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# If r and t are three-digit positive integers, is r greater than t ?

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Re: If r and t are three-digit positive integers, is r greater than t ? [#permalink]
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Why is C correct? Why are we taking "either of the other two digits" as each of the other two digits?

Doesn't either of the two means either y<x or y<z( for r=xyz), and not y<x and y<z?
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700ABOVE wrote:
Why is C correct? Why are we taking "either of the other two digits" as each of the other two digits?

Doesn't either of the two means either y<x or y<z( for r=xyz), and not y<x and y<z?

No. Either of the other two means both.
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Re: If r and t are three-digit positive integers, is r greater than t ? [#permalink]
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AbdurRakib wrote:
If r and t are three-digit positive integers, is r greater than t ?

(1) The tens digit of r is greater than each of the three digits of t .

(2) The tens digit of r is less than either of the other two digits of r .

We are given that r and t are three-digit positive integers and need to determine whether r is greater than t.

Statement One Alone:

The tens digit of r is greater than each of the three digits of t.

Statement one alone is not sufficient to answer the question. For example, if r = 190 and t = 180, then r is greater than t; however, if r = 190 and t = 210, then r is less than t.

Statement Two Alone:

The tens digit of r is less than either of the other two digits of r.

Since we know nothing about t, statement two alone is not sufficient to answer the question.

Statements One and Two Together:

Using statements one and two, we see that the tens digit of r is greater than each of the three digits of t and that the tens digit of r is less than either of the other two digits of r.

Thus, we can say that all digits of r must be greater than all digits of t, so we can say that r is greater than t.

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Re: If r and t are three-digit positive integers, is r greater than t ? [#permalink]
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Hi All,

We're told that R and T are both three-digit positive integers. We're asked if R is greater than T. This is a YES/NO question. We can solve it by TESTing VALUES.

1) The TENS digit of R is greater than each of the three digits of T.

IF....
R = 191 and T = 111, then the answer to the question is YES.
R = 191 and T = 888, then the answer to the question is NO.
Fact 1 is INSUFFICIENT

2) The TENS digit of R is less than either of the other two digits of R.

Fact 2 tells us NOTHING about the value of T, so there's no way to determine whether R is greater than T or not.
Fact 2 is INSUFFICIENT

Combined, we know:
-The TENS digit of R is greater than each of the three digits of T.
-The TENS digit of R is less than either of the other two digits of R.

With Fact 1, we know that the TENS digit of R is greater than all 3 digits in T, but we had no way to compare the HUNDREDS digits of R and T (so we didn't know which number was bigger). With the information in Fact 2 though, we know that the TENS digit of R is LESS than the HUNDREDS digit of R. Thus, we can create the following inequality:

(Hundreds digit of R) > (Tens digit of R) > (ANY digit in T)

eg. R = 870 and T = 512

Since the HUNDREDS digit of R is greater than EACH digit in T (including the HUNDREDS digit of T), R will ALWAYS be greater than T.
Combined, SUFFICIENT

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Re: If r and t are three-digit positive integers, is r greater than t ? [#permalink]
Can someone explain how "either of the 2" means both in Statement 2 ?
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Re: If r and t are three-digit positive integers, is r greater than t ? [#permalink]
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rhushishah wrote:
Can someone explain how "either of the 2" means both in Statement 2 ?

rhushishah

STatement 2: (2) The tens digit of r is less than either of the other two digits of r .

WHich MEans, if the number is abc then b must be less than a as well as less than c

b < a and b < c
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Re: If r and t are three-digit positive integers, is r greater than t ? [#permalink]
rhushishah wrote:
Can someone explain how "either of the 2" means both in Statement 2 ?

Hi rhushishah,

If you find that you're not clear on the meaning of a particular phrase in a Quant question, then it can sometimes help to come up with a simple example - to see if your example matches the 'description' in the text. With Fact 2, we're dealing with the phrase:

"The TENS digit of r is less than either of the other two digits of r."

In the prompt, we're told that R is a 3-digit number (any number from 100 to 999, inclusive). With the extra information in Fact 2, we have a 'restriction' that we have to deal with.

Does the number 123 fit that description? NO, because the TENS digit ("2") is NOT less than either the of the other two digits (the "1" and the "3") - it's only less than one of them.

Does the number 523 fit that description? YES, because the TENS digit ("2") IS less than either the of the other two digits (the "5" and the "3").

Thus, with Fact 2, we can only use certain types of 3-digit numbers (ones that have a 'middle digit' that is smaller than the other two digits).

