arjtryarjtry wrote:
If z is a three-digit positive integer, what is the value of the tens digit of z ?
(1) The tens digit of z - 91 is 3
(2) The units digit of z + 9 is 5
\(\underline a \,\,\underline b \,\,\underline c \,\,\,\left\{ \matrix{\\
\,\,a\,\, \in \,\,\,\left\{ {\,1,2, \ldots ,9\,} \right\} \hfill \cr \\
\,\,b\,\, \in \,\,\,\left\{ {\,0,1,2, \ldots ,9\,} \right\} \hfill \cr \\
\,\,c\,\, \in \,\,\,\left\{ {\,0,1,2, \ldots ,9\,} \right\} \hfill \cr} \right.\)
\(? = b\)
\(\left( 1 \right)\,\,\, \Rightarrow \,\,\,\left\{ \matrix{\\
\,\underline a \,21 - 91 = {\rm{tens}}\,\,{\rm{digit}}\,\,3\,\,\,\,;\,\,\, \ldots \,\,\,;\,\,\,\,\,\underline a \,29 - 91 = {\rm{tens}}\,\,{\rm{digit}}\,\,3\,\,\,\,;\,\,\,\,\,\underline a \,30 - 91 = \,\,{\rm{tens}}\,\,{\rm{digit}}\,\,3 \hfill \cr \\
{\rm{where}}\,\,a\,\, \in \,\,\,\left\{ {\,1,2, \ldots ,9\,} \right\}\,\,\, \hfill \cr} \right.\,\,\,\)
\(\,\left\{ \matrix{\\
\,{\rm{Take}}\,\,\left( {a,b,c} \right) = \left( {1,2,1} \right)\,\,\,\, \Rightarrow \,\,\,\,\,? = \,\,2\,\, \hfill \cr \\
\,{\rm{Take}}\,\,\left( {a,b,c} \right) = \left( {1,3,0} \right)\,\,\,\, \Rightarrow \,\,\,\,\,? = \,\,3\,\, \hfill \cr} \right.\)
\(\left( 2 \right)\,\,\, \Rightarrow \,\,\,c = 6\,\,\,\,\left\{ \matrix{\\
\,{\rm{Take}}\,\,\left( {a,b,c} \right) = \left( {1,1,6} \right)\,\,\,\, \Rightarrow \,\,\,\,\,? = \,\,1\,\, \hfill \cr \\
\,{\rm{Take}}\,\,\left( {a,b,c} \right) = \left( {1,2,6} \right)\,\,\,\, \Rightarrow \,\,\,\,\,? = \,\,2\,\, \hfill \cr} \right.\)
\(\left( {1 + 2} \right)\,\,\,c = 6\,\,\,\,\mathop \Rightarrow \limits^{\left( 1 \right)} \,\,\,\,\underline a \,26 - 91 = {\rm{tens}}\,\,{\rm{digit}}\,\,3\,\,\,\,\left( {a\,\, \in \,\,\,\left\{ {\,1,2, \ldots ,9\,} \right\}\,} \right)\,\,\,\,\, \Rightarrow \,\,\,\,b = 2\,\,\,\, \Rightarrow \,\,\,\,{\rm{SUFF}}.\)
This solution follows the notations and rationale taught in the GMATH method.
Regards,
Fabio.
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Fabio Skilnik :: GMATH method creator (Math for the GMAT)