Simplify fraction - Where did the rest go?

Question

While studying maths I encountered the following fraction:$$\frac{5ab}{10b},$$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.

Answer 1

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 $\frac{a}{2} \times 1 = \frac{a}{2}$.

Answer 2

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

Answer 3

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.

Answer 4

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 $b \not = 0$).