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Re: If r and t are three-digit positive integers, is r greater than t ? [#permalink]
Video solution from Quant Reasoning:
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Re: If r and t are three-digit positive integers, is r greater than t ? [#permalink]
Forget the conventional way to solve DS questions.

We will solve this DS question using the variable approach.

DS question with 2 variables: Let the original condition in a DS question contain 2 variables. Now, 2 variables would generally require 2 equations for us to be able to solve for the value of the variable.

We know that each condition would usually give us an equation, and Since we need 2 equations to match the numbers of variables and equations in the original condition, the logical answer is C.

To master the Variable Approach, visit https://www.mathrevolution.com and check our lessons and proven techniques to score high in DS questions.

Let’s apply the 3 steps suggested previously. [Watch lessons on our website to master these 3 steps]

Step 1 of the Variable Approach: Modifying and rechecking the original condition and the question.

Let us assignt he variables:

=> 'r' is three digit positive integer: Let (Hundred's digit 'a'),(Ten's digit 'b'), and (Unit's digit 'c')

=> 't' is three digit positive integer: Let (Hundred's digit 'p'),(Ten's digit 'q'), and (Unit's digit 'r')

We have to find 'Is r > t'.

'r' will be greater than 't' if a > p

Second and the third step of Variable Approach: From the original condition, we have 2 variables (a and p).To match the number of variables with the number of equations, we need 2 equations. Since conditions (1) and (2) will provide 2 equations , C would most likely be the answer.

Let’s take look at both the conditions combined together.

Condition(1) tells us that the tens digit of r is greater than each of the three digits of t .

=> b > (p, q, r)

Condition(2) tells us that the tens digit of r is less than either of the other two digits of r.

=> b < a and b < c

From both we get, a > b, which is greater than p.Therefore, is 'r' greater than 't' - YES

Since the answer is unique YES , Both the conditions combined together are sufficient by CMT 2.

Both conditions combined together are sufficient.

So, C is the correct answer.

SAVE TIME: By Variable Approach, when we need 2 equations we combined and check for C to be an answer. Thus we save our time in not checking the individual conditions.
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Re: If r and t are three-digit positive integers, is r greater than t ? [#permalink]
AbdurRakib wrote:
If r and t are three-digit positive integers, is r greater than t ?

(1) The tens digit of r is greater than each of the three digits of t .

(2) The tens digit of r is less than either of the other two digits of r .

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If r and t are three-digit positive integers, is r greater than t ? [#permalink]
If r and t are three-digit positive integers, is r greater than t ?

(1) The tens digit of r is greater than each of the three digits of t .

(2) The tens digit of r is less than either of the other two digits of r .

I needed to see examples with numbers in order to be sure of the answer so:
(1) r= 109 t=111 is valid but if t= 222 it's not --> NOT SUFFICIENT
(2) r= 231. We don't know about t. --> NOT SUFFICIENT

(1)+(2): r=998 t=111 --> SUFFICIENT. Try again r=743 t=121, r=894 t=132, and so on.­
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Re: If r and t are three-digit positive integers, is r greater than t ? [#permalink]
AbdurRakib wrote:
If r and t are three-digit positive integers, is r greater than t ?

(1) The tens digit of r is greater than each of the three digits of t .

(2) The tens digit of r is less than either of the other two digits of r .

­I am confused with the language of the question.
tens digit of r is less than either of the other two digits of r.

Here Either means one digit will be bigger other will be smaller in r. I doesn't say that Tens digit is bigger that BOTH, it say EITHER OF TWO
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Re: If r and t are three-digit positive integers, is r greater than t ? [#permalink]
Apeksha2101 wrote:
AbdurRakib wrote:
If r and t are three-digit positive integers, is r greater than t ?

(1) The tens digit of r is greater than each of the three digits of t .

(2) The tens digit of r is less than either of the other two digits of r .

­I am confused with the language of the question.
tens digit of r is less than either of the other two digits of r.

Here Either means one digit will be bigger other will be smaller in r. I doesn't say that Tens digit is bigger that BOTH, it say EITHER OF TWO

­The phrase "either of the two" is used to refer to each of two possibilities. It means that what is being stated applies independently to both options presented. Hence, 'either of the other two' means the tens digit of r is less than each of the other two digits, so it is less than both the hundreds and the units digits.
Re: If r and t are three-digit positive integers, is r greater than t ? [#permalink]
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