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Ordering Rational Numbers

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Learning text on the topic Ordering Rational Numbers

Ordering Rational Numbers – Definition

Rational numbers include all integers, fractions and decimals that can be expressed as a fraction of two integers, where the denominator is not zero.

Before you begin this learning text about ordering rational numbers, make sure you are familiar with Decimals Greater than 1 as Fractions, Ordering Fractions on a Number Line, and Comparing Decimals on a Number Line: Up to Thousandths as all of these topics are crucial for understanding how to order rational numbers.

Ordering rational numbers means arranging them from smallest to largest or largest to smallest. Rational numbers include fractions, decimals and whole numbers.

To order rational numbers, we need to compare their sizes. This can sometimes be tricky, especially with fractions and decimals, but don't worry, we've got some easy methods to follow!

Comparing Rational Numbers – Methods

  • Comparing Whole Numbers: This is straightforward. Larger numbers are greater (e.g., 5 is greater than 3).

27229_INTL_Math_Ordering_Rational_Numbers-01.svg

  • Comparing Fractions: To compare fractions, they must have the same denominator. If they don’t, you will need to find their equivalent fractions. Then, the fraction with the larger numerator is greater.

27229_INTL_Math_Ordering_Rational_Numbers-02.svg

  • Comparing Decimals: Line up the decimal points and compare digits from left to right, as the value of the digits from left to right decreases with each place value we move along (5 hundreds is a greater value than 5 tens). The first different digit determines which is greater or smaller based on the value of the digit.

27229_INTL_Math_Ordering_Rational_Numbers-03.svg

  • Comparing Fractions and Decimals: Convert them to the same type (either both fractions or both decimals) and then compare them as usual. View the video on Decimals Greater than 1 as Fractions to get an idea of how this can work.

27229_INTL_Math_Ordering_Rational_Numbers-04.svg

To order rational numbers, you can follow these steps:

Step Description
1. Convert Number Convert fractions and decimals to the same form if needed, for easier comparison.
2. Understand Relationship Determine the relationship between numbers (which is larger, smaller, or if they are equal).
3. Order Rational Numbers Arrange the numbers in the required order (smallest to largest or largest to smallest), by converting them back to their original expression.
4. Convert to Original Expressions Now the rational numbers are ordered, it’s important to convert any changed values back to their original expression.

Common Fractions and Equivalent Decimals

Here is a table you can reference that shows some commonly used fractions and their decimal equivalents. This is not a complete list, and you can copy this and add any additional ones you know of!

Fraction Decimal Equivalent
$\frac{1}{2}$ 0.5
$\frac{1}{3}$ 0.33
$\frac{2}{3}$ 0.67
$\frac{1}{4}$ 0.25
$\frac{2}{4}$ 0.5
$\frac{3}{4}$ 0.75
$\frac{1}{5}$ 0.2
$\frac{2}{5}$ 0.4
$\frac{3}{5}$ 0.6
$\frac{4}{5}$ 0.8
$\frac{1}{10}$ 0.1
$\frac{3}{10}$ 0.3
$\frac{7}{10}$ 0.7
$\frac{9}{10}$ 0.9

Ordering Rational Numbers – Detailed Examples

Now, let's try some examples together. Pay close attention to each step.

Order these rational numbers from smallest to largest: 4, $\frac{5}{2}$, 2.3

Convert numbers: $\frac{5}{2}$ as a decimal is 2.5.

Understand the Relationship: Notice that $\frac{5}{2}$ is 2.5, which is greater than 2.3, but less than 4.

Order the Numbers: Since 2.3 is less than 2.5, and both are smaller than 4, the order is 2.3, 2.5, 4.

Convert to Original Expressions: The order with original expressions is 2.3, $\frac{5}{2}$, 4.

27229_INTL_Math_Ordering_Rational_Numbers-05.svg

Arrange these rational numbers from largest to smallest: $\frac{4}{5}$, 0.9, $\frac{3}{6}$

Convert numbers: $\frac{4}{5}$ is 0.8, and $\frac{3}{6}$ is 0.5.

Understand the Relationship: Notice that $\frac{4}{5}$ is larger than $\frac{3}{6}$, but 0.9 is larger than both.

Order the Numbers: From largest to smallest, the order is 0.9, 0.8, 0.5.

Convert to Original Expressions: The order with original expressions is 0.9, $\frac{4}{5}$, $\frac{3}{6}$.

27229_INTL_Math_Ordering_Rational_Numbers-06.svg

Ordering Rational Numbers on a Number Line

Rational numbers can often be close in value, which is why it is important to be accurate with your conversions of these numbers. Let’s look at the first problem from above, where the order from largest to smallest was 0.9, $\frac{4}{5}$, $\frac{3}{6}$. These values in decimal were 0.9, 0.8, 0.5. Let’s arrange these numbers on a number line.

27229_INTL_Math_Ordering_Rational_Numbers-07.svg

Ordering Rational Numbers – Guided Practice

Let's practise ordering rational numbers together.

Arrange these numbers from largest to smallest: 0.6, $\frac{1}{5}$, 0.15

Ordering Rational Numbers – Application

Try solving these exercises on your own to practise ordering rational numbers!

Ordering Rational Numbers – Summary

Key Learnings from this Text:

  • Rational numbers include whole numbers, fractions and decimals.

  • Convert fractions and decimals to the same form to compare them easily.

  • Remember to align decimal points when comparing decimals.

  • When fractions have the same denominator, the larger numerator means a larger fraction.

Explore more with interactive exercises and quizzes on our platform. Keep practising, and you'll be a pro at ordering rational numbers in no time! If you're looking for more challenges, keep exploring our other resources including Sofahero!

Ordering Rational Numbers – Frequently Asked Questions

How do you compare a fraction and a decimal?
What's the easiest way to compare decimals?
Why is it important to know how to order rational numbers?
Can negative numbers be rational numbers?
What do you do if two fractions have different denominators?
How do you decide which of two decimals is larger?
Is there a quick way to order a mix of fractions and whole numbers?
What happens when two rational numbers are identical in a sequence?
Are all whole numbers also rational numbers?
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