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

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

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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

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AbhishekRyu 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
$$\underline a \,\,\underline b \,\,\underline c \,\,\,\left\{ \matrix{
\,\,a\,\, \in \,\,\,\left\{ {\,1,2, \ldots ,9\,} \right\} \hfill \cr
\,\,b\,\, \in \,\,\,\left\{ {\,0,1,2, \ldots ,9\,} \right\} \hfill \cr
\,\,c\,\, \in \,\,\,\left\{ {\,0,1,2, \ldots ,9\,} \right\} \hfill \cr} \right.$$
$$? = b$$
$$\left( 1 \right)\,\,\,\left\{ \matrix{
\,\underline a \,21 - 91 = {\rm{tens}}\,\,{\rm{digit}}\,\,3\,\,\,\,;\,\,\, \ldots \,\,\,;\,\,\,\,\,\underline a \,29 - 91 = {\rm{tens}}\,\,{\rm{digit}}\,\,3\,\,\,\,;\,\,\,\,\,\underline a \,30 - 91 = \,\,{\rm{tens}}\,\,{\rm{digit}}\,\,3 \hfill \cr
{\rm{where}}\,\,a\,\, \in \,\,\,\left\{ {\,1,2, \ldots ,9\,} \right\}\,\,\, \hfill \cr} \right.\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,\left( {a,b,c} \right) = \left( {1,2,1} \right)\,\,\,\, \Rightarrow \,\,\,\,\,? = \,\,2\,\, \hfill \cr
\,{\rm{Take}}\,\,\left( {a,b,c} \right) = \left( {1,3,0} \right)\,\,\,\, \Rightarrow \,\,\,\,\,? = \,\,3\,\, \hfill \cr} \right.$$
$$\left( 2 \right)\,\,\, \Rightarrow \,\,\,c = 6\,\,\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,\left( {a,b,c} \right) = \left( {1,1,6} \right)\,\,\,\, \Rightarrow \,\,\,\,\,? = \,\,1\,\, \hfill \cr
\,{\rm{Take}}\,\,\left( {a,b,c} \right) = \left( {1,2,6} \right)\,\,\,\, \Rightarrow \,\,\,\,\,? = \,\,2\,\, \hfill \cr} \right.$$
$$\left( {1 + 2} \right)\,\,\,c = 6\,\,\,\,\mathop \Rightarrow \limits^{\left( 1 \right)} \,\,\,\,\underline a \,26 - 91 = {\text{tens}}\,\,{\text{digit}}\,\,3\,\,\,\,\left( {a\,\, \in \,\,\,\left\{ {\,1,2, \ldots ,9\,} \right\}\,} \right)\,\,\,\,\, \Rightarrow \,\,\,\,b = 2\,\,\,\, \Rightarrow \,\,\,\,{\text{SUFF}}.$$


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)
English-speakers :: https://www.gmath.net
Portuguese-speakers :: https://www.gmath.com.br

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AbhishekRyu 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
Given: z is a three-digit positive integer.
Question: What is the value of the tens digit of z?

Let's take each statement one by one.

(1) The tens digit of z - 91 is 3.

Case 1: Say z = 930; 930 - 91 = 839. Tens digit of z = 930 is 3.
Case 2: Say z = 926; 926 - 91 = 835. Tens digit of z = 926 is 2.

No unique value. Insufficient.

(2) The units digit of z + 9 is 5.

Case 1: Say z = 936; 936 + 9 = 945. Tens digit of z = 936 is 3.
Case 2: Say z = 926; 926 + 9 = 935. Tens digit of z = 920 is 2.

No unique value. Insufficient.

(1) and (2) together

Only Case 2 from both the statements are common. z = x26; the tens digit is 2. Sufficient.

The correct answer: C

Hope this helps!

-Jay
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AbhishekRyu 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
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 [tens digit is 3]
z - 91 = 39 [tens digit is 3]
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 [units digit is 5]
z + 9 = 145 [units digit is 5]

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

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Brent Hanneson – Creator of GMATPrepNow.com
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