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

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 *and*

**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, 12 and 1.**

Because the number 7, which is the numerator in the fraction

In the fraction,

The number

In the fraction,

The numerator of 29 in the fraction

**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:**

**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

From this illustration, it is easy to see that

**The answer is**

**Here is another one.**

Compare *and*

**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

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

**In order to find an equivalent fraction for**23 with a denominator of 21, we multiply both the numerator and denominator of 23 by 7. We get an equivalent fraction of 1421 . In order to find an equivalent fraction for 57 with a denominator of 21, we multiply both the numerator and denominator of 57 by 3. We get an equivalent fraction of 1521 .

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

**The answer is that 23<57.**

Compare using <, > or =

#### Example A

*and*

**Solution: <**

#### Example B

*and*

**Solution: <**

#### Example C

*and*

**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

Who baked more cookies?

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

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*}8 \frac{9}{12}\end{align*}

Sam baked more cookies than Kelly.

**This is our answer.**

### Vocabulary

- Fraction
- a part of a whole.

- Numerator
- the top number in a fraction.

- Denominator
- the bottom number in a fraction. It tells you how many parts the whole is divided into.

- Mixed Number
- a whole number with a fraction

- Improper Fraction
- when the numerator is greater than the denominator in a fraction

### 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*}

**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*}

**The answer is Sharon \begin{align*}6 \frac{3}{5} \ ft\end{align*} 635 ft., Juan \begin{align*}6 \frac{2}{7} \ ft\end{align*}627 ft., Peter \begin{align*}5 \frac{3}{8} \ ft\end{align*}538 ft.**

### Video Review

- This is a James Sousa video on comparing fractions.

### Practice

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

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

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

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

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

5. \begin{align*}\frac{2}{5}\end{align*}

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

7. \begin{align*}\frac{12}{15}\end{align*}

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

9. \begin{align*}\frac{3}{8}\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?