Topics
2. Fractions, Decimals, & Percents
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United States of AmericaVirginia
Grade 6

2.04 Compare and order fractions, decimals, and percents

Lesson
Practice

Compare and order fractions, decimals, and percents

We can use the relationship between fractions, decimals, and percents to compare and order them based on their size. But first, let's review a few important terms:

Greater than

the symbol is written as \gt and will have the larger number on the left

Example:

5\gt2

Less than

the symbol is written as \lt and will have the smaller number on the left

Example:

2\lt5

Ascending order

smallest to largest

Example:

-2, 3, 5, 8

Descending order

largest to smallest

Example:

10, 5, 2, -1

Let's compare the numbers:

10\%, \, 0.6, \, \dfrac{5}{10}

To get an idea of their size we can plot each on a number line.

An image depicting three number lines. The first number line ranges from 0 percent to 100 percent, with increments of 10 percent, displaying the length of 10 percent intervals. The second number line ranges from 0 to 1, with increments of 0.1, indicating a length of 0.6. The third number line ranges from 0 to 1, with increments of 1 over ten, and displays a plotted length of 5 over ten.

A number line isn't always the most efficient method to compare a list of numbers. If our denominator of the fraction was very large we would need a lot more divisions on the fraction number line.

To avoid this, we can compare fractions, decimals and percents algebraically if we convert them to be in the same form. Begin by looking at the given list and deciding which form would be most helpful to make the comparisons.

10\%, \text{ } 0.6, \text{ } \dfrac{5}{10}

You may think it makes the most sense to make comparisons as decimals, changing them to be 0.1, 0.6, and 0.5. While someone else might think it makes the most sense to convert them all to percentages, such as 10\%, 60\%, and 50\%. Which form you choose doesn't matter as long as it is the same for each number in the list.

Examples

Example 1

Fill in the ⬚ with < or > to make a correct comparison between the numbers.

a

3.5 ⬚ 35\%

Worked Solution
Create a strategy

Write the numbers in the same form to easily see which is larger.

Apply the idea

To convert decimal to percentage, we just multiply by 100 \%

\displaystyle 3.5\cdot 100 \%\displaystyle =\displaystyle 350\%Multiply by 100
\displaystyle 350\%\displaystyle >\displaystyle 35\%Compare

This means that:

3.5 > 35\%

Reflect and check

We could have converted the percentage to a decimal and compared the values. To convert a percentage to a decimal, divide by 100.

\displaystyle 35\%\displaystyle =\displaystyle \dfrac{35}{100}Divide by 100
\displaystyle 35\%\displaystyle =\displaystyle 0.35Convert to a decimal

Comparing the decimal values, we still find that 3.5 is greater than 0.35, confirming our previous solution.

3.5 > 0.35

b

35\% ⬚ \dfrac{3}{5}

Worked Solution
Create a strategy

Write the numbers in the same form to easily see which is larger.

Apply the idea

To compare the numbers, we first need to convert both values to the same form. We can convert both values to fractions with a denominator of 100.

\displaystyle 35\%\displaystyle =\displaystyle \dfrac{35}{100}Convert 35\% to a fraction with a denominator of 100
\displaystyle \dfrac{3}{5}\cdot\dfrac{20}{20}\displaystyle =\displaystyle \dfrac{60}{100}Convert \dfrac{3}{5} to a fraction with a denominator of 100

Now that both values are in the same form, we can easily compare them:

\dfrac{35}{100}<\dfrac{60}{100}

Therefore, 35\% < \dfrac{3}{5}.

Reflect and check

We can also compare these numbers using benchmark values. Let's compare each value to \dfrac{1}{2}.

We know 35 \% is less than 50 \% = \dfrac{1}{2}.

We also know \dfrac{3}{5} is larger than \dfrac{1}{2} since 3 is more than half of 5.

Since 35 \% is less than \dfrac{1}{2} and \dfrac{3}{5} is more than \dfrac{1}{2}, we know 35 \% < \dfrac{3}{5}.

c

\dfrac{3}{5} ⬚ 0.8

Worked Solution
Create a strategy

Write the numbers in the same form to easily see which is larger.

