For clarity, here is the conclusion in fraction notation:

$\frac{a}{b}\cdot \frac{c}{d}=\frac{a\cdot c}{b\cdot d}$

This theorem states that when we are multiplying fractions, the numerator is the product of numerators and the denominator is the product of denominators.

Before you read the proof, we encourage you to try to prove this theorem on your own.

This proof begins with the following theorem:

$\frac{1}{b}\cdot \frac{1}{d}=\frac{1}{b\cdot d}$

The proof multiplies both sides of this equation by a ⋅ c. It then shows that the left side is equal to

$\frac{a}{b}\cdot \frac{c}{d}$

and the right side is equal to

$\frac{a\cdot c}{b\cdot d}$

Quiz (1 point)

• a ⋅ (1 / b) = a / b
• (a ⋅ c) ⋅ ((1 / b) ⋅ (1 / d)) = (a / b) ⋅ (c / d)

• if the following are true:

  • not (a = 0)
  • not (b = 0)
  • not (a ⋅ b = 0)

  then (1 / a) ⋅ (1 / b) = 1 / (a ⋅ b)

• if (1 / b) ⋅ (1 / d) = 1 / (b ⋅ d), then (a ⋅ c) ⋅ ((1 / b) ⋅ (1 / d)) = (a ⋅ c) ⋅ (1 / (b ⋅ d))

• if the following are true:

  • a = b
  • a = c
  • b = d

  then c = d

Please write your proof in the table below. Each row should contain one claim. The last claim is the statement that you are trying to prove.