2. Zadatak

Prvi način:

$$\frac{a^5b^2}{c^3}:\frac{a^3b^5}{c^4}=\frac{a^5b^2}{c^3}\cdot\frac{c^4}{a^3b^5}=\frac{a^2c}{b^3}.$$

Iz omjera $(ab):(ac):(bc)=5:3:1$ dobijemo

$$(ac):(bc)=3:1,\quad (ab):(ac)=5:3$$

odnosno

$$\frac{ac}{bc}=\frac{3}{1},\quad \frac{a}{b}=\frac{3}{1};$$ $$\frac{ab}{ac}=\frac{5}{3},\quad \frac{b}{c}=\frac{5}{3},\quad \frac{c}{b}=\frac{3}{5}.$$

Uvrstimo vrijednosti $\frac{a}{b}$ i $\frac{c}{b}$ u početni izraz

$$\frac{a^2c}{b^3}=\frac{a^2}{b^2}\cdot\frac{c}{b}=\left(\frac{a}{b}\right)^2\cdot\frac{c}{b}=\left(\frac{3}{1}\right)^2\cdot\frac{3}{5}=\frac{9}{1}\cdot\frac{3}{5}=\frac{27}{5}.$$

Vrijednost zadanog izraza je $\frac{27}{5}$ ili $5\frac{2}{5}$ ili $5.4$.

Drugi način:

$$\frac{a^5b^2}{c^3}:\frac{a^3b^5}{c^4}=\frac{a^5b^2}{c^3}\cdot\frac{c^4}{a^3b^5}=\frac{a^2c}{b^3}.$$

Iz omjera $(ab):(ac):(bc)=5:3:1$ dobijemo

$$(ac):(bc)=3:1\qquad (ab):(ac)=5:3$$ $$\frac{ac}{bc}=\frac{3}{1}\qquad \frac{ab}{ac}=\frac{5}{3}$$ $$\frac{a}{b}=\frac{3}{1}\qquad \frac{b}{c}=\frac{5}{3}$$ $$a:b=3:1\qquad b:c=5:3$$

Omjer $a:b=3:1$ proširimo brojem 5 pa vrijedi $a:b=15:5$.

Iz dva jednostavna omjera dobijemo prošireni omjer

$$a:b=15:5\quad\text{i}\quad b:c=5:3\quad\Longrightarrow\quad a:b:c=15:5:3.$$

Slijedi da je $a=15k$, $b=5k$, $c=3k$. Uvrstimo $a$, $b$ i $c$ u zadani izraz

$$\frac{a^5b^2}{c^3}:\frac{a^3b^5}{c^4}=\frac{a^2c}{b^3}=\frac{(15k)^2\cdot3k}{(5k)^3}=\frac{15\cdot15\cdot3\cdot k^3}{5\cdot5\cdot5\cdot k^3}=\frac{27}{5}.$$

Treći način:

$$\frac{a^5b^2}{c^3}:\frac{a^3b^5}{c^4}=\frac{a^5b^2}{c^3}\cdot\frac{c^4}{a^3b^5}=\frac{a^2c}{b^3}.$$

Iz omjera $(ab):(ac):(bc)=5:3:1$ dobijemo

$$(ac):(bc)=3:1\qquad (ab):(ac)=5:3$$ $$\frac{ac}{bc}=\frac{3}{1}\qquad \frac{ab}{ac}=\frac{5}{3}$$ $$\frac{a}{b}=\frac{3}{1}\qquad \frac{b}{c}=\frac{5}{3}$$ $$a=3b\qquad c=\frac{3b}{5}$$

Uvrstimo $a$ i $c$ u zadani izraz

$$\frac{a^5b^2}{c^3}:\frac{a^3b^5}{c^4}=\frac{a^2c}{b^3}=\frac{(3b)^2\cdot\frac{3b}{5}}{b^3}=\frac{9b^2\cdot3b}{5b^3}=\frac{27}{5}.$$