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3.1: Comparing and Ordering Fractions

Created by: CK-12

Introduction

Cookies for the Bake Sale

The Seventh grade class is having a bake sale to raise money for class projects and trips. Madison has decided to bake her favorite kind of cookie for the sale. She loves linzer cookies with jam inside.

“What are you doing?” asks her brother Kyle, coming into the kitchen.

“I’m making cookies for the bake sale,” Madison explains as she takes out the flour and the measuring cups.

“Can I help?” Kyle asks.

“Sure, now we need 2 \frac{1}{2} cups of flour. Here is the \frac{1}{2} cup measuring cup, but I can’t find the 1 cup measuring cup. That’s okay though because I can measure five \frac{1}{2} cups full of flour and that will be 2 \frac{1}{2} cups,” She explains to Kyle.

“You could also use the \frac{1}{3} measuring cup and fill it up 8 times,” Kyle says picking up the \frac{1}{3} measuring cup.

“I don’t think so,” Madison says. “I think that is too much flour.”

“No it isn’t,” Kyle argues.

Who is correct? To figure out whether 2 \frac{1}{2} cups of flour is equal to eight \frac{1}{3} cups of flour, you will need to understand how to compare and figure out equivalent fractions. Pay attention during this lesson and by the end of it, you will know who is correct and who needs to rethink their figuring.

What You Will Learn

By the end of this lesson, you will be able to complete the following:

  • Identify equivalent proper fractions, mixed numbers and improper fractions.
  • Approximate fractions and mixed numbers using common benchmarks.
  • Compare and order fractions and mixed numbers with and without approximation.
  • Describe real-world portion or measurement situations by comparing and ordering fractions with and without approximation.

Teaching Time

I. Identify Equivalent Proper Fractions, Mixed Numbers and Improper Fractions

This lesson is all about fractions. To understand fractions, you will need to think about whole numbers too. Without whole numbers, it is impossible to understand fractions because a fraction is a part of a whole.

Whole numbers are numbers like 1, 8, 56, and 278—numbers that don’t contain fractional parts. Not all numbers are whole.

A Fraction describes a part of a whole number. You are certainly familiar with fractions in your everyday dealings with cooking. Consider a recipe that calls for \frac{1}{2} cup of chocolate chips. You know that \frac{1}{2} cup represents one-half of a whole cup.

A fraction has certain parts. What are those parts?

The number written below the bar in a fraction is the denominator, which tells how many parts the whole is divided into. The numerator is the number above the bar in a fraction, which tells how many parts of the whole you have. In the recipe that calls for \frac{1}{2} cup, the denominator is 2, so we know that one whole cup is divided into 2 parts. The numerator is 1, so we know that we need 1 of the 2 parts of the whole cup. Notice that the fraction can be written up and down with one number on top of the other, or they may be written using a slash. With a slash, the first number is the top number or numerator and the bottom number is the second number.

A whole can be divided into an infinite number of parts. You can divide 1 cup of flour into thirds, sixths, tenths, and so on. Fractions which describe the same part of a whole are called equivalent fractions. Remember that the word equivalent means equal. For instance, if you measure out \frac{2}{4} cup of flour, \frac{3}{6} cup of flour or \frac{1}{2} cup of flour, you will have the same amount of flour.

Therefore \frac{2}{4}, \frac{3}{6} and \frac{1}{2} are all equivalent fractions.

When we have a fraction, we can create a new fraction that is equivalent to that fraction. We call this making equal fractions or making equivalent fractions.

How do we do make equivalent fractions?

The first way is to work on simplifying a fraction to make it smaller. To simplify a fraction, we can reduce the number in the numerator and denominator by dividing them by the same number. For example, \frac{4}{8} can be rewritten as \frac{1}{2} by dividing both the numerator and the denominator by 4. Note that not all fractions can be rewritten by dividing. If the only number that both the numerator and denominator are divisible by is 1, then the fraction is said to be in its simplest form.

