2. Zadatak

Prvi način:

Razmjer $a:b=2:5$ možemo napisati u obliku $\frac{a}{b}=\frac{2}{5}$.

Podijelimo li brojnik i nazivnik zadanog izraza s $b^2$ i uvrstimo da je $\frac{a}{b}=\frac{2}{5}$, dobivamo

$$\begin{aligned} \frac{a^2}{ab+b^2} &=\frac{\frac{a^2}{b^2}}{\frac{ab}{b^2}+\frac{b^2}{b^2}} =\frac{\left(\frac{a}{b}\right)^2}{\frac{a}{b}+1}\\ &=\frac{\left(\frac{2}{5}\right)^2}{\frac{2}{5}+1} =\frac{\frac{4}{25}}{\frac{7}{5}} =\frac{4\cdot5}{25\cdot7} =\frac{4}{35}. \end{aligned}$$

Drugi način:

Razmjer $a:b=2:5$ možemo napisati u obliku $a=\frac{2}{5}b$.

$$\begin{aligned} \frac{a^2}{ab+b^2} &=\frac{\left(\frac{2}{5}b\right)^2}{\frac{2}{5}b\cdot b+b^2} =\frac{\frac{4}{25}b^2}{\frac{2}{5}b^2+b^2}\\ &=\frac{\frac{4}{25}b^2}{\frac{7}{5}b^2} =\frac{\frac{4}{25}}{\frac{7}{5}} =\frac{4\cdot5}{25\cdot7} =\frac{4}{35}. \end{aligned}$$

Treći način:

Razmjer $a:b=2:5$ možemo napisati u obliku $b=\frac{5}{2}a$.

$$\begin{aligned} \frac{a^2}{ab+b^2} &=\frac{a^2}{a\cdot\frac{5}{2}a+\left(\frac{5}{2}a\right)^2} =\frac{a^2}{\frac{5}{2}a^2+\frac{25}{4}a^2}\\ &=\frac{a^2}{\frac{35}{4}a^2} =\frac{1}{\frac{35}{4}} =\frac{4}{35}. \end{aligned}$$

Četvrti način:

Razmjer $a:b=2:5$ možemo napisati u obliku $\frac{a}{b}=\frac{2}{5}$.

$$\begin{aligned} \frac{a^2}{ab+b^2} &=\frac{a^2}{b(a+b)} =\frac{a}{b}\cdot\frac{a}{a+b} =\frac{2}{5}\cdot\frac{\frac{a}{b}}{\frac{a}{b}+1}\\ &=\frac{2}{5}\cdot\frac{\frac{2}{5}}{\frac{2}{5}+1} =\frac{2}{5}\cdot\frac{\frac{2}{5}}{\frac{7}{5}} =\frac{2}{5}\cdot\frac{2}{7} =\frac{4}{35}. \end{aligned}$$

Peti način:

Budući da vrijedi $a:b=2:5$, postoji racionalan broj $k$ takav da je $a=2k$ i $b=5k$.

$$\begin{aligned} \frac{a^2}{ab+b^2} &=\frac{(2k)^2}{2k\cdot5k+(5k)^2} =\frac{4k^2}{10k^2+25k^2}\\ &=\frac{4k^2}{35k^2} =\frac{4}{35}. \end{aligned}$$