Question

divida o número 3161 em partes inversamente a 2/3, 4/5 e 7/8.​

Answer 1

Seja "x" o coeficiente de proporcionalidade, temos:

$\rightarrow~''Parte~inversamente~proporcional~a~\frac{2}{3}'':~x~.~\frac{1}{\frac{2}{3}}~=~\boxed{\frac{3}{2}x}\\\\\\\rightarrow~''Parte~inversamente~proporcional~a~\frac{4}{5}'':~x~.~\frac{1}{\frac{4}{5}}~=~\boxed{\frac{5}{4}x}\\\\\\\rightarrow~''Parte~inversamente~proporcional~a~\frac{7}{8}'':~x~.~\frac{1}{\frac{7}{8}}~=~\boxed{\frac{8}{7}x}$

A soma das 3 partes deve valer 3161, logo:

$\frac{3}{2}x~+~\frac{5}{4}x~+~\frac{8}{7}x~=~3161\\\\\\MMC~=~56\\\\\\\frac{28~.~3x~+~14~.~5x~+~8~.~8x}{56}~=~3161\\\\\\84x~+~70x~+~64x~=~3161~.~56\\\\\\218x~=~177016\\\\\\x~=~\frac{177016}{218}\\\\\\\boxed{x~=~812}$

Com o "x" determinado podemos achar as partes:

$\rightarrow~''Parte~inversamente~proporcional~a~\frac{2}{3}'':~\frac{3}{2}~.~812~=~\boxed{1218}\\\\\\\rightarrow~''Parte~inversamente~proporcional~a~\frac{4}{5}'':~\frac{5}{4}~.~812~=~\boxed{1015}\\\\\\\rightarrow~''Parte~inversamente~proporcional~a~\frac{7}{8}'':~\frac{8}{7}~.~812~=~\boxed{928}$