Why does multiplying by $5/5$ or $4/4$ keep a fraction equivalent?

Conditions

• The factor used is a fraction whose numerator and denominator are equal and nonzero.
• The original fraction is being rewritten to obtain a specific common denominator.

Reasoning, step by step

1. Recognize that $\frac{5}{5} = 1$ and $\frac{4}{4} = 1$ because any nonzero number divided by itself equals 1.
2. Apply the multiplicative identity property: multiplying any number by 1 leaves its value unchanged.
3. Multiply the numerator and denominator of the original fraction by the same factor (e.g., $\frac{6}{4} \cdot \frac{5}{5} = \frac{30}{20}$).
4. Observe that both the numerator and denominator are scaled by the same amount, preserving the ratio.
5. Conclude that the new fraction is equivalent to the original fraction, differing only in form.

Example

The instructor explicitly justifies the denominator change by stating, 'if you're multiplying by five over five, that's one, you're not changing the value, and you're not changing the value by multiplying by four over four.' This converts $\frac{6}{4}$ to $\frac{30}{20}$ and $\frac{2}{5}$ to $\frac{8}{20}$ without altering their values.

Common misconceptions

• Believing that changing the denominator of a fraction necessarily changes its numerical value.
• Confusing equivalent fractions (same value, different form) with equal fractions in a different sense.

Watch the explanation

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