How to Simplify Multi-Step Rational Number Expressions
Simplifying multi-step rational number expressions can feel overwhelming at first, especially when fractions, negative numbers, and multiple operations are all involved. The good news is that there are strategies to help make these expressions easier to work with. In this blog, we’ll walk through the core ideas behind simplifying rational numbers, explain what are rational numbers, and work through step-by-step examples to show how it’s done.
What Are Rational Numbers?
Before jumping into complex expressions, it helps to understand the building blocks. A rational number is any number that can be written as a fraction where the numerator and denominator are integers and the denominator is not zero. In other words:
Rational Number = a ⁄ b, where a and b are integers and b ≠ 0
Examples of rational numbers include:
- Fractions: 1/2, 3/4, −5/6
- Integers: 5 = 5/1, −3 = −3/1
- Terminating decimals: 0.25 = 1/4, 1.5 = 3/2
- Repeating decimals: 0.333… = 1/3
Why Follow the Order of Operations?
When working with multi-step rational number expressions, completing the steps in the wrong order can lead to incorrect results. That’s why it’s important to follow the order of operations. The order of operations, often remembered by PEMDAS tells us to:
- Evaluate Parentheses
- Simplify Exponents
- Do Multiplication and Division (from left to right)
- Finish with Addition and Subtraction (from left to right)
Following this rule is essential when simplifying expressions that involve rational numbers and more than one operation.
How to Simplify the Expression
Let’s break down several kinds of steps you may see in multi-step expressions.
Example 1: Rational Number Operations
Suppose you have an expression like: (3/4) + (2/3 × 6/1)
- Multiply before adding: 2/3 × 6/1 = 12/3 = 4
- Now add: 3/4 + 4 = 3/4 + 16/4 = 19/4
So the simplified expression is 19/4.
This follows the order of operations, multiplying before adding, and demonstrates how rational numbers interact in steps.
Example 2: Complex Multi-Step Rational Expression
You might see something like: 1 1/2 – (3/4 × 8) ÷ 2
- Convert mixed numbers to improper fractions if needed.
- Do the multiplication first: 3/4 × 8 = 24/4 = 6
- Then division: 6 ÷ 2 = 3
- Finally perform subtraction: 1½ – 3 = 3/2 – 3 = 3/2 – 6/2 = −3/2
By applying order of operations, we get the correct result.
Tips for Simplifying Expressions with Rational Numbers
- Always simplify fractions as soon as you can to keep numbers manageable.
- If an expression has parentheses, simplify inside them first.
- Convert mixed numbers to improper fractions early so all numbers are in the same form before operating.
- Be consistent: multiplication and division stay on the same level, as do addition and subtraction.
Why This Matters
Mastering the simplification of multi-step rational number expressions goes beyond finding the correct answer. It deepens your understanding of how numbers work together and lays the foundation for more advanced topics such as algebra and rational expressions.
Want to see these strategies in action?
Watch our full video on YouTube for a step-by-step walkthrough and more helpful tips to support your learning:
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