If z is a three-digit positive integer, what is the value of

Question

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

Bunuel · Answer

fskilnik · Answer

\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.

KanishkM · Answer

Have to think very fast in such questions

from 1) 221 - 91, or 230 - 91

need to look for a Tens digit value as 3, we are getting 2 values here.

from 2) this is clearly insufficient, z +9 = 146 + 9, 176 +9, 186+9

Combination is a trick, But since i know from 2, UD of z has to end on a 6

for value of the tens digit of z, We can always get the value as 2

C

BrentGMATPrepNow · Answer

Given : z is a three-digit positive integer

Target question :    What is the value of the tens digit of z ?

Statement 1 : The tens digit of z - 91 is 3   
Let's examine two EXTREME cases
z - 91 = 30    
z - 91 = 39    
NOTE: These are extreme cases, because 30 is the smallest 2-digit number with tens digit 3, and 39 is the biggest 2-digit number with tens digit 3.

If z - 91 = 30, then z = 121. 
So, z COULD be 121, in which case the answer to the target question is  the tens digit of z is 2

If z - 91 = 39, then z = 130. 
So, z COULD be 130, in which case the answer to the target question is  the tens digit of z is 3

Since we cannot answer the  target question  with certainty, statement 1 is NOT SUFFICIENT

Statement 2 : The units digit of z + 9 is 5 
Let's examine two cases
z + 9 = 135    
z + 9 = 145

If z + 9 = 135, then z = 126. 
So, z COULD be 126, in which case the answer to the target question is  the tens digit of z is 2

If z + 9 = 145, then z = 136. 
So, z COULD be 136, in which case the answer to the target question is  the tens digit of z is 3

Since we cannot answer the  target question  with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined    
Statement 2 indirectly tells us that the UNITS digit of z is 6
Statement 1 tells us that the tens digit of z - 91 is 3

Now that the units digit of z MUST be 6, let's find some values of z that satisfy  statement 1 . 
Now that we know the units digit of z is 6, we know that z - 91 will have UNITS digit 5. 
Also, statement 1 tells us that z - 91 will have TENS digit 3. 
So, we know that: z - 91 = ?35

If z - 91 = 35, then z = 126. So, the answer to the target question is  z has units digit 2 
If z - 91 = 135, then z = 226. So, the answer to the target question is  z has units digit 2 
If z - 91 = 235, then z = 326. So, the answer to the target question is  z has units digit 2 
If z - 91 = 335, then z = 426. So, the answer to the target question is  z has units digit 2 
As you can see, the answer to the target question will ALWAYS be  z has units digit 2

Since we can answer the  target question  with certainty, the combined statements are SUFFICIENT

Answer: C

Cheers,
Brent

Rakesh1987 · Answer

From 1) if the unit digit of z is >=1, the tens digit is 2 (121-91=30) and if the unit digit of z is 0, tens digit is 3 (130-91=39).  Insufficient

From 2) z+9 unit digit is 5 only gives that unit digit of z is 6 (6+9=15, 16+9=25, so on).  Insufficient.

From 1+2) we know unit digit is 6, i.e. >=1, therefore tens digit of z is 2.  Sufficient.

bumpbot · Answer

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If z is a three-digit positive integer, what is the value of

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