Question

Divida 560 em partes diretamente proporcionais a 3, 6 e 7 e inversamente
proporcionais 5, 4 e 2.

Answer 1

Resposta:

Diretamente proporcional:

partes: x, y e z

x + y + z = 560

x/3, y/6 e z/7

$\frac{x + y + z}{3 + 6 + 7} = \frac{x}{3} \\ \frac{560}{16} = \frac{x}{3} \\ 16x = 1680 \\ x = \frac{1680}{16} \\ x = 105$

$\frac{560}{16} = \frac{y}{6} \\ 16y = 3360 \\ y = \frac{3360}{16} \\ y = 210$

$\frac{560}{16} = \frac{z}{7} \\ 16z = 3920 \\ z = \frac{3920}{16} \\ z = 245$

Inversamente proporcional:

partes: x, y e z

x + y + z = 560

x/(⅓), y/(1/6) e z/(1/7)

$\frac{x + y + z}{ \frac{1}{3} + \frac{1}{6} + \frac{1}{7} } = \frac{x}{ \frac{1}{3} } \\ \frac{560}{ \frac{14 + 7 + 6}{42} } = x.3 \\ \frac{560}{ \frac{27}{42} } = 3x \\ \frac{560}{ \frac{9}{14} } = 3x \\ 560 \times \frac{14}{9} = 3x \\ \frac{7840}{9} = 3x \\ 27x = 7840 \\ x = \frac{7840}{27}$

$\frac{7840}{9} = 6y \\ 54y = 7840 \\ y = \frac{7840}{54} \\ y = \frac{3920}{27}$

$\frac{7840}{9} = 7z \\ 63z = 7840 \\ z = \frac{7840}{63} \\ z = \frac{1120}{9}$