### Let’s Think About It

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 \begin{align*}1 \frac{1}{2}\end{align*} teaspoons of salt and another calls for \begin{align*}1 \frac{5}{8}\end{align*} teaspoons of salt. How can Margaret help her dad to determine which recipe to use if he wants to use the least amount of salt possible?

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

### Guidance

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: \begin{align*}0, \frac{1}{2}\end{align*}, and \begin{align*}1\end{align*}. Then, compare the approximations.

Here is an example.

Use approximation to order \begin{align*}\frac{7}{8}, \frac{2}{5}, 3 \frac{5}{8}, \frac{1}{29}\end{align*}, and \begin{align*}\frac{29}{30}\end{align*} greatest to least.

First, approximate each fraction using the fraction benchmarks.

- \begin{align*}\frac{7}{8}:7\end{align*} out of 8 is almost 8 out of 8 which would be a whole, so \begin{align*}\frac{7}{8}\end{align*} is approximately 1.
- \begin{align*}\frac{2}{5} : 2\end{align*} is a little less than half of 5, so \begin{align*}\frac{2}{5}\end{align*} is approximately \begin{align*}\frac{1}{2}\end{align*}.
- \begin{align*}3 \frac{5}{8}\end{align*} is a mixed number greater than 1, so it will automatically be the greatest number in our list.
- \begin{align*}\frac{1}{29}\end{align*} the denominator of 29 is much larger than the numerator of 1, so \begin{align*}\frac{1}{29}\end{align*} is approximately 0.
- \begin{align*}\frac{29}{30} : 29\end{align*} out of 30 is almost 30 out of 30 which would be a whole, so \begin{align*}\frac{29}{30}\end{align*} is approximately 1.

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

\begin{align*}3 \frac{5}{8},\ \frac{7}{8} (1), \ \frac{29}{30} (1), \ \frac{2}{5} \left( \frac{1}{2} \right), \ \frac{1}{29} (0)\end{align*}

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 \begin{align*}\frac{7}{8}\end{align*} compares to \begin{align*}\frac{29}{30}\end{align*}.

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 \begin{align*}\frac{7}{8}\end{align*}. Divide the bottom number line into thirtieths and find \begin{align*}\frac{29}{30}\end{align*}. Look to see which value is closest to 1.

Now you can see that \begin{align*}\frac{29}{30}\end{align*} is closer to 1, so it is the greater number.

The answer is that the numbers ordered from greatest to least are \begin{align*}3 \frac{5}{8}, \frac{29}{30}, \frac{7}{8}, \frac{2}{5}, \frac{1}{29}\end{align*}.

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 \begin{align*}\frac{2}{3}\end{align*} and \begin{align*}\frac{5}{7}\end{align*}.

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.

\begin{align*}\begin{matrix} \frac{2}{3} & = & \frac{2 \times 7}{3 \times 7} & = & \frac{14}{21} \\ \frac{5}{7} & = & \frac{5 \times 3}{7 \times 3}& = & \frac{15}{21} \end{matrix}\end{align*}

Next, compare the rewritten fractions. Now that they have the same denominator, you can see that \begin{align*}\frac{14}{21}\end{align*} is less than \begin{align*}\frac{15}{21}\end{align*}, so \begin{align*}\frac{2}{3}\end{align*} is less than \begin{align*}\frac{5}{7}\end{align*}.

The answer is \begin{align*}\frac{2}{3} < \frac{5}{7}\end{align*}.

### Guided Practice

In the long jump contest, Peter jumped \begin{align*}5 \frac{3}{8}\end{align*} feet, Sharon jumped \begin{align*}6 \frac{3}{5}\end{align*} feet, and Juan jumped \begin{align*}6 \frac{2}{7}\end{align*} feet. Order their jump distances from greatest to least.

First, notice that \begin{align*}5 \frac{3}{8}\end{align*} is less than 6 while \begin{align*}6 \frac{3}{5}\end{align*} and \begin{align*}6 \frac{2}{7}\end{align*} are both greater than 6. That means \begin{align*}5 \frac{3}{8}\end{align*} is the smallest number.