Apply the idea
\displaystyle \dfrac{3}{5} \cdot \dfrac{20}{20}\displaystyle =\displaystyle \dfrac{60}{100}Multiply by \dfrac{20}{20} to get a denominator of 100
\displaystyle 0.8\displaystyle =\displaystyle \dfrac{8}{10}Convert 0.8 to fraction
\displaystyle \dfrac{8}{10}\cdot \dfrac{10}{10}\displaystyle =\displaystyle \dfrac{80}{100}Convert \dfrac{8}{10} to a fraction with a denominator of 100
\displaystyle \dfrac{60}{100}\displaystyle <\displaystyle \dfrac{80}{100}Compare the two fractions

Therefore, \dfrac{3}{5}< 0.8

Reflect and check

We can also compare these values using a benchmark by seeing how far each is from 1 whole.

We know \dfrac{3}{5} is \dfrac{2}{5} less than 1, and 0.8 is 0.2 less than 1.

Since \dfrac{2}{5} = 0.4 > 0.2, we know that \dfrac{3}{5} < 0.8.

Example 2

Arrange \dfrac {9}{10} , 40\% and 0.5 in descending order using percentages.

a

First, convert \dfrac {9}{10} to a percentage.

Worked Solution
Create a strategy

Convert \dfrac{9}{10} to be a fraction out of 100.

Apply the idea
\displaystyle \dfrac{9}{10}\displaystyle =\displaystyle \dfrac{9}{10} \cdot \dfrac{10}{10}Multiply the numerator and denominator by 10
\displaystyle =\displaystyle \dfrac{9 \cdot 10}{10 \cdot 10}Multiply
\displaystyle =\displaystyle \dfrac{90}{100}Write as a fraction
\displaystyle =\displaystyle 90\%Rewrite as a percentage
b

Now convert 0.5 to a percentage.

Worked Solution
Create a strategy

Multiply the decimal by 100\%.

Apply the idea
\displaystyle 0.5\displaystyle =\displaystyle 0.5 \cdot 100\%Multiplying by 100\%
\displaystyle =\displaystyle 50\%Evaluate
c

Which of the following arranges \dfrac{9}{10}, \enspace 40\% and 0.5 in descending order?

A
40\%, \, \dfrac{9}{10}, \, 0.5
B
\dfrac{9}{10}, \, 0.5, \, 40\%
C
\dfrac{9}{10}, \, 40\%, \, 0.5
D
0.5, \, 40\%, \, \dfrac{9}{10}
Worked Solution
Create a strategy

Compare the percentages.

Apply the idea

The percentages in order are: 90\%, \, 50\%, \, 40\%

So the original numbers in order are: \dfrac{9}{10}, \, 0.5, \, 40\%

The answer is option B.

Reflect and check

Another way to compare and arrange the numbers \dfrac {9}{10}, 40\%, and 0.5 in descending order is by using a 10\times 10 grid to represent each number as a percentage. Each grid square represents 1\%. Fill in the appropriate number of squares for each percentage.

Three 10 by 10 square grids illustrating different shaded areas. The first grid depicts 9 over 10, with 90 grids shaded out of 100 squares. The second grid represents 40 percent, with 40 squares shaded out of 100 squares. The third grid shows 0.5, with 50 squares shaded out of 100 squares.

This confirms that the numbers arranged in descending order are:

\dfrac{9}{10} > \, 0.5, >\, 40\%

Idea summary

When we need to compare a list of fractions, decimals and percents, converting them to be in the same form can make it easier to compare. Look at the given list and decide which form would be most helpful to make the comparisons in.

Outcomes

6.NS.1

The student will reason and use multiple strategies to express equivalency, compare, and order numbers written as fractions, mixed numbers, decimals, and percents.

6.NS.1e

Use multiple strategies (e.g., benchmarks, number line, equivalency) to compare and order no more than four positive rational numbers expressed as fractions (proper or improper), mixed numbers, decimals, and percents (decimals through thousandths, fractions with denominators of 12 or less or factors of 100), with and without models. Justify solutions orally, in writing or with a model. Ordering may be in ascending or descending order.*

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