We have previously compared and ordered fractions, decimals, and percents. We have also compared and ordered integers. All of those are types of rational numbers, and they can all be represented on the same number line.

A number line from -4 to 4. The following are plotted: -3.12, -3/2 , 50% , 1.3 , 2.6, 13/3.

As we move to the right on the number line the numbers get larger.

As we move to the left on the number line the numbers get smaller.

For example, -3.12, is to the left of - \dfrac{3}{2}, so -3.12 is smaller than - \dfrac{3}{2}.

We can write this statement as the following inequalities:

\displaystyle -3.12	\displaystyle <	\displaystyle - \dfrac32	because -3.12 is further to the left on the number line
\displaystyle - \dfrac{3}{2}	\displaystyle >	\displaystyle -3.12	because - \dfrac{3}{2} is further to the right on the number line

Sometimes, it's easiest to compare rational numbers if they are written in the same form. For example, by converting decimal values to fractions or fractions to decimals.

For -3.12 and - \dfrac{3}{2} we can write - \dfrac{3}{2} as the decimal -1.5 to more easily see that -1.5 is closer to 0 on the negative side of the number line, so it must be larger than -3.12.

This also shows how we can use benchmarks, or commonly known values, to approximate the size of a number. The most commonly used benchmarks are 0, \dfrac{1}{2}, and 1 . Some other common benchmarks are:

Integer	Fraction	Decimal	Percent
0	\dfrac{0}{1}	0	0\%
\text{N/A}	\dfrac{1}{10}	0.1	10\%
\text{N/A}	\dfrac{1}{4}	0.25	25\%
\text{N/A}	\dfrac{1}{3}	0.33	33.33\%
\text{N/A}	\dfrac{1}{2}	0.5	50\%
\text{N/A}	\dfrac{2}{3}	0.67	66.67\%
\text{N/A}	\dfrac{3}{4}	0.75	75\%
1	\dfrac{1}{1}	1.0	100\%

We can also use multiples of those benchmarks to create other benchmarks to get closer to the numbers we're using. For example: 6 \cdot \dfrac{1}{10}=\dfrac{6}{10}=0.6=60 \%.

Knowing the relative size of numbers also allows us to order them in ascending order (least to greatest) or descending order (greatest to least).

A number line automatically puts numbers in ascending order.

Examples

Example 1

Which is the largest rational number marked on the number line?

Worked Solution

Create a strategy

Recall that the further a number is to the right on a number line, the larger the number is. Notice this number line is split into thirds.

Apply the idea

The rational number farthest to the right on the number line is 2 \dfrac {1}{3}. So the largest number is 2 \dfrac {1}{3}

Example 2

Consider the values 1.25 and 0.75.

Plot the pair of numbers on a number line.

Worked Solution

Create a strategy

We can see that 1.25 and 0.75 are both positive and so will be to the right of 0.

Apply the idea

To plot the point 1.25, start at 0 and count right 1 place and then move to the right 0.25 one time past 1. To plot the point 0.75, we can start at 0 and jump to the right by 0.25 three times.

State the inequality sign that makes the statement true.

1.25 \,⬚\, 0.75

Worked Solution

Create a strategy

Compare the plotted decimals given in the answer found from part (a).

Apply the idea

The rational number farther to the right on the number line is 1.25 so the larger number is 1.25

\displaystyle 1.25	\displaystyle >	\displaystyle 0.75	1.25 is greater than 0.75
\displaystyle 1.25	\displaystyle >	\displaystyle 0.75	Complete the inquality with the greater than symbol

Example 3

Consider the statement:

\dfrac{67}{50} \,>\, 154\%

Convert \dfrac{67}{50} to a percentage.

Worked Solution

Create a strategy

Multiply the numerator and denominator by 2, then the result by 100\%.

Apply the idea

\displaystyle \dfrac{67}{50}	\displaystyle =	\displaystyle \dfrac{67 \cdot 2}{50 \cdot 2}	Multiply both numerator and denominator by 2
\displaystyle \text{ }	\displaystyle =	\displaystyle \dfrac{134}{100}	Evaluate
	\displaystyle =	\displaystyle \dfrac{134}{100} \cdot 100\%	Multiply by 100\%
\displaystyle \text{ }	\displaystyle =	\displaystyle 134\%	Evaluate

Is the statement true or false?

Worked Solution

Create a strategy

Use the answer from part (a) to compare the given numbers.

Apply the idea

Let's start with the original statement to see if it is true.

\displaystyle \dfrac{67}{50}	\displaystyle >	\displaystyle 164\%	Original statement
\displaystyle 134\%	\displaystyle >	\displaystyle 154\%	Substitute 134\% for \dfrac{67}{50}

The statement is false. 134\% is not greater than 164\%.

Example 4

Write the numbers in ascending order: 71\%,\, \dfrac{4}{6},\, \, 0.7,\, 0.99

Worked Solution

Create a strategy

We know 0.99 is the largest because it is almost 1, which is 100\%.

71\% is a little bigger than 0.7 which is equal to 70\%.

For \dfrac{4}{6}, simplify to \dfrac{2}{3}, which we know is 66.67\% because it's a common benchmark.

Apply the idea

List the numbers from smallest to the largest: \dfrac{4}{6},\,0.7,\,71\%,\,0.99

Idea summary

The symbol \lt represents the phrase "is less than".

The symbol \gt represents the phrase "is greater than".

The symbol = represents the phrase "is equal to".

Descending rational numbers get smaller as we move to the left on the number line.

Ascending rational numbers get larger as we move to the right on the number line.

Outcomes

7.NS.2

The student will reason and use multiple strategies to compare and order rational numbers.

7.NS.2a

Use multiple strategies (e.g., benchmarks, number line, equivalency) to compare (using symbols <, >, =) and order (a set of no more than four) rational numbers expressed as integers, fractions (proper or improper), mixed numbers, decimals, and percents. Fractions and mixed numbers may be positive or negative. Decimals may be positive or negative and are limited to the thousandths place. Ordering may be in ascending or descending order. Justify solutions orally, in writing or with a model.*

1.01 Compare and order rational numbers

Ideas

Compare and order rational numbers

Examples

Example 1

Example 2

Example 3

Example 4

Outcomes

7.NS.2

7.NS.2a

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