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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 organize all kinds of baked goods for a bake sale? Well, this is the task that Tracy is taking on.

Tracy has organized all of the baked goods that students have made. She has brownies, pies, a couple of cakes and lots of cookies. Sam baked \begin{align*}8 \frac{9}{12}\end{align*} batches of cookies. Kelly baked \begin{align*}8 \frac{6}{12}\end{align*} batches of cookies.

Who baked more cookies?

In order to figure this out, you will need to know how to compare and order fractions and mixed numbers. Pay attention to this Concept and you will see this problem again at the end of it.

### Guidance

Previously we worked on fractions and approximation, you can easily compare and order fractions using this technique. By figuring out the benchmark, you can determine which fractions are larger or smaller than each other. This is the best way to approximate fractions as you compare and order them.

Sometimes, however, you can’t always rely on the approximation technique when comparing and ordering fractions. This is true when two fractions have the same benchmark, or when they have different denominators. In order to be exact when comparing and ordering fractions, you have to find a common denominator for all of the fractions. Then, compare or order the fractions by looking at the value of the numerator. This will give you an exact comparison.

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

We begin by getting an approximate sense of the value of each of the fractions in the group by comparing each fraction with the common benchmarks 0, \begin{align*}\frac{1}{2}\end{align*} and 1.

Because the number 7, which is the numerator in the fraction \begin{align*}\frac{7}{8}\end{align*} is very close in value to the denominator (8), we say that \begin{align*}\frac{7}{8}\end{align*} is approximately 1.

In the fraction, \begin{align*}\frac{2}{5}\end{align*}, the numerator is approximately \begin{align*}\frac{1}{2}\end{align*} of the denominator. So, we say that \begin{align*}\frac{2}{5}\end{align*} is about \begin{align*}\frac{1}{2}\end{align*}.

The number \begin{align*}3 \frac{5}{8}\end{align*} is the only mixed number in the group, so we can see immediately that this number is larger than all of the other numbers in the group because it is greater than 1.

In the fraction, \begin{align*}\frac{1}{29}\end{align*}, the denominator is much greater than the numerator, so \begin{align*}\frac{1}{29}\end{align*} is closest to the benchmark 0.

The numerator of 29 in the fraction \begin{align*}\frac{29}{30}\end{align*} is close in value to the denominator, 30, so \begin{align*}\frac{29}{30}\end{align*} is approximately 1.

Now that we have the approximate values of each fraction in the group, we write the fractions in a preliminary greatest to least order with the benchmarks in parentheses: \begin{align*}3 \frac{5}{8} \left(3 \frac{1}{2} \right), \frac{7}{8} (1), \frac{29}{30} (1), \frac{2}{5} \left(\frac{1}{2}\right), \frac{1}{29} (0)\end{align*}.

This approximation technique helps with most of the fractions in the group, but there are two fractions which are close to 1. We know that both \begin{align*}\frac{7}{8}\end{align*} and \begin{align*}\frac{29}{30}\end{align*} are less than 1, but which of the two fractions is closest to 1? One helpful 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 eight equal parts (eighths) and the bottom number line into thirty equal parts (thirtieths).

From this illustration, it is easy to see that \begin{align*}\frac{29}{30}\end{align*} is closer to 1 than \begin{align*}\frac{7}{8}\end{align*} and is therefore greater than \begin{align*}\frac{7}{8}\end{align*}.

The answer is \begin{align*}3 \frac{5}{8}, \frac{29}{30}, \frac{7}{8}, \frac{2}{5}, \frac{1}{29}\end{align*}.

Here is another one.

Compare \begin{align*}\frac{2}{3}\end{align*} and \begin{align*}\frac{5}{7}\end{align*}. Write >, < or =.

At first glance, it is hard to compare the two fractions because they have different denominators. Remember the first step in comparing fractions is to find the common denominator. Look at the two denominators. Sometimes the larger denominator is a multiple of the smaller denominator. For example, if we were comparing the fractions \begin{align*}\frac{2}{3}\end{align*} and \begin{align*}\frac{5}{7}\end{align*}, we can see that the denominator in \begin{align*}\frac{5}{7}\end{align*}, is a multiple of 3, which is the denominator of \begin{align*}\frac{2}{3}\end{align*}. This makes it easier to find a common denominator.

In this problem, 7 is not a multiple of 3. The lowest common denominator in this instance can only be the product of the two denominators \begin{align*}(3 \times 7 = 21)\end{align*}. In order to find an equivalent fraction for \begin{align*}\frac{2}{3}\end{align*} with a denominator of 21, we multiply both the numerator and denominator of \begin{align*}\frac{2}{3}\end{align*} by 7. We get an equivalent fraction of \begin{align*}\frac{14}{21}\end{align*}. In order to find an equivalent fraction for \begin{align*}\frac{5}{7}\end{align*} with a denominator of 21, we multiply both the numerator and denominator of \begin{align*}\frac{5}{7}\end{align*} by 3. We get an equivalent fraction of \begin{align*}\frac{15}{21}\end{align*}.

