Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

As much as it embarasses me to say it, but I always had a hard time understanding the following equality:

$$ \frac{a}{\frac{b}{x}} = x \times \frac{a}{b} $$

I always thought that the left-hand side of the above equation was equivalent to

$$ \frac{a}{\frac{b}{x}} = \frac{a}{b} \div \frac{x}{1} = \frac{a}{b} \times \frac{1}{x} $$

What am I doing wrong, here?

share|cite|improve this question
Do a few calculations where the fractions collapse, like $a=48$, $b=20$, $x=5$. – André Nicolas Dec 5 '12 at 5:33
You have to distinguish between $\dfrac{a}{\left(\dfrac{b}{x}\right)}$ and $\dfrac{\left(\dfrac{a}{b}\right)}{x}$. E.g., $8/(4/2)\neq (8/4)/2$. – Jonas Meyer Dec 5 '12 at 5:34
up vote 7 down vote accepted

$\frac{a}{b}\times\frac{1}{x}=\frac{a}{b}\div x=\frac{\frac{a}{b}}{x}\neq\frac{a}{\frac{b}{x}}=a\div\frac{b}{x}=a\times \frac{x}{b}$

share|cite|improve this answer
Delectably symmetric answer. – Thomas Dec 5 '12 at 9:28

You’re treating the fraction $$\frac{a}{b/x}$$ as if it were $$\frac{a/b}{1/x}\;.$$ If you apply the rule invert the denominator and multiply to $$\frac{a}{\frac{b}x}=\frac{a}{b/x}\;,$$ you get $$a\cdot\frac{x}b=\frac{ax}b\;.$$

To see that this really is correct, remember that the statement that $\dfrac{p}q=r$ means that $p=qr$. Thus, if $\dfrac{a}{b/x}$ really is $\dfrac{ax}b$, we should find that


which you can check is indeed the case.

share|cite|improve this answer

The problem is that $\frac{a}{\frac{b}{x}}=\frac{\frac{a}{b}}{\frac{1}{x}}=x\frac{a}{b}$ because $\frac{1}{\frac{1}{x}}=x$.

share|cite|improve this answer

$\frac{a}{\frac{b}{x}}$ means a divided by $\frac{b}{x}$.

Note that

$$\frac{b}{x}\frac{x}{b} =1$$

This means that

$$a \frac{b}{x}\frac{x}{b} = a$$

Now divide both sides by $\frac{b}{x}$ and you get

$$a\frac{x}{b} = \frac{a}{ \frac{b}{x}}$$

The mistake you make is confusing $\frac{a}{\frac{b}{x}}$ with $\frac{\frac{a}{b}}{x}$. In general

$$\frac{a}{\frac{b}{x}}\neq \frac{\frac{a}{b}}{x}$$

as they have different meanings.

share|cite|improve this answer

It might help to use the definition of equality of fractions, which says that two fractions $a/b$ and $c/d$ are equal if and only if $ad = bc$.

EDIT: I think that it is always a bad idea to use the notation $\frac{a}{\frac{b}{x}}$, even though the relative length of the bars supposedly makes it unambiguous. In my experience there is a strong correlation between people who use this notation and people who are confused. So it may help to avoid writing such things.

share|cite|improve this answer

$$ \frac{a}{\large\frac{b}{x}} = \large\frac {x}{\not x} \frac{a}{\frac{b}{\not x}} = x\cdot \frac{a}{b} $$

Multiplying by $\dfrac xx = 1$ does not change the expression; but by multiplying numerator and denominator by $x$, the numerator becomes $ax$ and the denominator becomes $x\cdot \dfrac{b}{x} = b$

share|cite|improve this answer

It seems that your problem is mostly notational. For example,

$\dfrac{\big(\frac{a}{b}\big)}{c} = \dfrac{a}{bc}$, but $\dfrac{a}{\big( \frac{b}{c} \big)} = \dfrac{ac}{b}$, and these mean different things.

As for the intuition, I have always found that in doubt, you should try with 'natural' feeling numbers. For example, try $a =1, b = c = 2$, and you should be convinced of the differences.

share|cite|improve this answer

It's really a question of which fraction is "inside" another fraction. Another way of looking at your top equation is:

$$\frac{a}{\frac{b}{x}}=a \div \frac{b}{x} = a \times \frac{x}{b} = \frac{ax}{b}$$

Hope that helps.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.