Does a/(bc) = (a/b) * (a/c)?

This is part of a series on common misconceptions.

True or False?

$\frac{a}{bc}$ is equal to $\frac{a}{b}$ x $\frac{a}{c}.$

Why some people say it's true: You can multiply fractions like this to get the denominator.

Why some people say it's false: The numerator is $a,$ not $a^2.$

The statement is $\color{#D61F06}{\textbf{false}}$.

Proof:

When multiplying the second fraction, we get $\frac{a^2}{bc}.$ However, $\frac{a}{bc}$=$\frac{1}{bc}$ x a or $\frac{1}{b}$ x $\frac{a}{c}$ or $\frac{\hspace{2mm} \frac a2\hspace{2mm} }{b}$ x $\frac{2}{c},$ etc. $_\square$

Rebuttal: But if $a=1,$ it works.

Reply: But it doesn’t work for all values of $a.$

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Rebuttal: What if $a$ stands for a different number each time?

Reply: That would break the rules of algebra.

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Rebuttal: What if $a$ stands for the square root of anything? $i$ can be $+i$ or $- i$ and then $a$ can be different values.

Reply: $i$ is just $+i$ and $a$ can stand for one value only.

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Rebuttal: So it never works?

Reply: It works if $a=1$ or $a=0.$

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Rebuttal: How can we judge the value if $b=0$ or $c=0?$

Reply: No value could be given to the final fraction, so we can’t do anything, although if $a=2, b=6,$ and $c=4,$ then the statement doesn’t work anyway.

See Also

• List of Common Misconceptions