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

Now that we have a common denominator between the two fractions, we can simply compare the numerators.

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

Compare using <, > or =

#### Example A

\begin{align*}\frac{1}{3}\end{align*} and \begin{align*}\frac{5}{6}\end{align*}

Solution: <

#### Example B

\begin{align*}\frac{2}{9}\end{align*} and \begin{align*}\frac{7}{11}\end{align*}

Solution: <

#### Example C

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

Solution: >

Remember the bake sale? Here is the original problem once again.

Tracy has organized all of the baked goods that students have made. She has brownies, pies, a couple of cakes and lots of cookies. Sam baked \begin{align*}8 \frac{9}{12}\end{align*} batches of cookies. Kelly baked \begin{align*}8 \frac{6}{12}\end{align*} batches of cookies.

Who baked more cookies?

To figure out who baked more cookies, let's write out a problem comparing the two mixed numbers.

\begin{align*}8 \frac{9}{12}\end{align*} _____ \begin{align*}8 \frac{6}{12}\end{align*}

The fraction parts of these two mixed numbers have a common denominator. Because baking cookies involves dozens, both Sam and Kelly wrote the fraction part of their cookies in twelfths. Therefore, we can compare the numerators.

\begin{align*}9 > 6\end{align*}

\begin{align*}8 \frac{9}{12}\end{align*} > \begin{align*}8 \frac{6}{12}\end{align*}

Sam baked more cookies than Kelly.

This is our answer.

### Guided Practice

Here is one for you to try on your own.

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. Now order their jump distances from greatest to least.

Answer

The problem asks us to order the jump distances from greatest to least. We have three mixed numbers, so we should look first at the whole number parts of the mixed numbers to see if we can compare the jump distances.

Peter jumped more than 5 feet, but less than 6 feet. Sharon jumped more than 6 feet, but less than 7 feet. Juan also jumped more than 6 feet, but less than 7 feet.

Simply by comparing the whole numbers, we can see that Peter jumped the shortest distance because he jumped less than 6 feet. Because Sharon and Juan both jumped between 6 and 7 feet, we need to compare the fractional part of their jumps. Sharon jumped \begin{align*}\frac{3}{5}\end{align*} of a foot more than 6 feet and Juan jumped \begin{align*}\frac{2}{7}\end{align*} of a foot more than 6 feet. In order to compare these two fractions, we have to find a common denominator. The lowest common denominator for these two fractions is 35. We get an equivalent fraction of \begin{align*}\frac{21}{35}\end{align*} for \begin{align*}\frac{3}{5}\end{align*} when we multiply both the numerator and denominator by 7. We get an equivalent fraction of \begin{align*}\frac{10}{35}\end{align*} for \begin{align*}\frac{2}{7}\end{align*} when we multiply both the numerator and denominator by 5. Now we can order the distances.

The answer is Sharon \begin{align*}6 \frac{3}{5} \ ft\end{align*}., Juan \begin{align*}6 \frac{2}{7} \ ft\end{align*}., Peter \begin{align*}5 \frac{3}{8} \ ft\end{align*}.

### Explore More

Directions: Compare each pair of fractions or mixed numbers. Write >, < or =.

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. \begin{align*}\frac{1}{5}\end{align*} and \begin{align*}\frac{1}{7}\end{align*}.

7. \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*}.

Directions: 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*}

Directions: Solve these two problems.

17. Brantley is making an asparagus souffle, which calls for \begin{align*}3 \frac{3}{7}\end{align*} cup of cheese, \begin{align*}3 \frac{2}{3}\end{align*} cup of asparagus and \begin{align*}2 \frac{2}{5}\end{align*} cup 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 \begin{align*}A\end{align*} is \begin{align*}12 \frac{4}{9}\end{align*} feet and wall \begin{align*}B\end{align*} is \begin{align*}12 \frac{2}{5}\end{align*} feet. Against which wall will Geraldine put her pool table?

### Vocabulary Language: English

Denominator

Denominator

The denominator of a fraction (rational number) is the number on the bottom and indicates the total number of equal parts in the whole or the group. $\frac{5}{8}$ has denominator $8$.
fraction

fraction

A fraction is a part of a whole. A fraction is written mathematically as one value on top of another, separated by a fraction bar. It is also called a rational number.
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.
Mixed Number

Mixed Number

A mixed number is a number made up of a whole number and a fraction, such as $4\frac{3}{5}$.
Numerator

Numerator

The numerator is the number above the fraction bar in a fraction.

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