Lesson 42: Renaming Fractions by Multiplying by 1

New Concept

We create different names for a fraction, called equivalent fractions, by multiplying it by a fraction equal to 1. This changes the fraction's name, not its value.

What’s next

Now, let’s see this in action. You'll walk through worked examples of finding equivalent fractions and using them to solve addition problems.

Property

Fractions with the same value but different names are called equivalent fractions. For example, the fractions $\frac{2}{4}$, $\frac{3}{6}$, and $\frac{4}{8}$ are all equivalent to $\frac{1}{2}$.

Examples

To find an equivalent fraction for $\frac{1}{3}$: $\frac{1}{3} \times \frac{2}{2} = \frac{2}{6}$
To find an equivalent fraction for $\frac{2}{5}$: $\frac{2}{5} \times \frac{3}{3} = \frac{6}{15}$
To find a fraction equal to $\frac{1}{2}$ with a denominator of 20: $\frac{1}{2} \times \frac{10}{10} = \frac{10}{20}$

Explanation

Think of it like giving a fraction a disguise! By multiplying it by a special form of 1, like $\frac{2}{2}$ or $\frac{10}{10}$, you change its numerator and denominator but not its actual value. The fraction just gets a new name and look!

Property

The Identity Property of Multiplication states that if one of two factors is 1, the product equals the other factor. So, $a \times 1 = a$.

Examples

Multiplying by 1 directly: $\frac{3}{4} \times 1 = \frac{3}{4}$
Multiplying by a fraction equal to 1: $\frac{1}{2} \times \frac{2}{2} = \frac{2}{4}$
Using the property to rename a fraction: $\frac{3}{5} \times \frac{4}{4} = \frac{12}{20}$

Explanation

Multiplying any number by 1 is like looking in a mirror—it doesn't change! The fun trick is that '1' can dress up as any fraction where the top and bottom match, like $\frac{5}{5}$. This helps us rename other fractions without changing their value.

Property

To add or subtract fractions with different denominators, you must first rename them to have a common denominator. For example, to solve $\frac{1}{2} + \frac{1}{3}$, you first find equivalent fractions with a denominator of 6.

Examples

Add $\frac{1}{2} + \frac{1}{3}$: Rename to $\frac{3}{6} + \frac{2}{6} = \frac{5}{6}$
Add $\frac{1}{3} + \frac{2}{3}$ after renaming to a denominator of 6: $\frac{2}{6} + \frac{4}{6} = \frac{6}{6} = 1$
Subtract $\frac{5}{12} - \frac{1}{6}$: Rename to $\frac{5}{12} - \frac{2}{12} = \frac{3}{12}$

Explanation

You can't add fractions with different denominators, just like you can't add cats and dogs. You first need a common name, or 'common denominator'! We use multiplication to give each fraction a new name so they can finally be added together properly.