Example

Simplify \frac{6}{18}

To simplify this fraction, we look for a number that we can divide into both the numerator and the denominator. In this case, the number is 6. We call 6 the Greatest Common Factor (GCF) of the numerator and the denominator. To simplify, we divide the numerator and the denominator by 6.

\frac{6 \div 6}{18 \div 6}= \frac{1}{3}

The simplified answer is \frac{1}{3}.

The second way to create an equivalent fraction is by multiplying. We can create an equivalent fraction by multiplying the numerator and denominator by the same number. It doesn’t matter which number you choose, as long as the numbers are the same numbers.

Example

Create an equivalent fraction for \frac{7}{8}.

To do this, we need to multiply the numerator and the denominator by the same number. Let’s choose 2.

\frac{7 \times 2}{8 \times 2}= \frac{14}{16}

The answer is \frac{14}{16}.

Example

Write four equivalent fractions for \frac{8}{12}.

First, let’s see if we can reduce the numbers in the numerator and denominator. Are there any numbers that can be divided into both 8 and 12? 8 and 12 are both divisible by 2 and 4. So, the fraction \frac{8}{12} is not in its simplest form.

\frac{8}{12} &= \frac{8 \div 2}{12 \div 2}=\frac{4}{6}\\\frac{8}{12} &= \frac{8 \div 4}{12 \div 4}=\frac{2}{3}

When we divide both the numerator and the denominator by 2, we get \frac{4}{6} as an equivalent fraction. When we divide the numerator and the denominator by 4, we get \frac{2}{3} as an equivalent fraction.

To find more equivalent fractions, we can multiply the numerator and denominator of \frac{8}{12} by any number. Let’s multiply by 3. We get \frac{24}{36} as an equivalent fraction to \frac{8}{12}. If we multiply the numerator and denominator by 5, we get \frac{40}{60} as an equivalent fraction to \frac{8}{12}.

\frac{8}{12} &= \frac{8 \times 3}{12 \times 3}=\frac{24}{36}\\\frac{8}{12} &= \frac{8 \times 5}{12 \times 5}=\frac{40}{60}

The answers, \frac{2}{3}, \frac{4}{6}, \frac{24}{36}, \frac{40}{60}, are all equivalent fractions of \frac{8}{12}.

Notice that creating equivalent fractions in this example involved both simplifying and multiplying!!

There are other types of fractions too.

Sometimes when working with fractions, you use numbers which consist of a whole number and a fraction. This is called a mixed number. For example, if a recipe calls for more than 1 cup of flour but less than 2 cups of flour, you need to use a mixed number to describe exactly how much flour you need. A mixed number is written as a whole number with a fraction to the right of it. Some common mixed numbers include: 1 \frac{1}{2} or 2 \frac{2}{3}.

When the numerator of a fraction is greater than or equal to the denominator, you have an improper fraction. Improper fractions are greater than or equal to 1.

\frac{2}{2}, \frac{3}{3}, and \frac{10}{10} are all improper fractions that equal 1.

Why is this? Well, to understand this, you have to think about what the numerator and the denominator mean. The denominator is how many parts the whole is divided into. The numerator is how many of those parts you have. If you have two out of two parts, then you have the whole thing.

\frac{5}{2}, \frac{8}{3}, and \frac{11}{4} are all fractions that are greater than 1. These are called improper fractions.

Mixed numbers and Improper Fractions can be equivalent or equal to each other.

Improper fractions can be written as mixed numbers by dividing the numerator by the denominator and keeping the remainder as the numerator (while keeping the same denominator). Mixed numbers can be rewritten as improper fractions by multiplying the whole number in the mixed number by the denominator and adding the product to the numerator.

Example

\frac{9}{2}=4 \frac{1}{2}

These two quantities are equal. This improper fraction is equal to the mixed number.

