In each of the division problems above, observe how we found the answer. Can we frame a rule that tells us how to divide two fractions?

Question

Homi · Accepted Answer

Understand the Question

Division by a number is the same as multiplying by its reciprocal (multiplicative inverse).
To divide one fraction by another, we keep the first fraction unchanged, replace the division sign with a multiplication sign, and invert (flip) the second fraction.

Step 1 · Understanding Reciprocals

The reciprocal of a fraction is obtained by swapping its numerator and denominator. The product of a non-zero fraction and its reciprocal is always $1$ . Diagram 1

For example, the reciprocal of $\dfrac{2}{3}$ is $\dfrac{3}{2}$ :

\begin{aligned} \dfrac{2}{3} \times \dfrac{3}{2} &= \dfrac{2 \times 3}{3 \times 2} \\[0.6em] &= \dfrac{6}{6} \\[0.6em] &= 1 \end{aligned}

Step 2 · Formulate the Division Rule

To divide two fractions, multiply the first fraction by the reciprocal of the second fraction.

For example, dividing $\dfrac{1}{2}$ by $\dfrac{3}{4}$ :

\begin{aligned} \dfrac{1}{2} \div \dfrac{3}{4} &= \dfrac{1}{2} \times \dfrac{4}{3} \\[0.6em] &= \dfrac{1 \times 4}{2 \times 3} \\[0.6em] &= \dfrac{4}{6} \\[0.6em] &= \dfrac{2}{3} \end{aligned}

In general: $\dfrac{a}{b} \div \dfrac{c}{d} = \dfrac{a}{b} \times \dfrac{d}{c}$

Answer

To divide two fractions, multiply the first fraction by the reciprocal of the second fraction: $\dfrac{a}{b} \div \dfrac{c}{d} = \dfrac{a}{b} \times \dfrac{d}{c}$

Common Mistakes

Inverting the Dividend: Flipping the first fraction instead of the second fraction (divisor). Only the fraction after the $\div$ symbol should be inverted.
Forgetting to Change the Sign: Taking the reciprocal of the divisor but keeping the division sign instead of changing it to multiplication ( $imes$ ).

Question 2

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