, rewrite each fraction as an equivalent fraction with a denominator of 36.

\begin{align*}\begin{matrix} \frac{8}{9} & = & \frac{8 \times 4}{9 \times 4} & = & \frac{32}{36} \\ \frac{3}{4} & = & \frac{3 \times 9}{4 \times 9} & = & \frac{27}{36} \end{matrix}\end{align*}

Next, compare the rewritten fractions. Now that they have the same denominator, you can see that \begin{align*}\frac{32}{36}\end{align*} is greater than \begin{align*}\frac{27}{36}\end{align*}, so \begin{align*}\frac{8}{9}\end{align*} is greater than \begin{align*}\frac{3}{4}\end{align*}.

The answer is \begin{align*}\frac{8}{9} > \frac{3}{4}\end{align*}.

### Review

Compare each pair of fractions or mixed numbers using an inequality symbol or equals sign.

- \begin{align*}\frac{2}{5} \text{ and } \frac{3}{5}\end{align*}
- \begin{align*}\frac{2}{6} \text{ and } \frac{1}{6}\end{align*}
- \begin{align*}\frac{4}{5} \text{ and } \frac{3}{5}\end{align*}
- \begin{align*}\frac{2}{15} \text{ and } \frac{13}{15}\end{align*}
- \begin{align*}\frac{2}{5} \text{ and } \frac{3}{7}\end{align*}
- \begin{align*}\frac{1}{5} \text{ and } \frac{1}{7}\end{align*}
- \begin{align*}\frac{12}{15} \text{ and } \frac{13}{30}\end{align*}
- \begin{align*}\frac{4}{5} \text{ and } \frac{7}{9}\end{align*}
- \begin{align*}\frac{3}{8} \text{ and } \frac{4}{7}\end{align*}
- \begin{align*}\frac{1}{2} \text{ and } \frac{3}{6}\end{align*}

Write each set in order from least to greatest.

- \begin{align*}\frac{5}{6}, \frac{1}{3}, \frac{4}{9}\end{align*}
- \begin{align*}\frac{6}{7}, \frac{1}{4}, \frac{2}{3}\end{align*}
- \begin{align*}\frac{6}{6}, \frac{4}{5}, \frac{2}{3}\end{align*}
- \begin{align*}\frac{1}{2}, \frac{3}{5}, \frac{2}{3}\end{align*}
- \begin{align*}\frac{2}{7}, \frac{1}{4}, \frac{3}{6}\end{align*}
- \begin{align*}\frac{1}{6}, \frac{2}{9}, \frac{2}{5}\end{align*}
- Brantley is making an asparagus soufflé which calls for \begin{align*}3 \frac{3}{7}\end{align*} cups of cheese, \begin{align*}3 \frac{2}{3}\end{align*} cups of asparagus, and \begin{align*}2 \frac{2}{5}\end{align*} cups of parsley. Using approximation, order the ingredients from largest amount used to least amount used.
- Geraldine is putting in a pool table in her living room. She wants to put it against the longest wall of the room. Wall A is \begin{align*}12 \frac{4}{9}\end{align*} feet and wall B is \begin{align*}12 \frac{2}{5}\end{align*} feet. Against which wall will Geraldine put her pool table?

### Review (Answers)

To see the Review answers, open this PDF file and look for section 3.3.