Example

Write 3 \frac{2}{3} as an improper fraction.

Remember, to write a mixed number as an improper fraction, we first multiply the whole number (3) by the denominator in the fraction, 3 \times 3 = 9. Next, we add this number to the numerator of the fraction, 9 + 2 = 11. We put this new number over the original denominator and we have our improper fraction.

Our answer is that 3 \frac{2}{3} can be written as the improper fraction \frac{11}{3}. It may help you to think about it this way: \frac{3}{3} = 1, so if we have a whole number of 3's, we also have \frac{3}{3} + \frac{3}{3} + \frac{3}{3} + \frac{2}{3} or \frac{11}{3}

Example

Write \frac{7}{3} as a mixed number.

To write an improper fraction as a mixed number, we divide the numerator by the denominator. 7 \div 3 = 2R1. To finish, we write the remainder above the original denominator and write the whole number part of the quotient to the left of this new fraction.

Our answer is that \frac{7}{3} can be written as the mixed number 2 \frac{1}{3}.

2A. Lesson Exercises

Practice working with equivalent fractions.

  1. Simplify \frac{10}{12}
  2. Create an equivalent fraction for \frac{5}{6}
  3. Write \frac{15}{2} as a mixed number

Take a few minutes to check your work with a friend.

II. Approximate Fractions and Mixed Numbers Using Common Benchmarks

Because a whole can be divided into an infinite number of parts, it is sometimes difficult to get a good sense of the value of a fraction or mixed number when the denominator of the fraction is large. In order to get an approximate sense of the value of a fraction, we compare the complicated fraction with several simpler fractions, or benchmarks. The three basic fraction benchmarks are: 0, \frac{1}{2} and 1.

When approximating the value of a fraction or mixed number, ask yourself which of these benchmarks is the number closest to?

Let’s look at how to apply benchmarks to an example.

Example

What is the approximate size of \frac{17}{18}?

To begin with, we need to determine whether the fraction is closest to 0, one-half or 1 whole. The denominator is 18 and the numerator is 17. The numerator is close in value to the denominator. The value of \frac{17}{18} is closest to 1 because \frac{18}{18} would be equal to one.

Our answer is 1.

That’s right. When you are looking for a benchmark, you want to choose the one that makes the most sense.

Example

What is the benchmark for \frac{24}{49}?

First, we can look at the relationship between the numerator and the denominator. The numerator in this case is almost half the denominator. Therefore the correct benchmark is one-half.

What about mixed numbers?

We can identify benchmarks for mixed numbers too. The difference is that rather than zero, we look to the whole number of the mixed number, the half and the whole number next in consecutive order.

Example

What is the benchmark for 7 \frac{1}{8}?

Here we have 7 and one-eighth. Is this closer to 7, 7 \frac{1}{2} or 8? If you think about it logically, one-eighth is a very small fraction. There is only one part out of eight. Therefore, it makes sense for our benchmark to be 7.

3B. Lesson Exercises

Choose the correct benchmark for each example.

  1. \frac{1}{12}
  2. \frac{5}{6}
  3. 9 \frac{4}{9}

Now check your answers with your neighbor. Did you select the most accurate benchmark?

III. Compare and Order Fractions and Mixed Numbers with and without Approximation

Now that you are able to look at a fraction and get a sense of its value by using 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.

Example

Use approximation to order \frac{7}{8}. \frac{2}{5}, 3 \frac{5}{8}, \frac{1}{29} and \frac{29}{30} 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, \frac{1}{2} and 1.

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

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

The number 3 \frac{5}{8} 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, \frac{1}{29}, the denominator is much greater than the numerator, so \frac{1}{29} is closest to the benchmark 0.

The numerator of 29 in the fraction \frac{29}{30} is close in value to the denominator, 30, so \frac{29}{30} 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: 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).

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 \frac{7}{8} and \frac{29}{30} 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 \frac{29}{30} is closer to 1 than \frac{7}{8} and is therefore greater than \frac{7}{8}.

