Problema 2

Problemas Resueltos - Razonamiento matemático

La expresión numérica:

\[ \begin{aligned} &\frac{1}{2} - \left[ \frac{1}{2} - \left\{ \frac{1}{4} - \left( 2 \cdot \left( \frac{1}{2} \right) - \frac{1}{4} \right) \right\} \right. \\[.5em] &\hspace{9em} \left. + \frac{1}{2} - \left( - \frac{1}{4} \right) \right] \end{aligned} \]

\[ \frac{1}{2} - \left[ \frac{1}{2} - \left\{ \frac{1}{4} - \left( 2 \cdot \left( \frac{1}{2} \right) - \frac{1}{4} \right) \right\} + \frac{1}{2} - \left( - \frac{1}{4} \right) \right] \]

es equivalente a:

La expresión numérica: \( \cfrac{1}{2} - \left[ \cfrac{1}{2} - \left\{ \cfrac{1}{4} - \left( 2 \cdot \left( \cfrac{1}{2} \right) - \cfrac{1}{4} \right) \right\} + \cfrac{1}{2} - \left( - \cfrac{1}{4} \right) \right] \), es equivalente a:

\(\cfrac{3}{4}\)
\(-\cfrac{5}{4}\)
\(-\cfrac{7}{4}\)
\(-\cfrac{1}{4}\)
\(\cfrac{3}{2}\)

Intenta resolverlo antes de ver la respuesta...

Respuesta

\(-\cfrac{5}{4}\)

Solución Paso a Paso

Eliminando los signos de la agrupación de afuera hacia adentro:

\[ \begin{aligned} &\frac{1}{2} – \left[ \frac{1}{2} – \left\{ \frac{1}{4} – \left( 2 \cdot \left( \frac{1}{2} \right) – \frac{1}{4} \right) \right\} \right. \\[.5em] &\hspace{1em} \left. + \frac{1}{2} – \left( – \frac{1}{4} \right) \right] = \\[.5em] &\hspace{1em} \frac{1}{2} – \frac{1}{2} + \left\{ \frac{1}{4} – \left( 2 \cdot \left( \frac{1}{2} \right) – \frac{1}{4} \right) \right\} \\[.5em] &\hspace{1em}- \frac{1}{2} + \left( – \frac{1}{4} \right) = \\[.5em] &\hspace{1em} \frac{1}{2} – \frac{1}{2} + \frac{1}{4} – 2 \cdot \left( \frac{1}{2} \right) + \frac{1}{4} \\[.5em] &\hspace{3em} – \frac{1}{2} – \frac{1}{4} = \\[.5em] & \frac{1}{2} – \frac{1}{2} + \frac{1}{4} – \frac{2}{2} + \frac{1}{4} – \frac{1}{2} – \frac{1}{4} \\[.5em] &\hspace{7em} = -\frac{3}{2} + \frac{1}{4} \\[.5em] &\hspace{7em} = \frac{ -6 + 1 }{4} \\[.5em] &\hspace{7em} = \frac{-5}{4} \\[.5em] &\hspace{7em} = \boldsymbol{ -\frac{5}{4} } \end{aligned} \]

\[ \begin{aligned} &\frac{1}{2} – \left[ \frac{1}{2} – \left\{ \frac{1}{4} – \left( 2 \cdot \left( \frac{1}{2} \right) – \frac{1}{4} \right) \right\} + \frac{1}{2} – \left( – \frac{1}{4} \right) \right] \\[.5em] &\hspace{1em} = \frac{1}{2} – \frac{1}{2} + \left\{ \frac{1}{4} – \left( 2 \cdot \left( \frac{1}{2} \right) – \frac{1}{4} \right) \right\} – \frac{1}{2} + \left( – \frac{1}{4} \right) \\[.5em] &\hspace{1em} = \frac{1}{2} – \frac{1}{2} + \frac{1}{4} – 2 \cdot \left( \frac{1}{2} \right) + \frac{1}{4} – \frac{1}{2} – \frac{1}{4} \\[.5em] &\hspace{1em} = \frac{1}{2} – \frac{1}{2} + \frac{1}{4} – \frac{2}{2} + \frac{1}{4} – \frac{1}{2} – \frac{1}{4} \\[.5em] &\hspace{1em} = -\frac{3}{2} + \frac{1}{4} \\[.5em] &\hspace{1em} = \frac{ -6 + 1 }{4} \\[.5em] &\hspace{1em} = \frac{-5}{4} \\[.5em] &\hspace{1em} = \boldsymbol{ -\frac{5}{4} } \end{aligned} \]

\[ \begin{aligned} \frac{1}{2} – \left[ \frac{1}{2} – \left\{ \frac{1}{4} – \left( 2 \cdot \left( \frac{1}{2} \right) – \frac{1}{4} \right) \right\} + \frac{1}{2} – \left( – \frac{1}{4} \right) \right] & = \frac{1}{2} – \frac{1}{2} + \left\{ \frac{1}{4} – \left( 2 \cdot \left( \frac{1}{2} \right) – \frac{1}{4} \right) \right\} – \frac{1}{2} + \left( – \frac{1}{4} \right) \\[.5em] & = \frac{1}{2} – \frac{1}{2} + \frac{1}{4} – 2 \cdot \left( \frac{1}{2} \right) + \frac{1}{4} – \frac{1}{2} – \frac{1}{4} \\[.5em] & = \frac{1}{2} – \frac{1}{2} + \frac{1}{4} – \frac{2}{2} + \frac{1}{4} – \frac{1}{2} – \frac{1}{4} \\[.5em] & = -\frac{3}{2} + \frac{1}{4} \\[.5em] & = \frac{ -6 + 1 }{4} \\[.5em] & = \frac{-5}{4} \\[.5em] & = \boldsymbol{ -\frac{5}{4} } \end{aligned} \]