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.

\frac{2}{3}\div\frac{7}{5}=\frac{2}{3}\times\frac{5}{7}=\frac{10}{21}

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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