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

Example

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

At first glance, it is hard to compare the two fractions because they have different denominators. Remember the second mentioned method in comparing fractions is to find the common denominator. Look at the two denominators. Sometimes when comparing fractions we see fractions that have denominators that are multiples of the smaller denominator, and thus could be simplified so that both fractions have common denominators, or that could be multiplied to match a larger denominator.

For example, consider \frac{3}{9} and \frac{2}{3}. We see that in the fraction \frac{3}{9} the 9 is a multiple of 3 and could be simplified by dividing the top and the bottom by 3. When we do this we get \frac{1}{3}. Now we can compare the two fractions: \frac{1}{3} and \frac{2}{3}, so \frac{1}{3} < \frac{2}{3}.

Also, we could have changed \frac{2}{3} to \frac{6}{9} and compare the fractions in this way also.

\frac{3}{9} and \frac{6}{9} - our comparison is the same, \frac{3}{9} < \frac{6}{9}

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 (3 \times 7 = 21). In order to find an equivalent fraction for \frac{2}{3} with a denominator of 21, we multiply both the numerator and denominator of \frac{2}{3} by 7. We get an equivalent fraction of \frac{14}{21}. In order to find an equivalent fraction for \frac{5}{7} with a denominator of 21, we multiply both the numerator and denominator of \frac{5}{7} by 3. We get an equivalent fraction of \frac{15}{21}.

\frac{2}{3} \times \frac{7}{7}=\frac{14}{21}\\\frac{5}{7} \times \frac{3}{3}=\frac{15}{21}

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

The answer is that \frac{2}{3} < \frac{5}{7}.

3C. Lesson Exercises

Compare using <, >, or =

  1. \frac{1}{3} and \frac{5}{6}
  2. \frac{2}{9} and \frac{7}{11}
  3. \frac{8}{9} and \frac{3}{4}

Check your work. Be sure that your answers are accurate.

IV. Describe Real-World Portion or Measurement Situations by Comparing and Ordering Fractions with and without Approximation

With common measurement units, such as feet, cups, inches and ounces, we often use fractions to describe more precise measurements in relation to other measurements. You saw an example like this in our introductory problem. For example, if a recipe calls for \frac{2}{3} cups of sugar, in proportion to the other measurements, 1 whole cup of sugar will make it too sweet, \frac{1}{2} cup of sugar won’t make it sweet enough.

Example

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

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 \frac{3}{5} of a foot more than 6 feet and Juan jumped \frac{2}{7} 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 \frac{21}{35} for \frac{3}{5} when we multiply both the numerator and denominator by 7. We get an equivalent fraction of \frac{10}{35} for \frac{2}{7} when we multiply both the numerator and denominator by 5. Now we can order the distances.

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

Examples like this one use fractional measurements. You can compare them and figure out which are equal and which are greater or less than each other. Now let’s go back to our original problem and help Madison with her measurement dilemma.

Real Life Example Completed

The Bake Sale Cookies

Here is the original problem once again. Reread it and underline any important information.

The Seventh grade class is having a bake sale to raise money for class projects and trips. Madison has decided to bake her favorite kind of cookie for the sale. She loves linzer cookies with jam inside.

“What are you doing?” asks her brother Kyle, coming into the kitchen.

“I’m making cookies for the bake sale,” Madison explains, taking out the flour and the measuring cups.

“Can I help?” Kyle asks.

“Sure, now we need 2 \frac{1}{2} cups of flour. Here is the \frac{1}{2} measuring cup, but I can’t find the 1 cup measuring cup. That’s okay though because I can measure five \frac{1}{2} cups full of flour and that will be 2 \frac{1}{2} cups,” She explains to Kyle.

“You could also use the \frac{1}{3} measuring cup and fill it up 8 times,” Kyle says picking up the \frac{1}{3} measuring cup.

