When dividing fractions, why can we flip one fraction and then multiply them together?

Question

When dividing fractions, why can we flip one fraction and then multiply them together?

Answer 1

When dividing fractions, we can flip one fraction and then multiply them together because division is essentially the same as multiplying by the reciprocal.

For example, when dividing fractions 𝑎/𝑏 by 𝑐/𝑑, we can flip the second fraction to get the reciprocal 𝑑/𝑐, and then multiply the fractions together:
(𝑎/𝑏) ÷ (𝑐/𝑑) = (𝑎/𝑏) × (𝑑/𝑐)

This is based on the concept of equivalent fractions. When we flip a fraction, we are essentially finding an equivalent fraction with the numerator and denominator swapped. And when we multiply fractions, we can simply multiply the numerators together and the denominators together to find the product.

Therefore, flipping one fraction and multiplying is a convenient way to handle division of fractions, and it provides the same result as dividing them directly.

Answer 2

When dividing fractions, we can flip one fraction and then multiply them together to find the quotient. This method is also known as "multiplying by the reciprocal" or "cross multiplying."

To understand why this works, let's consider an example.

Suppose we have two fractions: a/b and c/d. To find the quotient of these two fractions, we need to perform the division operation (a/b) / (c/d).

Now, if we multiply the first fraction by the reciprocal of the second fraction, we get (a/b) * (d/c) instead.

Using the definition of multiplication, we multiply the numerators and denominators: (a*d) / (b*c).

By multiplying the numerators, we combine the values of a and d, and by multiplying the denominators, we similarly combine the values of b and c.

So, when we flip the second fraction and multiply the two fractions together, the result is a new fraction with the product of the numerators as the numerator and the product of the denominators as the denominator.

In other words, flipping the second fraction effectively turns the division operation into a multiplication operation. This method is a convenient way to divide fractions and allows us to avoid complex division calculations.