If $a^2$% of $b$ is equal to $b^3$% of $c$ and $c^4$ % of $a$ is equal to $b$ % of $b$, then what is the relation between $a$ and $b?$
According to my way of doing it: $a^2b = b^3c = c^4a = b^2$
Equating $a^2b=b^2$ I got $a^2=b$. But the correct answer is $a^9=b^{10}$. How?
From the hypotheses you have:
$$\frac{a^2}{100}\cdot b=\frac{b^3}{100}\cdot c\\ \frac{c^4}{100}\cdot a=\frac{b}{100}\cdot b$$
Or, simplifying:
$$a^2b=b^3c\\ c^4a=b^2$$
Notice that you do not have equality between the upper and lower expressions (from the hypotheses). Using the upper equality, you get that $c=a^2/b^2$; and substituting this into the lower equality yields
$$\frac{a^8}{b^8}\cdot a=b^2\implies a^9=b^{10}$$
as desired.