If $a$ is three times $b$, $b$ is twice $c$, and $c$ is four times $d$, what is the value of $\frac{a\cdot c}{b\cdot d}$?
I can use the given relationships to express $a, b, c, d$ in terms of one variable, say $d$. If $c$ is four times $d$, then $c = 4d$. If $b$ is twice $c$, then $b = 2c = 2(4d) = 8d$. If $a$ is three times $b$, then $a = 3b = 3(8d) = 24d$. Now I can plug these expressions into the fraction $\frac{a\cdot c}{b\cdot d}$ and simplify. I get $\frac{a\cdot c}{b\cdot d} = \frac{(24d)\cdot (4d)}{(8d)\cdot d} = \frac{96d^2}{8d^2} = \frac{96}{8} = 12$. 