$\begin{gathered}\begin{array}{ccc}\\\hspace{80pt}\sf{\dfrac{m}{2}= \dfrac{n}{6} = \blue{\boldsymbol{\sf{k}}}}\\\\\boldsymbol{\bigcirc \kern-11.5pt \blacktriangleright} \:\: \sf{\dfrac{m}{2} = k}&\hspace{-20pt}\boldsymbol{\bigcirc \kern-11.5pt \blacktriangleright} \:\: \sf{\dfrac{n}{6} = k}\\\\\\\boxed{\sf{m=2k}}&\hspace{-20pt}\boxed{\sf{n=6k}}\end{array}\end{gathered}$

Pero del problema tenemos que:

$\begin{gathered}\begin{array}{c}\sf{m + n =12 }\\\\\sf{(2k)+(6k) = 12}\\\\\sf{8k = 12}\\\\\sf{k = \dfrac{\not\!\!12}{\not\!8}}\\\\\sf{k = \dfrac{3}{2}}\end{array}\end{gathered}$

Reemplazamos "k" para determinar los valores de "m" y "n"

$\begin{gathered}\begin{array}{cccccccc}\\\sf{m=2k }&&&&&&&\sf{n=6k }\\\\\sf{m= 2\left(\dfrac{3}{2}\right)}&&&&&&&\sf{n= 6\left(\dfrac{3}{2}\right)}\\\\\boxed{\boxed{\boldsymbol{\sf{m = 3}}}}&&&&&&&\boxed{\boxed{\boldsymbol{\sf{n = 9}}}}\end{array}\end{gathered}$

277. Otra manera de resolver es despejando una variable de la siguiente manera

$\begin{gathered}\begin{array}{c}\sf{\dfrac{8}{m}=\dfrac{12}{n}}\\\\\sf{n=\dfrac{12m}{8}}\\\\\sf{n=\dfrac{3m}{2}}\end{array}\end{gathered}$

Reemplazamos "n"

$\begin{gathered}\begin{array}{c}\sf{m+n = 35}\\\\\sf{m+\left(\dfrac{3m}{2}\right)=35}\\\\\sf{\dfrac{2m+3m}{2}=35}\\\\\sf{\dfrac{5m}{2}=35}\\\\\sf{5m=70}\\\\\sf{m=\dfrac{70}{5}}\\\\\boxed{\boxed{\boldsymbol{\sf{m=14}}}}\end{array}\end{gathered}$

Para determinar "m" reemplazamos "n"

$\begin{gathered}\begin{array}{c}\sf{n=\dfrac{3m}{2}}\\\\\sf{n=\dfrac{3(14)}{2}}\\\\\boxed{\boxed{\boldsymbol{\sf{n=21}}}}\end{array}\end{gathered}$

278. Realizamos el mismo procedimiento anterior

$\begin{gathered}\begin{array}{c}\sf{\dfrac{10}{m}=\dfrac{15}{n}}\\\\\sf{n=\dfrac{15m}{10}}\\\\\sf{n=\dfrac{3m}{2}}\end{array}\end{gathered}$

Reemplazamos "n"

$\begin{gathered}\begin{array}{c}\sf{m+n = 48}\\\\\sf{m+\left(\dfrac{3m}{2}\right)=48}\\\\\sf{\dfrac{2m+3m}{2}=48}\\\\\sf{\dfrac{5m}{2}=48}\\\\\sf{5m=96}\\\\\sf{m=\dfrac{96}{5}}\\\\\boxed{\boxed{\boldsymbol{\sf{m=19.2}}}}\end{array}\end{gathered}$

Para determinar "m" reemplazamos "n"

$\begin{gathered}\begin{array}{c}\sf{n=\dfrac{3m}{2}}\\\\\sf{n=\dfrac{3(19.2)}{2}}\\\\\boxed{\boxed{\boldsymbol{\sf{n=28.8}}}}\end{array}\end{gathered}$

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