“I don’t think so,” Madison says. “I think that is too much flour.”

“No it isn’t,” Kyle argues.

To solve this problem, we need to compare Madison’s measurement with Kyle’s measurement.

Madison’s measurement is 2 \frac{1}{2}.

Kyle’s measurement is 8 one-third cups which is \frac{8}{3}.

Next, we compare the two quantities. Kyle thinks that his measurement is equal to Madison’s. To see if he is correct, we convert the improper fraction to a mixed number. That will make our comparison much easier.

\frac{8}{3}=2 \frac{2}{3}

Now we compare 2 \frac{1}{2} < 2 \frac{2}{3}.

Madison is correct. If she uses Kyle’s measurement technique, she will have too much flour! Kyle needs to remember that two-thirds is greater than one-half.

You can check this by taking the fraction part of each and rewriting them with a common denominator.

\frac{1}{2} &= \frac{3}{6}\\ \frac{2}{3} &= \frac{4}{6}

You can see that two-thirds is greater than one-half.

Vocabulary

Here are the vocabulary words that are found in this lesson.

Whole Number
a number that is a counting number like 5, 7, 10, or 22.
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.
Equivalent Fractions
equal fractions
Equivalent
equal
Simplifying
making a fraction smaller
Greatest Common Factor
the largest number that will divide into two numbers.
Mixed Number
a whole number with a fraction
Improper Fraction
when the numerator is greater than the denominator in a fraction

Technology Integration

Khan Academy Mixed Numbers and Improper Fractions

James Sousa, Comparing Fractions with Different Denominators Using Inequality Symbols

James Sousa, Example of Ordering Fractions with Different Denominators from Least to Greatest

James Sousa, Converting Between Improper Fractions and Mixed Numbers

James Sousa, Example of Converting a Mixed Number to an Improper Fraction

James Sousa, Example of Converting an Improper Fraction to a Mixed Number

Other Videos:

http://www.mathplayground.com/howto_comparefractions.html – This is a video on comparing and ordering fractions.

Time to Practice

1. Write four equivalent fractions for \frac{6}{8}.

Directions: Write the following mixed numbers as improper fractions

2. 2 \frac{5}{8}

3. 3 \frac{2}{5}

4. 1 \frac{1}{7}

5. 5 \frac{4}{9}

Directions: Write the following improper fractions as mixed numbers.

6. \frac{29}{28}

7. \frac{12}{5}

8. \frac{9}{2}

9. \frac{17}{8}

10. \frac{22}{3}

Directions: Approximate the value of the following fractions using the benchmarks 0, \frac{1}{2} and 1.

11. \frac{9}{10}

12. \frac{11}{20}

13. \frac{2}{32}

14. \frac{21}{22}

Directions: Approximate the value of the following mixed numbers.

15. 2 \frac{79}{80}

16. 6 \frac{1}{10}

17. 43 \frac{7}{15}

18. 8 \frac{7}{99}

19. Use approximation to order \frac{1}{9}, 2 \frac{7}{15}, 2 \frac{5}{8}, \frac{16}{17} and \frac{5}{9} from greatest to least.

20. Compare \frac{2}{5} and \frac{3}{7}. Write >, <, or =.

21. Compare \frac{5}{6} and \frac{7}{9}. Write >, <, or =.

22. Brantley is making an asparagus souffle, which calls for 3 \frac{3}{7} cup of cheese, 3 \frac{2}{3} cup of asparagus and 2 \frac{2}{5} cup of parsley. Using approximation order the ingredients from the largest amount used to the least amount used

23. Geraldine is putting a pool table in her living room. She wants to put it against the longest wall of the room. Wall A is 12 \frac{4}{9} feet and wall B is 12 \frac{2}{5} feet. Against which wall will Geraldine put her pool table?

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CK.MAT.ENG.SE.1.Math-Grade-7.3.1

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