Lesson 10: Rational Numbers and Equivalent Fractions

New Concept

This course builds your mathematical foundation, starting with a key idea: Rational numbers are numbers that can be expressed as a ratio of two integers.

What’s next

Now, let’s begin our journey. In this first lesson, you'll see how rational numbers work through examples on equivalent fractions and comparing values.

Property

Rational numbers are numbers that can be expressed as a ratio of two integers. Any integer can be written as a fraction with a denominator of 1, so integers are rational numbers.

Examples

• $2$ is a whole number, an integer, and a rational number.
• $-3$ is an integer and a rational number.
• $\frac{3}{4}$ is a rational number.

Explanation

Think of number sets like clubs! The 'Whole Numbers' club handles adding and multiplying. The 'Integers' club adds subtraction. But the 'Rationals' club can do it all, including division (just not by zero), making it the most powerful and versatile number team around. It includes whole numbers, integers, and fractions.

Property

We can form equivalent fractions by multiplying a fraction by a fraction equal to 1. Multiplying by a fraction equal to 1 does not change the size of the fraction, but it changes the name of the fraction.

Examples

• To give $\frac{1}{2}$ a new denominator of 100, we multiply by $\frac{50}{50}$: $\frac{1}{2} \cdot \frac{50}{50} = \frac{50}{100}$.
• To compare $\frac{2}{3}$ and $\frac{3}{5}$, find a common denominator of 15: $\frac{2}{3} \cdot \frac{5}{5} = \frac{10}{15}$ and $\frac{3}{5} \cdot \frac{3}{3} = \frac{9}{15}$.

Explanation

This is like giving a fraction a secret identity! By multiplying it by a form of 1, like $\frac{5}{5}$, you change its name without changing its value. It's the perfect disguise for when you need to find a common denominator to compare two different fractions or add them together.

Property

We reduce a fraction by removing pairs of factors that the numerator and denominator have in common. To do this, we can write the prime factorization of the terms and then simplify.

Examples

• Reduce $\frac{12}{18}$: $\frac{2 \cdot 2 \cdot 3}{2 \cdot 3 \cdot 3} = \frac{\cancel{2} \cdot 2 \cdot \cancel{3}}{\cancel{2} \cdot \cancel{3} \cdot 3} = \frac{2}{3}$.
• Reduce $\frac{72}{108}$ using prime factorization: $\frac{\cancel{2} \cdot \cancel{2} \cdot 2 \cdot \cancel{3} \cdot \cancel{3}}{\cancel{2} \cdot \cancel{2} \cdot 3 \cdot \cancel{3} \cdot \cancel{3}} = \frac{2}{3}$.

Explanation

Reducing a fraction is like spring cleaning for numbers. You break down the numerator and denominator into their prime factors, find any matching pairs, and cancel them out! This leaves you with the simplest, tidiest version of the fraction, which is much easier to work with. It's the ultimate decluttering tool!