Flip and Multiply

What is \frac{2}{3} \div \frac{7}{5} ?

Some of you may remember the simple rule for dealing with fraction division. If you encounter a question about the division of fractions, simply flip the second fraction and change the division to multiplication. Flip and multiply.


Why does this work? To justify this technique, we need to remember a crucial rule in math. If we see a division, we can change the question into a fraction, or vise-versa. For example:

4\div3=\frac{4}{3} and \frac{11}{5}=11\div 5

Here is the argument:

\frac{2}{3}\div\frac{7}{5}=\cfrac{\frac{2}{3}}{\frac{7}{5}}    because division can be converted to a fraction

\cfrac{\frac{2}{3}}{\frac{7}{5}}=\cfrac{\frac{2}{3}\times\frac{5}{7}}{\frac{7}{5}\times\frac{5}{7}}    because we can multiply the numerator and the denominator of a fraction by the same quantity

\cfrac{\frac{2}{3}\times\frac{5}{7}}{\frac{7}{5}\times\frac{5}{7}}=\cfrac{\frac{2}{3}\times\frac{5}{7}}{\frac{35}{35}}    by multiplying out the denominator

\cfrac{\frac{2}{3}\times\frac{5}{7}}{\frac{35}{35}}=\cfrac{\frac{2}{3}\times\frac{5}{7}}{1}     by simplifying the denominator

\cfrac{\frac{2}{3}\times\frac{5}{7}}{1}=\frac{2}{3}\times\frac{5}{7}\div 1    because a fraction can be converted to division

\frac{2}{3}\times\frac{5}{7}\div 1=\frac{2}{3}\times\frac{5}{7}    because anything divided by 1 remains the same]

Thus \frac{2}{3}\div\frac{7}{5}=\frac{2}{3}\times\frac{5}{7}

The above technique works with any fraction division question. Hence, the flip and multiply shortcut will work for any fraction division question. QED


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