6. The sum of three fractions is $\frac{83}{72}$, where their numerators are in the ratio $5: 7: 1$. The denominator of the third fraction is in the ratio $1: 4$ to the denominator of the first fraction, and the denominator of the second fraction to the denominator of the third fraction is in the ratio $3: 2$. Determine these fractions if they are fully simplified.
6. Let $\frac{x}{a}, \frac{y}{b}$, and $\frac{z}{c}$ be the sought fractions.

Then $\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=\frac{83}{72}$ and $x: y: z=5: 7: 1$,

so it follows that $x=5k, y=7k, z=k$.

1 POINT

Furthermore, from $c: a=1: 4$ it follows that $a=4c$.

1 POINT

From $b: c=3: 2$ it follows that $b=\frac{3}{2}c$.

1 POINT

By substituting the obtained expressions into the initial equation, we get:

$\frac{5k}{4c}+\frac{7k}{\frac{3}{2}c}+\frac{k}{c}=\frac{83}{72}$

$\frac{5k}{4c}+\frac{14k}{3c}+\frac{k}{c}=\frac{83}{72}$

1 POINT

$\frac{15k+56k+12k}{12c}=\frac{83}{72}$

1 POINT

$\frac{83k}{12c}=\frac{83}{72}$

The fractions are reduced to their simplest form, so it follows 1 POINT

$83k=83$ and $12c=72$,

i.e., $k=1$ and $c=6$.

1 POINT

It follows that $x=5, y=7, z=1$

1 POINT

and $c=6, a=24, b=9$.

1 POINT

The sought fractions are $\frac{5}{24}, \frac{7}{9}$, and $\frac{1}{6}$.

1 POINT

TOTAL 10 POINTS