Fraction Flipping

Edit Notice

2025-07-23

• Moved fraction flipping from Divisibility Theorems

Many of you have probably been taught the trick of flipping the right fraction in a division to instead use simpler multiplication. This is how it works:

We start with the division:

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

From there, we can reconstruct the two fractions as the dividend over the divisor with a horizontal line, for simplicity’s sake, like so:

$$\frac{3}{2}/\frac{4}{5}=\frac{\frac{3}{2}}{\frac{4}{5}}$$

After that, the “trick” can begin. First, we multiply both the dividend and the divisor with the inverse of the divisor. As long as we treat the dividend and the divisor the same way, this is fine. However, keep PEMDAS in mind! If any of the dividend or the divisor would have been an addition or subtraction, you would have to multiply both terms by $x$, either by $x(a+b)$ or $xa+xb$.

$$\frac{3}{2}/\frac{4}{5}=\frac{\frac{3}{2}}{\frac{4}{5}}=\frac{\frac{3}{2}\times \frac{5}{4}}{\frac{4}{5}\times \frac{5}{4}}=\frac{\frac{3}{2}\times \frac{5}{4}}{\frac{4\times 5}{5\times 4}}=\frac{\frac{3}{2}\times \frac{5}{4}}{\frac{20}{20}}=\frac{\frac{3}{2}\times \frac{5}{4}}{1}=\frac{3}{2}\times \frac{5}{4}$$

As you can see, the right fraction has now “flipped”, and not by magic, but with logic and reason. So, as division by 1 is equal to the dividend, we can then solve the expression, like so:

$$\frac{3}{2}\times \frac{5}{4}=\frac{3\times 5}{2\times 4}=\frac{15}{18}$$

As the final cherry on top, we can prove the procedure by dividing 1 by 2, as we know this should result in one half.

$$1/2=\frac{1}{1}/\frac{2}{1}=\frac{\frac{1}{1}}{\frac{2}{1}}$$ $$\frac{1}{1}/\frac{2}{1}=\frac{\frac{1}{1}}{\frac{2}{1}}=\frac{\frac{1}{1}\times \frac{1}{2}}{\frac{2}{1}\times \frac{1}{2}}=\frac{\frac{1}{1}\times \frac{1}{2}}{\frac{2\times 1}{1\times 2}}=\frac{\frac{1}{1}\times \frac{1}{2}}{\frac{2}{2}}=\frac{\frac{1}{1}\times \frac{1}{2}}{1}=\frac{1}{1}\times \frac{1}{2}$$ $$\frac{1}{1}\times \frac{1}{2}=\frac{1\times 1}{1\times 2}=\frac{1}{2}$$ $$\boxed{\frac{a}{b}/\frac{c}{d}=\frac{a}{b}\times \frac{d}{c}}$$