Dividing Fractions

At first glance, dividing fractions might seem complicated. However, beneath its surface lies a straightforward and logical process. You don’t need to reinvent the wheel or get caught up in lengthy calculations. The key idea is this:

Dividing by a fraction is the same as multiplying by its reciprocal.  $$ \frac{a}{b} \div \frac{c}{d} \Leftrightarrow  \frac{a}{b} \cdot \frac{d}{c} $$

But what does that actually mean? Let’s break it down with a practical example.

Imagine you need to divide these two fractions:

$$ \frac{3}{4} \div \frac{2}{5} $$

The first step is to find the reciprocal of the second fraction.

The reciprocal of a fraction is simply the fraction flipped—swap the numerator and denominator. In this case, the reciprocal of \(\frac{2}{5}\) is \(\frac{5}{2}\).

Next, turn the division into multiplication. Replace the division symbol (\(÷\)) with a multiplication symbol (\( \cdot \)) and use the reciprocal of the second fraction:

$$ \frac{3}{4} \div \frac{2}{5} = \frac{3}{4} \cdot \frac{5}{2} $$

Now, multiply the fractions by multiplying the numerators together and the denominators together:

$$ \frac{3 \cdot 5}{4 \cdot 2} = \frac{15}{8} $$

And that’s it! The result of the division is \(\frac{15}{8}\), which is an improper fraction. You can also express it as a mixed number: \(1 + \frac{7}{8}\).

$$ \frac{3 \cdot 5}{4 \cdot 2} = \frac{15}{8} = \frac{8+7}{8} = \frac{8}{8} + \frac{7}{8} = 1 + \frac{7}{8} $$

Once you get the hang of this rule, dividing fractions will feel simple and intuitive.

Why Does This Rule Work?

To understand why dividing by a fraction is equivalent to multiplying by its reciprocal, let’s revisit the mathematical definition of division.

$$ \frac{a}{b} \div \frac{c}{d} = x $$

Here, $ \frac{a}{b} $ is the dividend, $ \frac{c}{d} $ is the divisor, and $ x $ is the quotient.

In division, the quotient ($ x $) is the number that, when multiplied by the divisor ($ \frac{c}{d} $), equals the dividend ($ \frac{a}{b} $):

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

To solve for $ x $, multiply both sides of the equation by the reciprocal of the divisor, $ \frac{d}{c} $:

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

Since the product of a fraction and its reciprocal is always 1, we get:

$$ x \cdot  1 = \frac{a}{b} \cdot \frac{d}{c} $$

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

This shows that the quotient ($ x $) is the product of the dividend ($ \frac{a}{b} $) and the reciprocal of the divisor ($ \frac{d}{c} $).

Now that you understand how it works, there’s no reason to let fractions intimidate you. Tackle them with confidence, and you’ll find they’re not so tricky after all!