Interpreting $a^2b \div \frac13a^2b^3$. The maths book I'm using shows:
$$ a^2b \div \frac13a^2b^3 $$
Which would be something like:
$$ a^2 \cdot b ÷ \frac13 \cdot a^2 \cdot b^3 $$
My understanding of order of operations it would equal:
$$ 3a^4b^4 $$
However in the book the second term is evaluated before the division (switching it to multiplication) and equaling:
$$ a^2b \cdot \frac{3}{a^2b^3} = \frac{3a^2b}{a^2b^3} = \frac{3}{b^2} $$
If a multiplication or division signs shows between terms, should that be taken as multiplying or dividing one term by the other?

In this form
$$a^2b \div \frac13a^2b^3$$
my interpretation is to consider the two terms as a whole, that is
$$\left(a^2b\right) \div \left(\frac13a^2b^3\right)=\frac{3a^2b}{a^2b^3}=\frac 3 {b^2}$$
Otherwise, in this form
$$a^2 \times b ÷ \frac13 \times a^2 \times b^3$$
whithout parenthesis my interpretation would be
$$a^2 \times \left(b ÷ \frac13\right) \times a^2 \times b^3=3a^4b^4$$
