Question

Escreva maneiras diferentes de agrupar dezenas. Exatas. Para obter uma centena

Answer 1

Olá.

 

Basicamente, temos que agrupar dezenas (ou seja, 10 unidades) para que, no final, essas dezenas resultem em um centena (ou seja, 10 unidades ou 100 unidades). 

Demonstro a relação entre centenas, dezenas e unidades:

$\boxed{\begin{array}{r|c|c|c} ~&~&~&~\\ &\textsf{Centena}&\textsf{Dezena}&\textsf{Unidade}\\ &\textsf{---------------}&\textsf{---------------}&\textsf{---------------}\\ \textsf{1 centena}\rightarrow\mathsf{100\rightarrow}&1&0&0\\ \textsf{----------------------------}&\textsf{---------------}&\textsf{---------------}&\textsf{---------------}\\ \textsf{1 dezena}\rightarrow\mathsf{10\rightarrow}&0&1&0\\ \textsf{----------------------------}&\textsf{---------------}&\textsf{---------------}&\textsf{---------------}\\ \textsf{1 unidade}\rightarrow\mathsf{1\rightarrow}&1&0&0\\ ~&~&~& \end{array}}$

Abaixo, demonstro alguns modos de combinar e permutar dezenas.

Como estou usando LaTeX, por conveniência e necessidade, simplifico "dezena" por "d". Teremos:

 

$\begin{array}{rl} \mathsf{100=}&\mathsf{1d+1d+1d+1d+1d+1d+1d+1d+1d+1d}\\\\ \mathsf{100=}&\mathsf{2d+1d+1d+1d+1d+1d+1d+1d+1d}\\\\ \mathsf{100=}&\mathsf{3d+1d+1d+1d+1d+1d+1d+1d}\\\\ \mathsf{100=}&\mathsf{4d+1d+1d+1d+1d+1d+1d}\\\\ \mathsf{100=}&\mathsf{5d+1d+1d+1d+1d+1d}\\\\ \mathsf{100=}&\mathsf{6d+1d+1d+1d+1d}\\\\ \mathsf{100=}&\mathsf{7d+1d+1d+1d}\\\\ \mathsf{100=}&\mathsf{8d+1d+1d}\\\\ \mathsf{100=}&\mathsf{9d+1d} \end{array}$

 

Podem haver permutações em todas as combinações, ou seja, os termos de uma combinação podem ser trocados. Como exemplo, segue:

 

$\begin{array}{rl} \mathsf{100=}&\mathsf{2d+1d+1d+1d+1d+1d+1d+1d+1d}\\\\ \mathsf{100=}&\mathsf{1d+2d+1d+1d+1d+1d+1d+1d+1d}\\\\ \mathsf{100=}&\mathsf{1d+1d+2d+1d+1d+1d+1d+1d+1d}\\\\ \mathsf{100=}&\mathsf{1d+1d+1d+2d+1d+1d+1d+1d+1d}\\\\ \mathsf{100=}&\mathsf{1d+1d+1d+1d+2d+1d+1d+1d+1d}\\\\ \mathsf{100=}&\mathsf{1d+1d+1d+1d+1d+2d+1d+1d+1d}\\\\ \mathsf{100=}&\mathsf{1d+1d+1d+1d+1d+1d+2d+1d+1d}\\\\ \mathsf{100=}&\mathsf{1d+1d+1d+1d+1d+1d+1d+2d+1d}\\\\ \mathsf{100=}&\mathsf{1d+1d+1d+1d+1d+1d+1d+1d+2d} \end{array}$

 

Outros diversos eventos são possíveis. Abaixo demonstro alguns exemplos/eventos:

 

$\begin{array}{rl} \mathsf{100=}&\mathsf{2d+1d+1d+1d+1d+1d+1d+1d+1d}\\\\ \mathsf{100=}&\mathsf{2d+2d+1d+1d+1d+1d+1d+1d}\\\\ \mathsf{100=}&\mathsf{2d+2d+2d+1d+1d+1d+1d}\\\\ \mathsf{100=}&\mathsf{2d+2d+2d+2d+1d+1d}\\\\ \mathsf{100=}&\mathsf{2d+2d+2d+2d+2d}\\\\ \mathsf{100=}&\mathsf{4d+2d+2d+2d}\\\\ \mathsf{100=}&\mathsf{4d+4d+2d}\\\\ \mathsf{100=}&\mathsf{8d+2d}\end{array}$

 

$\textsf{--------------------}$

$\begin{array}{rl} \mathsf{100=}&\mathsf{3d+1d+1d+1d+1d+1d+1d+1d}\\\\ \mathsf{100=}&\mathsf{3d+2d+1d+1d+1d+1d+1d}\\\\ \mathsf{100=}&\mathsf{3d+3d+1d+1d+1d+1d}\\\\ \mathsf{100=}&\mathsf{3d+3d+2d+1d+1d}\\\\ \mathsf{100=}&\mathsf{3d+3d+3d+1d}\\\\ \mathsf{100=}&\mathsf{6d+3d+1d}\\\\ \mathsf{100=}&\mathsf{6d+4d} \end{array}$

$\textsf{--------------------}$

 

$\begin{array}{rl} \mathsf{100=}&\mathsf{4d+1d+1d+1d+1d+1d+1d}\\\\ \mathsf{100=}&\mathsf{4d+2d+1d+1d+1d+1d}\\\\ \mathsf{100=}&\mathsf{4d+2d+2d+1d+1d}\\\\ \mathsf{100=}&\mathsf{4d+2d+2d+2d}\\\\ \mathsf{100=}&\mathsf{4d+4d+2d}\\\\ \mathsf{100=}&\mathsf{8d+2d} \end{array}$

$\textsf{--------------------}$

 

$\begin{array}{rl} \mathsf{100=}&\mathsf{5d+1d+1d+1d+1d+1d}\\\\ \mathsf{100=}&\mathsf{5d+2d+1d+1d+1d}\\\\ \mathsf{100=}&\mathsf{5d+2d+2d+1d}\\\\ \mathsf{100=}&\mathsf{5d+4d+1d}\\\\ \mathsf{100=}&\mathsf{5d+5d} \end{array}$

$\textsf{--------------------}$

 

$\begin{array}{rl} \mathsf{100=}&\mathsf{6d+1d+1d+1d+1d}\\\\ \mathsf{100=}&\mathsf{6d+2d+2d}\\\\ \mathsf{100=}&\mathsf{6d+4d} \end{array}$

$\textsf{--------------------}$

 

$\begin{array}{rl} \mathsf{100=}&\mathsf{7d+1d+1d+1d}\\\\ \mathsf{100=}&\mathsf{7d+2d+1d}\\\\ \mathsf{100=}&\mathsf{7d+3d} \end{array}$

$\textsf{--------------------}$

 

$\begin{array}{rl} \mathsf{100=}&\mathsf{8d+1d+1d}\\\\ \mathsf{100=}&\mathsf{8d+2d} \end{array}$

 

Quaisquer dúvidas, deixe nos comentários.

Bons estudos$.$