From Förberedande kurs i matematik 1

Method 1

If we calculate the numerator in the main fraction first, we get

\displaystyle \frac{\frac{1}{4}-\frac{1}{5}}{\frac{3}{16}}=\frac{\frac{1\centerdot 5}{4\centerdot 5}-\frac{1\centerdot 4}{5\centerdot 4}}{\frac{3}{16}}=\frac{\frac{5}{20}-\frac{4}{20}}{\frac{3}{10}}=\frac{\frac{1}{20}}{\frac{3}{10}}

The double fraction on the right-hand side becomes, after multiplying top and bottom by \displaystyle {10}/{3}\; ,

\displaystyle \frac{\frac{1}{20}}{\frac{3}{10}}=\frac{\frac{1}{20}\centerdot \frac{10}{3}}{\frac{3}{10}\centerdot \frac{10}{3}}=\frac{1}{20}\centerdot \frac{10}{3}

Then, we remove the common factor 10:

\displaystyle \frac{1}{20}\centerdot \frac{10}{3}=\frac{1}{2\centerdot 10}\centerdot \frac{10}{3}=\frac{1}{2\centerdot 3}=\frac{1}{6}

Method 2

Another way to calculate the expression is to divide it up into two separate terms:

\displaystyle \frac{\frac{1}{4}-\frac{1}{5}}{\frac{3}{16}}=\frac{\frac{1}{4}}{\frac{3}{10}}-\frac{\frac{1}{5}}{\frac{3}{10}}

We simplify both double fractions on the right-hand side by multiplying top and bottom by 10/3:

\displaystyle \frac{\frac{1}{4}}{\frac{3}{10}}-\frac{\frac{1}{5}}{\frac{3}{10}}=\frac{\frac{1}{4}\centerdot \frac{10}{3}}{\frac{3}{10}\centerdot \frac{10}{3}}-\frac{\frac{1}{5}\centerdot \frac{10}{3}}{\frac{3}{10}\centerdot \frac{10}{3}}=\frac{1}{4}\centerdot \frac{10}{3}-\frac{1}{5}\centerdot \frac{10}{3}

Instead of multiplying, respectively, by \displaystyle 4\centerdot 3 and \displaystyle 5\centerdot 3 , we keep the numerators factorized and observe that if we multiply the top and bottom of the first fraction by \displaystyle 5 and the second by \displaystyle 4 , we obtain the common denominator:

\displaystyle \frac{10}{4\centerdot 3}-\frac{10}{5\centerdot 3}=\frac{10\centerdot 5}{4\centerdot 3\centerdot 5}-\frac{10\centerdot 4}{5\centerdot 3\centerdot 4}=\frac{50-40}{3\centerdot 4\centerdot 5}=\frac{10}{3\centerdot 4\centerdot 5}

Because \displaystyle 10=2\centerdot 5 and \displaystyle 4=2\centerdot 2 , we can cancel out the common factors \displaystyle 2 and \displaystyle 5 and obtain the answer:

\displaystyle \frac{10}{3\centerdot 4\centerdot 5}=\frac{2\centerdot 5}{3\centerdot 2\centerdot 2\centerdot 5}=\frac{1}{6}