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: $5 = \frac{5}{1}$, ; $-3 = \frac{-3}{1}$, ; $0 = \frac{0}{1}$
• Fractions: $\frac{2}{3}$, ; $-\frac{5}{8}$, ; $\frac{7}{4}$
• Terminating decimals: $0.5 = \frac{1}{2}$, ; $0.75 = \frac{3}{4}$
• Repeating decimals: $0.333\ldots = \frac{1}{3}$, ; $0.666\ldots = \frac{2}{3}$

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)

These are called irrational numbers. For this whole lesson, every number we touch will be rational, so we can always think of it as a fraction.

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

Step 1 — Find the LCD (Least Common Denominator). The smallest number both $3$ and $4$ divide into is $12$.

Step 2 — Rewrite each fraction with denominator $12$:

$\frac{1}{3} = \frac{4}{12} \qquad \frac{1}{4} = \frac{3}{12}$

Step 3 — Add the numerators:

$\frac{4}{12} + \frac{3}{12} = \frac{7}{12}$

Answer: $\frac{7}{12}$ ✅

Example 2: Subtracting a Negative

Compute $\frac{2}{3} - \left(-\frac{1}{3}\right)$.

Subtracting a negative is the same as adding a positive:

$\frac{2}{3} - \left(-\frac{1}{3}\right) = \frac{2}{3} + \frac{1}{3} = \frac{3}{3} = 1$

Answer: $1$ ✅

Example 3: Multiplying (Simplify First!)

Compute $\frac{4}{9} \times \frac{3}{8}$.

Tip: Cancel common factors before multiplying to keep numbers small.

• $4$ and $8$ share a factor of $4$: ; $\frac{4}{8} = \frac{1}{2}$
• $3$ and $9$ share a factor of $3$: ; $\frac{3}{9} = \frac{1}{3}$

Now multiply across:

$\frac{1}{3} \times \frac{1}{2} = \frac{1 \times 1}{3 \times 2} = \frac{1}{6}$

Answer: $\frac{1}{6}$ ✅

Example 4: Dividing — Keep, Change, Flip

Compute $\frac{2}{5} \div \frac{3}{4}$.

Keep the first fraction, change $\div$ to $\times$, and flip the second fraction (its reciprocal):

$\frac{2}{5} \div \frac{3}{4} = \frac{2}{5} \times \frac{4}{3} = \frac{8}{15}$

Answer: $\frac{8}{15}$ ✅

Your Turn ✏️

Work through this three-step addition: $\frac{5}{6} + \frac{1}{4}$.

• Box 1: What is the LCD of $6$ and $4$?
• Box 2: Rewrite $\frac{5}{6}$ with that denominator — type the new numerator only.
• Box 3: Write the final sum as a fraction (it does not need to be simplified).

🧭 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.

To find the total sugar, she adds:

$\frac{3}{4} + \frac{1}{4} = \frac{4}{4} = 1 \text{ cup}$

A Temperature Story 🌡️

At noon the temperature was $\frac{1}{2}$ degree above zero. By midnight it had dropped $\frac{5}{2}$ degrees. The new temperature is:

$\frac{1}{2} - \frac{5}{2} = \frac{1 - 5}{2} = \frac{-4}{2} = -2 \text{ degrees}$

So it ended at $-2$ degrees — below zero! Watch how subtraction can take us into negative numbers.

Word Problem Practice ✏️

A recipe calls for $\frac{2}{3}$ cup of flour. You want to make half of the recipe, so you multiply by $\frac{1}{2}$.

• Box 1: Multiply the numerators: $2 \times 1 = ?$
• Box 2: Multiply the denominators: $3 \times 2 = ?$
• Box 3: Write how many cups of flour you need as a fraction.

Real-World Check 🎯

🏆 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:

Remember: subtracting a negative is the same as adding a positive! Now let's mix it all together for a final challenge. 🚀

Mixed Mastery Challenge 🎯

These questions blend every operation. Read carefully and watch the signs!