Next, compare \begin{align*}6 \frac{3}{5}\end{align*} and \begin{align*}6 \frac{2}{7}\end{align*}. Because both numbers are between 6 and 7, you can compare the fractional parts of the mixed numbers, \begin{align*}\frac{3}{5}\end{align*} and \begin{align*}\frac{2}{7}\end{align*}, in order to determine which mixed number is greater. Because the denominators are different, you will need to find an equivalent fraction for each given fraction so that their denominators are the same.

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*}.

### Examples

#### Example 1

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 2

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 3

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*}.

### Follow Up

Remember Margaret and her dad who is making salsa? He found two recipes. One calls for \begin{align*}1 \frac{1}{2}\end{align*} teaspoons of salt and the other calls for \begin{align*}1 \frac{5}{8}\end{align*} teaspoons of salt. He wants to use the least amount of salt possible.

First, Margaret should notice that because both numbers are between 1 and 2, she can compare the fractional parts of the mixed numbers, \begin{align*}\frac{1}{2}\end{align*} and \begin{align*}\frac{5}{8}\end{align*}, in order to determine which mixed number is greater. Because the denominators are different, she will need to find an equivalent fraction for each given fraction so that their denominators are the same.

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 \begin{align*}\frac{5}{8}\end{align*} already has a denominator of 8, she will not need to rewrite that fraction!

\begin{align*}\begin{array}{rcl} \frac{1}{2} & = & \frac{1 \times 4}{2 \times 4} = \frac{4}{8}\\ \frac{5}{8} & = & \frac{5}{8} \end{array}\end{align*}

Finally, Margaret can compare the rewritten fractions. Now that they have the same denominator, she can see that \begin{align*}\frac{4}{8}\end{align*} is less than \begin{align*}\frac{5}{8}\end{align*}. This means \begin{align*}1 \frac{1}{2}\end{align*} teaspoons is less than \begin{align*}1 \frac{5}{8}\end{align*} teaspoons.

The answer is that Margaret’s dad should use the recipe that calls for \begin{align*}1 \frac{1}{2}\end{align*} teaspoons of salt.

### Explore More

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

1. \begin{align*}\frac{2}{5}\end{align*} and \begin{align*}\frac{3}{5}\end{align*}

2. \begin{align*}\frac{2}{6}\end{align*} and \begin{align*}\frac{1}{6}\end{align*}

3. \begin{align*}\frac{4}{5}\end{align*} and \begin{align*}\frac{3}{5}\end{align*}

4. \begin{align*}\frac{2}{15}\end{align*} and \begin{align*}\frac{13}{15}\end{align*}

5. \begin{align*}\frac{2}{5}\end{align*} and \begin{align*}\frac{3}{7}\end{align*}

6.

and7. \begin{align*}\frac{12}{15}\end{align*} and \begin{align*}\frac{13}{30}\end{align*}

8. \begin{align*}\frac{4}{5}\end{align*} and \begin{align*}\frac{7}{9}\end{align*}

9. \begin{align*}\frac{3}{8}\end{align*} and \begin{align*}\frac{4}{7}\end{align*}

10. \begin{align*}\frac{1}{2}\end{align*} and \begin{align*}\frac{3}{6}\end{align*}

Write each set in order from least to greatest.

11. \begin{align*}\frac{5}{6}, \frac{1}{3}, \frac{4}{9}\end{align*}

12. \begin{align*}\frac{6}{7}, \frac{1}{4}, \frac{2}{3}\end{align*}

13. \begin{align*}\frac{6}{6}, \frac{4}{5}, \frac{2}{3}\end{align*}

14. \begin{align*}\frac{1}{2}, \frac{3}{5}, \frac{2}{3}\end{align*}

15. \begin{align*}\frac{2}{7}, \frac{1}{4}, \frac{3}{6}\end{align*}

16. \begin{align*}\frac{1}{6}, \frac{2}{9}, \frac{2}{5}\end{align*}

17. 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.

18. 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?