The number A is a two-digit positive integer; the number B

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The number A is a two-digit positive integer; the number B is the two-digit positive integer formed by reversing the digits of A.

If Q=10B-A, what is the value of Q?

(1) The tens digit of A is 7
(2) The tens digit of B is 6

[spoiler]OA=B[/spoiler]

Source: Manhattan Prep

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by Vincen » Mon Apr 22, 2019 6:50 am

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The number A is a two-digit positive integer; the number B is the two-digit positive integer formed by reversing the digits of A.

If Q=10B-A, what is the value of Q?

(1) The tens digit of A is 7
(2) The tens digit of B is 6

[spoiler]OA=B[/spoiler]

Source: Manhattan Prep
Hi M7MBA.

Let's write \(A\) and \(B\) as follows:
\( A= xy=10x+y\)
and
\(B= yx=10y+x\),

where \(x,y\) are integer numbers from 0 to 9 with \(y\ne 0\).

Now, we just need to see if we can find the exact value of
\(Q= 10B-A \)
\(= 10(10y+x)-(10x+y) \)
\(= 100y+10x-10x-y\)
\(= 99y.\)

From the above, we can see that we just need to find the value of \(y\). In order to do that, let's check the given statements.

Statement 1:
(1) The tens digit of A is 7
This statement just tells us that \(x=7\) but it doesn't tell us anything about \(y\). So, this statement is NOT SUFFICIENT.

Statement 1:
(2) The tens digit of B is 6
If the tens digit of B is 6 then we are told that \(y=6\). Therefore, \(Q=99y=604.\)

So, this statement is SUFFICIENT.

Hence, the correct answer is the option _B_.

I hope it helps you.

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Re:

by ncktrn » Wed Sep 22, 2021 10:33 pm

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I am curious why the number 10 is added to formulate the equation 10x+y and 10y+x in these problems? The problem states that the number A and B are only two digits; I can see that using a place value with x,y is valid, but why do we have to put the number 10?
Vincen wrote:
Mon Apr 22, 2019 6:50 am
The number A is a two-digit positive integer; the number B is the two-digit positive integer formed by reversing the digits of A.

If Q=10B-A, what is the value of Q?

(1) The tens digit of A is 7
(2) The tens digit of B is 6

[spoiler]OA=B[/spoiler]

Source: Manhattan Prep
Hi M7MBA.

Let's write \(A\) and \(B\) as follows:
\( A= xy=10x+y\)
and
\(B= yx=10y+x\),

where \(x,y\) are integer numbers from 0 to 9 with \(y\ne 0\).

Now, we just need to see if we can find the exact value of
\(Q= 10B-A \)
\(= 10(10y+x)-(10x+y) \)
\(= 100y+10x-10x-y\)
\(= 99y.\)

From the above, we can see that we just need to find the value of \(y\). In order to do that, let's check the given statements.

Statement 1:
(1) The tens digit of A is 7
This statement just tells us that \(x=7\) but it doesn't tell us anything about \(y\). So, this statement is NOT SUFFICIENT.

Statement 1:
(2) The tens digit of B is 6
If the tens digit of B is 6 then we are told that \(y=6\). Therefore, \(Q=99y=604.\)

So, this statement is SUFFICIENT.

Hence, the correct answer is the option _B_.

I hope it helps you.