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. We know nothing about the hundreds digit of r. Not sufficient.

(2) The tens digit of r is less than either of the other two digits of r. We know nothing about t.

(1)+(2) From (2) we know that the tens digit of r, is smallest of the three and from (1) we know that the tens digit of r is greater than each of the digits of t. Thus, the hundreds digit of r must be greater than the hundreds digit of t, which means that r > t. Sufficient.

Answer: C.
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Visualise the number in this way:
r = xyz
t = ktm

1 - y>k nothing said about x. insuff
2 - y<x nothing said about k
1&2 k<y<x --> k<x --> r>t sufficient

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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If R = 100a + 10b + c, and T = 100d + 10e + f, then
From the 1st statement it follows that b>d, b>e, and b>f, however a and c are not given - NS
From the 2nd statement: a > b, and c > b, hence a > d, e, f and c > d, e, f but b is not given - NS
From both statements:
100a+10b+c > 100d + 10e + f

Use nos

Statement 1 -let Digit r=x9y and digit t=888 so X could be 9 or 7 or some other no hence not sufficient
Statement 2- 10's digit is less than either of the 2 digits of R eg 656 or 746 hence not sufficient

combined

eg r=898 and t=888 hence r>t and both statements together are sufficient.

I have a question:

If both r and t are 3-digits numbers, it means that they are >= 100 (at least 100).
So, if the tens digit of r is greater than each of t's digits, that means that the tens digit of r is at least 2 (since the smallest 3-digit number is 100).
If this case, r is at least 12X and t is at least 100. So why isn't statement 1 sufficient?

hi..

the statement should give yes clear answer to is r>t?

lets see..
t = 345, r can be 786 so r>t
t=345, r can be 286 now r<t
so different answers possible insyff
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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.

Answer: C
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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

Final Answer:

GMAT assassins aren't born, they're made,
Rich
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Statement One: R can be 245 and T can be 222 ( r is greater than t)

R can be 257 and T can be 333 ( r is less than t)

Hence St.(1) alone is insufficient


Statement Two: R can be 768 for example. But no information about T is given.

Hence St.(2) alone is insufficient


Combined (1)+(2)

(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

If R is 768 and tens digit of R is Greater than Each of the Three Digits of T , it means T can be 555, 545, 455, etc (in other words each digit of T will be less than 6)

Combined together sufficient i.e. YES ! R>T

C YAY!

I'm sorry. Why does "either" mean "both" in this case?

But why does either mean both? Please help me understand this logic.

[

Asked: If r and t are three-digit positive integers, is r greater than t ?
Let r = rHrTrU; where rH is hundredth digit of r, rT is tenth digit of r and rU is unit digit of r
Let t = tHtTtU; where tH is hundredth digit of t, tT is tenth digit of t and tU is unit digit of t

(1) The tens digit of r is greater than each of the three digits of t .
rT > {tH, tT, tU}
But rH may be less than or greater than tH
NOT SUFFICIENT

(2) The tens digit of r is less than either of the other two digits of r .
rT < {rH, rU}
The statement provides no information about t
NOT SUFFICIENT

(1) + (2)
(1) The tens digit of r is greater than each of the three digits of t .
rT > {tH, tT, tU}
(2) The tens digit of r is less than either of the other two digits of r .
rT < {rH, rU}
{rH, rU} > rT > {tH, tT, tU}
rHrTrU > tHtTtU
r>t
SUFFICIENT

IMO C
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Just like a few others on this thread, I misunderstood the usage of the word either and answered the question incorrectly.

This is what I found out about the usage of the word either in different context:

If there's a choice to be made - either = one or the other
For example: You may sit on either of the seats in the front row.

If it's a description of an existing state - either = both instances
For example: There are trees on either side of the road. (both sides of the road)

Hope this helps

Hi Bunuel,

Could you please clarify the usage of both ? If I consider either ..hundreds or units then I am getting incorrect answer? Please throw some light on this ?

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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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).

GMAT assassins aren't born, they're made,
Rich
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