Dividing fractions

Quotient of two rational numbers

The quotient of the integer $$-6$$ by the integer $$2$$ is the integer $$-3$$, since: $$-6=2 \cdot (-3)$$

This exercise of multiplying integers can be written as a division:$$$\begin{matrix}(-6) & :2 & =-3 \\\\ \nearrow & \uparrow &\nwarrow \\\\ \mbox{dividend }&\mbox{ divisor }& \mbox{quotient} \end{matrix}$$$

In the same way, the rational number$$\displaystyle \frac{3}{20}$$ can be expressed as the product of the rational number $$\displaystyle \frac{3}{4}$$ by another one. Which rational is this another one?

We can prove that this rational is $$\displaystyle \frac{1}{5}$$:$$$\frac{3}{20}=\frac{3}{4}\cdot \frac{1}{5}=\frac{3 \cdot 1}{4 \cdot 5}$$$And then we say that the quotient of the division of $$\displaystyle \frac{3}{20}$$ by $$\displaystyle \frac{3}{4}$$ is equal to $$\displaystyle \frac{1}{5}$$. In the same way as done with integers, the exercise of multiplying $$\displaystyle \frac{3}{20}=\frac{3}{4}\cdot$$ can be written as a division:$$\displaystyle \frac{3}{20}:\frac{3}{4}=?$$.

Calculating the quotient of two rational numbers

1. The exercise: $$$\displaystyle \frac{3}{20}:\frac{3}{4}=?$$$ can be written as: $$$\displaystyle ?\cdot \frac{3}{4}=\frac{3}{20}$$$

2. Multiplying both terms of the equality by the inverse of the divisor: $$\displaystyle \Big(?\cdot \frac{3}{4}\Big)\cdot \frac{4}{3}=\Big(\frac{3}{20}\cdot \frac{4}{3}\Big)$$

3. Bearing in mind the properties of the product of fractions, we obtain: $$$?\cdot \frac{3}{4}\cdot\frac{4}{3}=\frac{3}{20}\cdot\frac{4}{3}$$$ And as $$\displaystyle \frac{3}{4}\cdot\frac{4}{3}=1$$, we have$$$\displaystyle ?\cdot 1=\frac{3}{20}\cdot \frac{4}{3}=\frac{3\cdot4}{20\cdot 3}=\frac{4}{20}=\frac{1}{5}$$$Therefore:$$$\displaystyle \frac{3}{20}:\frac{3}{4}=\frac{3}{20}\cdot \frac{4}{3}=\frac{1}{5}$$$

Namely, to find the quotient of two rational numbers $$\displaystyle \frac{a}{b}$$ (dividend) and $$\displaystyle \frac{c}{d}$$ (divisor), the divisor being other than zero, it is necessary to multiply the dividend by the inverse of the divisor:$$$\frac{a}{b}:\frac{c}{d}=\frac{a}{b}\cdot\frac{d}{c}$$$

In the same way as with integers, when we have an expression with sums, subtractions, multiplications and divisions of fractions, we must operate first on the brackets, later the multiplications and the divisions and, finally, on the sums and subtractions.

Quotient of a rational number and an integer

To divide an integer $$a$$ by a rational number $$\displaystyle \frac{m}{n}$$ we have to express the integer $$a$$ as $$\displaystyle \frac{a}{1}$$ and proceed as in the previous case:$$$\displaystyle a:\frac{m}{n}=\frac{a}{1}:\frac{m}{n}=\frac{a}{1}\cdot \frac{n}{m}$$$

And, in the same way, to divide a rational number $$\displaystyle \frac{m}{n}$$ by an integer $$a$$, we do: $$$\displaystyle \frac{m}{n}:a=\frac{m}{n}:\frac{a}{1}=\frac{n}{m}\cdot \frac{1}{a}$$$