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While studying maths I encountered the following fraction :


Which I then had to simplify. The answer I came up with is:

$\frac{5ab}{10b} = \frac{ab}{2b}$

But the correct answer seemed to be:

$\frac{5ab}{10b} = \frac{a}{2} = \frac{1}{2}$a

Why is the above answer correct and mine wrong? I can't wrap my head around $b$ just disappearing like that.

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if $b \neq 0$ answer is correct – pedja Oct 20 '11 at 12:07
up vote 6 down vote accepted

It's ok for $b$ to disappear. You can divide your fraction $\frac{ab}{2b}$ into a product $\frac{a}{2} \times \frac{b}{b}$. Provided that $b \neq 0$, then $\frac{b}{b}$ will always be $1$, and any real number $x$ times $1$ will always equal $x$. So $\frac{a}{2} \times 1 = \frac{a}{2}$.

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Thanks. This really explains why $b$ can be left out. – Kevin Oct 20 '11 at 12:10

To get from $\dfrac{5ab}{10b}$ to $\dfrac{ab}{2b}$ you probably divided the numerator and denominator each by $5$.

Now divide them each by $b$ (if $b \not = 0$).

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If you are OK with cancelling the factors of 5 in the numerator and the denominator, well, this is just cancelling the factors of $b$ in the numerator and denominator.

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It is because you left $b$ which is an integer in the fraction.

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