Rational Number Operations

🔢 Rational Number Operations

Part 1 of 5 — Concept Introduction

Numbers are everywhere: temperatures below zero, slices of a pizza, money you owe a friend, the time you ran your last lap. To handle all of these, we need a family of numbers called rational numbers.

What Is a Rational Number?

A rational number is any number that can be written as a fraction $\frac{a}{b}$, where $a$ and $b$ are integers and $b \neq 0$ (you can never divide by zero!).

This one definition pulls together a lot of numbers you already know:

• Integers: $5 = \frac{5}{1}$, ; $-3 = \frac{-3}{1}$, ;

What Is NOT Rational? ⭐

A few special numbers can never be written as a clean fraction of integers. Their decimals go on forever without repeating:

• $\pi = 3.14159\ldots$ (never ends, never repeats)
• $\sqrt{2} = 1.41421\ldots$ (never ends, never repeats)

The Four Operations — A Preview

Because every rational number can be written as a fraction, the rules you learn for fractions work for all of them. Here is the big picture you'll master:

• Add / Subtract: get a common denominator, then combine the numerators.
• Multiply: multiply across — numerators together and denominators together (no common denominator needed!).
• Divide: keep, change, flip — multiply by the reciprocal of the second fraction.

And don't forget the sign rules you already know from integers:

• A negative times or divided by a negative gives a positive.
• A negative times or divided by a positive gives a negative.

We'll use all of these as we go. 💪

Concept Check 🎯

Make sure you can tell a rational number from an irrational one.

📝 Worked Examples

Part 2 of 5 — Worked Examples

Let's work through one example of each operation, slowly and carefully.

Example 1: Adding with Different Denominators

Compute $\frac{1}{3} + \frac{1}{4}$.

🧭 Guided Practice

Part 3 of 5 — Guided Practice

Time to practice all four operations with a little support. Take each problem one step at a time, and watch the signs!

Choose the Correct Answer 🎯

Fill in the Blanks ⬇️

Pick the correct value for each calculation.

🌍 Application & Word Problems

Part 4 of 5 — Application & Word Problems

Rational numbers show up in real life all the time — in recipes, money, distances, and temperatures. The trick is to spot which operation the story is asking for.

A Baking Story 🧁

Maria is making cookies. The recipe needs $\frac{3}{4}$ cup of sugar for the dough and $\frac{1}{4}$ cup more for the topping.

🏆 Review & Challenge

Part 5 of 5 — Review & Challenge

You've learned to add, subtract, multiply, and divide every kind of rational number. Here is the whole toolkit in one place: