2. If $(a b):(a c):(b c)=5: 3: 1$, calculate the value of the expression $\frac{a^{5} b^{2}}{c^{3}}: \frac{a^{3} b^{5}}{c^{4}}$.
First method:

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

1 POINT

From the ratio $(a b):(a c):(b c)=5: 3: 1$ we get

$(a c):(b c)=3: 1, (a b):(a c)=5: 3$

1 POINT

that is

$\frac{a c}{b c}=\frac{3}{1}, \quad \frac{a}{b}=\frac{3}{1}$;

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$\frac{a b}{a c}=\frac{5}{3}, \quad \frac{b}{c}=\frac{5}{3}, \quad \frac{c}{b}=\frac{3}{5}$.

1 POINT

Substitute the values of $\frac{a}{b}$ and $\frac{c}{b}$ into the initial expression

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

The value of the given expression is $\frac{27}{5}$ or $5 \frac{2}{5}$ or 5.4.