Question

Determinați numerele raționale pozitive a,b,c care sunt invers proporționale cu numerele 0,1(6); 0,125 și 0,08(3), iar 6abc+5bc-8ac= 1/3abc.

Answer 1

Explicație pas cu pas:

$0,1(6) = \frac{16 - 1}{90} = \frac{15}{90} = \frac{1}{6}$

$0,125 = \frac{125}{1000} = \frac{1}{8}$

$0,08(3) = \frac{83 - 8}{900} = \frac{75}{900} = \frac{1}{12}$

$\frac{a}{ \frac{1}{6} } = \frac{b}{ \frac{1}{8} } = \frac{c}{ \frac{1}{12} } \iff 6a = 8b = 12c \\ \iff 3a = 4b = 6c = k$

$a = \frac{k}{3} ; b = \frac{k}{4} ; c = \frac{k}{6}$

$6abc + 5bc - 8ac = \frac{abc}{3} \\18abc + 15bc - 24ac = abc \\ 17abc - 3c(8a - 5b) = 0$

c ≠ 0$\implies 17ab - 3(8a - 5b) = 0$

$\frac{17 {k}^{2} }{3 \times 4} - 3 (\frac{8k}{3} - \frac{5k}{4}) = 0 \\ \frac{17 {k}^{2} }{12} - \frac{3(32k - 15k)}{3 \times 4} = 0 \\ 17 {k}^{2} - 3 \times 17k = 0 \\ 17k(k - 3) = 0$

k ≠ 0$\implies k = 3$

$a = \frac{k}{3} = \frac{3}{3} = 1 \implies a = 1$

$b = \frac{k}{4} = \frac{3}{4}\implies b = \frac{3}{4}$

$c = \frac{k}{6} = \frac{3}{6} = \frac{1}{2}\implies c = \frac{1}{2}$