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# Fraction Ordering with Lowest Common Denominators

## Arranging numerators in ascending order

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Fraction Ordering with Lowest Common Denominators

Have you ever gone to a place where you could make your own sundae? It is a delicious idea.

The students in the sixth grade have decided to add a “Make Your Own Ice Cream Sundae” to the sixth grade social. They figure if they charge \$1.50 per sundae, then they can make some money for the next sixth grade social. On Friday night, the first eight students came in and made their sundaes. They looked delicious! Here is what the students chose for their ice cream and toppings.

\begin{align*}\frac{6}{8}\end{align*} chose vanilla ice cream

\begin{align*}\frac{1}{4}\end{align*} chose chocolate ice cream

\begin{align*}\frac{2}{8}\end{align*} chose sprinkles

\begin{align*}\frac{5}{8}\end{align*} chose hot fudge

\begin{align*}\frac{3}{8}\end{align*} chose caramel

\begin{align*}\frac{2}{4}\end{align*} chose nuts

Terrence wants to figure out which toppings were the most popular and which toppings were the least popular. You are going to help him do this.

In this Concept, you will learn all about ordering fractions. When you see this problem again at the end of the Concept, you will know how to help Terrence write the toppings in order from the most popular or greatest to the least popular.

### Guidance

In an earlier Concept, you learned how to compare fractions with different denominators.

Sometimes, we need to write fractions in order from least to greatest or from greatest to least.

If we have fractions with common denominators, this becomes very simple.

Write in order from least to greatest. \begin{align*}\frac{4}{9},\frac{2}{9},\frac{8}{9},\frac{3}{9},\frac{6}{9}\end{align*}

Since all of these fractions are written in ninths, the common denominator, we can use the numerators and arrange them in order from the smallest numerator to the largest numerator.

Our answer is \begin{align*}\frac{2}{9},\frac{3}{9},\frac{4}{9},\frac{6}{9},\frac{8}{9}\end{align*}.

How do we order fractions that do not have a common denominator?

To do this, we will need to rewrite the fractions using a common denominator. This is the best way to know how to order the fractions.

If we wanted to write these fractions in order from least to greatest, we would need to rewrite them so that they have a common denominator.

We can use the lowest common denominator (LCD) for 3, 4, 2 and 6. That number would be 12.

Next, we rewrite each fraction in terms of twelfths.

Our answer is \begin{align*}\frac{3}{12},\frac{6}{12},\frac{8}{12},\frac{10}{12}=\frac{1}{4},\frac{1}{2},\frac{2}{3},\frac{5}{6}\end{align*}.

Try a few of these on your own.

#### Example A

What is the LCD for 3, 5, and 6?

Solution: 30

#### Example B

Rename \begin{align*}\frac{4}{5},\frac{1}{5},\frac{2}{3}\end{align*}.

Solution: \begin{align*}\frac{24}{30},\frac{6}{30},\frac{20}{30}\end{align*}.

#### Example C

Next write them in order from greatest to least.

Solution: \begin{align*}\frac{6}{30},\frac{20}{30},\frac{24}{30}\end{align*}

Now let's go back to the ice cream sundaes at the sixth grade social.

Since we are only interested in ordering the toppings, we don’t need to underline the ice cream flavors. The topping that is the most popular is the greatest fraction and the topping that is the least popular is the smallest fraction.

To order these fractions, we will need to rewrite them all with the same lowest common denominator. The lowest common denominator for 4 and 8 is 8.

We only need to change \begin{align*}\frac{2}{4} = \frac{4}{8} \end{align*}.

Now we can write them in order.

Now we can write the toppings in order from the most popular to the least popular.

1. Hot fudge
2. Nuts
3. Caramel
4. Sprinkles

Terrence is surprised by his findings. He didn’t think that caramel would be more popular than sprinkles!

### Vocabulary

Equivalent Fractions
two equal fractions
Denominator
the bottom number of a fraction
Numerator
the top number of a fraction
Like Denominator
when two or more denominators are the same, can also be called common denominators.
Lowest Common Denominator
the least common multiple of two or more denominators.

### Guided Practice

Here is one for you to try on your own.

Write the following fractions in order from least to greatest.

\begin{align*}\frac{4}{7},\frac{2}{3},\frac{5}{7}\end{align*}

To complete this task, we have to rename the fractions in terms of a lowest common denominator. In this case, the lowest common denominator of 3 and 7 is 21.

\begin{align*}\frac{12}{21},\frac{14}{21},\frac{15}{21}\end{align*}

Now we can rewrite them in order from least to greatest.

\begin{align*}\frac{4}{7},\frac{2}{3},\frac{5}{7}\end{align*}

Notice that the original order was in order from least to greatest.

### Practice

Directions: Write each series in order from least to greatest.

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

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

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

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

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

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

7. \begin{align*}\frac{4}{16},\frac{4}{5},\frac{3}{7}\end{align*}

8. \begin{align*}\frac{9}{10},\frac{4}{5},\frac{3}{4}\end{align*}

9. \begin{align*}\frac{4}{5},\frac{1}{2},\frac{2}{3}\end{align*}

10. \begin{align*}\frac{9}{11},\frac{2}{3},\frac{3}{4}\end{align*}

11. \begin{align*}\frac{4}{7},\frac{1}{5},\frac{3}{8}\end{align*}

12. \begin{align*}\frac{6}{7},\frac{1}{3},\frac{2}{5}\end{align*}

13. \begin{align*}\frac{7}{8},\frac{4}{5},\frac{1}{3}\end{align*}

14. \begin{align*}\frac{1}{6},\frac{4}{5},\frac{2}{4}\end{align*}

15. \begin{align*}\frac{1}{9},\frac{4}{7},\frac{2}{9},\frac{7}{8}\end{align*}

### 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 $8$.
Equivalent Fractions

Equivalent Fractions

Equivalent fractions are fractions that can each be simplified to the same fraction. An equivalent fraction is created by multiplying both the numerator and denominator of the original fraction by the same number.
Like Denominators

Like Denominators

Two or more fractions have like denominators when their denominators are the same. "Common denominators" is a synonym for "like denominators".
Lowest Common Denominator

Lowest Common Denominator

The lowest common denominator of multiple fractions is the least common multiple of all of the related denominators.
Numerator

Numerator

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