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Fraction Ordering with Lowest Common Denominators

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Fraction Ordering with Lowest Common Denominators

Introduction

 

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

The students in the fifth grade have decided to add a “Make Your Own Pizza” to the fifth grade fundraiser. They figure if they charge $5.25 per pizza, then they can make some money for the Princeton Family Food Shelf. On Friday night, the first sixteen students came in and made their pizzas. They looked delicious! Here is what the students chose for their toppings.

\frac{7}{16} chose green peppers

\frac{1}{4} chose Canadian bacon

\frac{6}{8} chose onions

\frac{11}{16} chose sausage

\frac{2}{4} chose pepperoni

Guy 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 Guy write the toppings in order from the most popular or greatest to the least popular.

Guided Learning

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

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. \frac{4}{9},\frac{2}{9},\frac{8}{9},\frac{3}{9},\frac{6}{9}

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

Our answer is \frac{2}{9},\frac{3}{9},\frac{4}{9},\frac{6}{9},\frac{8}{9} .

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. We can use estimation to organize the fractions then use common denominators to see if we are correct. This is the best way to know how to order the fractions.

\frac{2}{3},\frac{1}{4},\frac{1}{2}, \frac{5}{6}

Looking at these fractions, we know that  \frac{1}{4}   is close to zero, then we have exactly \frac{1}{2}  . Lastly we have \frac{2}{3} and \frac{5}{6} . Well,   \frac{2}{3} and \frac{5}{6} are close to one, so then we will need to find a common denomiator for those two fractions so we can see which one is bigger. 

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

Next, we rewrite each fraction in terms of sixths, or equivalent fractions .

\\\frac{2}{3}=\frac{4}{6}\\\frac{5}{6}=\frac{5}{6}

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

Try a few of these on your own.

Example A

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

Example B

Organize from least to greatest.

    \frac{4}{5},\frac{1}{5},\frac{2}{3} .

Now let's go back to the make your own pizza at the fifth grade fundraiser. 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 want to organize them using estimation first. Then, we will determine which fractions that we need to find common denominators because they are estimated to the same number. Looking at the fractions, we see that  \frac{1}{4} is close to zero, \frac{7}{16}   is close to half, and \frac{2}{4}  is exactly half. So, looking at these three fractions, I know that  \frac{1}{4} is the smallest fraction,  \frac{7}{16} comes next because is close to half, but is less than half. I know it is less than half because 8 is half of 16 and 7 is less than 8. So, I can start my ordering these fractions like this: \frac{1}{4}, \frac{7}{16}, \frac{2}{4} .

 

Now, we look at the other two fractions, \frac{6}{8}   and \frac{11}{16}  . These fractions are both greater than half, so we will need to find common denomiators for these fractions. The lowest common denominator for 8 and 16 is 16.

We need to change     \frac{6}{8}=\frac{12}{16} ,  

Now we can write them in order.

\frac{1}{4}, \frac{7}{16}, \frac{2}{4}, \frac{11}{16}, \frac{12}{16}

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

  1. onions
  2. sausage
  3. pepperoni
  4. green peppers
  5. Canadian bacon

Guy is surprised by his findings. He didn’t think that onions would be more popular than sausage!


Video

Practice Set

Directions : Write each series in order from least to greatest. Use estimation first, then common denominators.

1. \frac{5}{6},\frac{1}{3},\frac{4}{9}

2. \frac{6}{8},\frac{1}{4},\frac{2}{3}

3. \frac{6}{6},\frac{4}{5},\frac{2}{3}

4. \frac{1}{2},\frac{3}{5},\frac{2}{3}

5. \frac{2}{7},\frac{1}{4},\frac{3}{6}

6. \frac{1}{3},\frac{2}{9},\frac{2}{5}

7. \frac{4}{16},\frac{4}{8},\frac{3}{4}

8. \frac{9}{10},\frac{4}{5},\frac{3}{4}

9. \frac{4}{5},\frac{1}{2},\frac{2}{3}

10. \frac{9}{12},\frac{2}{3},\frac{3}{4}

11. \frac{4}{10},\frac{1}{5},\frac{3}{8}

12. \frac{5}{6},\frac{1}{3},\frac{2}{5}

13. \frac{7}{8},\frac{4}{5},\frac{1}{3}

14. \frac{1}{6},\frac{4}{5},\frac{2}{4}

Justify: How could you organize the four fractions using only estimation? Show your work.

15. \frac{1}{9},\frac{4}{7},\frac{2}{9},\frac{7}{8}

Review

  • Using estimation helps organize the fractions we are ordering.
  • To order fractions, we will need to rewrite the fractions using a common denominator.

 

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