How do you order fractions with different denominators?

When comparing two fractions with same denominator, the larger fraction is the one with the greater numerator.

Let's illustrate through an example.

Example 1

 

of a birthday cake and your sister takes

 

The denominator of both the fractions is same i.e. 6.

To compare fractions when the denominators of two fractions are the same, the one with the greater numerator is larger.

Since 3 is greater than 1,

So, your sister has more cake than you.

Comparing Unlike Fractions

To compare fractions with different denominators, we must convert the fractions to equivalent fractions with a common denominator and then look for numerators.

 

Example 2

 

Convert these fractions to equivalent fractions with a common denominator in order to compare them.

Follow these steps:

List out the multiples.

Multiples of 6 = 6, 12, 18, 24, ......

Multiples of 3 = 3, 6, 9, 12, ......

The smallest multiple the two denominators have in common is 6.

The least common denominator (LCD) of the fractions is the least common multiple of their denominators.

Now, convert these fractions to equivalent fractions with a denominator 6. As we can notice the first fraction already has the denominator 6.

So, convert the other fraction with denominator 6.

with the same denominator.

Let's look for numerators to compare.

We know, 4 is greater than 3.

We can rewrite the answer as

If the denominator is the same, look at the numerators, and put the fractions in order.

If the denominator is different, we need to convert our fractions to equivalent fractions of the same denominator and then compare them to put in order.

 

Example 3

 

Order these fractions from least to greatest:

Here the denominator of all the fractions is same so we look at the numerators.

1 is the smallest number, followed by 7, 9, and 12.

1 < 7 < 9 < 12

Arrange the fractions from least to greatest:

 

Example 4

 

Order these fractions from least to greatest:

These fractions have unlike fractions. We will use the least common denominator (LCD) to write these fractions as equivalents fractions with same denominators, and then compare.

Example 1: An 8-ounce cup of milk was served to each of three children. Lisa drank 7 ounces of milk. Her sister Angie drank 3 ounces, and her brother Mark drank 5 ounces. What part of the cup did each child drink? Who drank the smallest part of the cup? Who drank the largest part of the cup? Who fell in the middle?

Analysis: Write the part of the cup that each child drank as a fraction, and then order them from least to greatest.

ChildMilk DrankFractionLisa7 oz.Angie3 oz.Mark5 oz.<and<ChildMilk DrankFractionOrderLisa7 oz.Angie3 oz.Mark5 oz.<<

Solution: Angie drank the smallest part of the cup. Lisa drank the largest part of the cup. Mark fell in the middle.

We were able to order these fractions from least to greatest because they have like denominators.

To order fractions with like denominators, look at the numerators and compare them two at a time. It is helpful to write a number in a circle next to each fraction to compare them more easily.

Let's look at another example of ordering fractions with like denominators.

 

Example 2: 

FractionOrder

Solution: 

Example 3: It takes Jack three-fifths of an hour to complete his math homework, five-sixths of an hour to complete his reading homework, and two-thirds of an hour to complete his science homework. Order the time spent to complete Jack's homework by subject from least to greatest.

Analysis: These fractions have unlike denominators. We will use the least common denominator (LCD) to write these fractions as equivalents fractions with like denominators, and then compare them two at a time.

Math:Reading:Science:Math:Reading:Science:

Solution: Ordering the time spent on Jack's homework from least to greatest, we get: Math, Science and Reading.

To order fractions with unlike denominators, use the LCD to write them as equivalent fractions with like denominators. Then compare two fractions at a time. It is helpful to write a number in a circle next to each fraction to compare them more easily. Let's look at another example.

Example 4:

 

Analysis:

Solution:

Example 5: Ned jogged for one-third of an mile, Moze jogged for one-half of a mile, and Cookie jogged for one-fifth of a mile. Order these distances from least to greatest.

Analysis: 

Since these fractions have like numerators, we will compare the denominators two at a time. The fraction with the smaller denominator is the larger fraction.

FractionOrder

<and<

Solution:

If you need a visual representation of example 5, look at the shaded rectangles below. These fractions are unit fractions: Each of them has the same numerator. You can see that as the denominator gets larger, the fraction gets smaller.To order fractions with like numerators, look at the denominators and compare them two at a time. The fraction with the smaller denominator is the larger fraction. Let's look at another example.Example 6:  FractionOrderSolution:Example 7: Analysis:Convert these fractions to equivalent fractions with a common denominator. and <<Solution:Our answer is three-eighths.In example 7, we did not find the LCD. If we had, then it would be difficult to name a fraction between one-fourth and two-fourths. Instead, we chose eighths as our common denominator. This allowed us to name a fraction between two-eighths and four-eighths, resulting in three-eighths as our answer. We could have also chosen larger common multiples of 2 and 4, such as 16, 24, 32, and so on. Since the number of common multiples of any two whole numbers is endless, there are many possible solutions to this problem.

Let's try to summarize what we have learned.

R U L E S  F O R  O R D E R I N G  F R A C T I O N SRelationshipHow To OrderLike DenominatorsCompare two fractions at a time. Look at the numerators. The larger fraction is the one with the greater numerator.Unlike DenominatorsConvert each fraction to an equivalent fraction with a common denominator. Then compare two fractions at a time. The larger fraction is the one with the greater numerator.Like NumeratorsCompare two fractions at a time. Look at the denominators. The fraction with the smaller denominator is the larger fraction.

Summary: When ordering three or more fractions from least to greatest, compare two fractions at a time. It is helpful to write a number in a circle next to each fraction to compare them more easily.

Exercises

In Exercises 1 through 5, click once in an ANSWER BOX and type in your answer; then click ENTER. After you click ENTER, a message will appear in the RESULTS BOX to indicate whether your answer is correct or incorrect. To start over, click CLEAR. Note: To write the fraction two-thirds, enter 2/3 into the form.

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