Integers x and y have three digits and z is the sum of x and y. Is the tens digit of z equal to the sum of the tens digits of x and y?
1). Both x and y have units digits greater than 6
2). The sum of tens digit of x and y is 7
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user123321
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1) we cannot determine 10s digits sum with units digit information. because whether the sum of 10s digits is greater than 9 or not cannot be determined. hence insuff.sud21 wrote:Integers x and y have three digits and z is the sum of x and y. Is the tens digit of z equal to the sum of the tens digits of x and y?
1). Both x and y have units digits greater than 6
2). The sum of tens digit of x and y is 7
2) with just 10s digits sum we cannot determine because we dont know information about units digit. hence insuff.
edited:
using both,
sum of units digit is greater than 12 =>
so we need to add 1 to sum of ten's digit.
But we know sum of tens digit as 7
so with 1 brought from unit's digit, sum of tens digit will become 8.
so we can say for sure that 10s digit of z is not equal to sum of 10s digit of x & 10s digit of y.
so should be sufficient.
IMO C
user123321
Last edited by user123321 on Thu Jan 12, 2012 4:32 pm, edited 2 times in total.
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sud21
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Can you explain a bit further when you are taking both statements.user123321 wrote:1) we cannot determine 10s digits sum with units digit information. because whether the sum of 10s digits is greater than 9 or not cannot be determined. hence insuff.sud21 wrote:Integers x and y have three digits and z is the sum of x and y. Is the tens digit of z equal to the sum of the tens digits of x and y?
1). Both x and y have units digits greater than 6
2). The sum of tens digit of x and y is 7
2) with just 10s digits sum we cannot determine because we dont know information about units digit. hence insuff.
using both,
sum of units digit is greater than 6 =>
say, if unit digit sum is 8 then tens digit sum will be 7..ok
say, if tens digit sum is 12 then tens digit sum will be 7+1 = 8...not ok
so still insuff.
user123321
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pemdas
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Both statements together must be sufficient, as the ten's digit of z will be different from the sum of the tens'digits of x and y, i.e. 7=!8
c
@user, it says in st(1) *Both x and y have units digits greater than 6* and you apply *sum of units digit is greater than 6*
c
@user, it says in st(1) *Both x and y have units digits greater than 6* and you apply *sum of units digit is greater than 6*
user123321 wrote:1) we cannot determine 10s digits sum with units digit information. because whether the sum of 10s digits is greater than 9 or not cannot be determined. hence insuff.sud21 wrote:Integers x and y have three digits and z is the sum of x and y. Is the tens digit of z equal to the sum of the tens digits of x and y?
1). Both x and y have units digits greater than 6
2). The sum of tens digit of x and y is 7
2) with just 10s digits sum we cannot determine because we dont know information about units digit. hence insuff.
using both,
sum of units digit is greater than 6 =>
say, if unit digit sum is 8 then tens digit sum will be 7..ok
say, if tens digit sum is 12 then tens digit sum will be 7+1 = 8...not ok
so still insuff.
user123321
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user123321
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@pemdas you're right. I mistook option (1). Will edit the post now.
Thanks,
user123321
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Mike@Magoosh
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Hi, there. I'm happy to put my 2 cents in here. 
The original question:
Integers x and y have three digits and z is the sum of x and y. Is the tens digit of z equal to the sum of the tens digits of x and y?
1) Both x and y have units digits greater than 6
2) The sum of tens digit of x and y is 7
Remember that the fundamental task on DS is not to answer the question, but rather to determine whether you have enough information to answer the question. If you can give the original question either a definite yes answer or a definite no answer, that's sufficient. Insufficient means no definitive answer is possible.
Statement #1: Both x and y have units digits greater than 6
If the units digits of x and of y are both more than 6, then their sum will be more than 12. Therefore, a digit from the one's place will carry into the tens place, and the tens digit of Z cannot possibly equal the tens digit of X plus the tens digit of Y. This determines that the answer to the question is: No. Thus, we have determined a definitive answer to the question, and that is sufficient.
Statement #2: The sum of tens digit of x and y is 7
Well, the fact that the sum of tens digit of x and y is a single digit means nothing will carry from the tens to the hundreds place, so that means it's possible that tens digit of Z equals the tens digit of X plus the tens digit of Y. Without information about the one's digits, though, and whether something in the one's digit would carry, we cannot answer this question with certainty. Numerical examples
143 + 134 = 247 (tens digit of X + tens digit of Y = tens digit of Z)
128 + 159 = 287 (tens digit of X + tens digit of Y =/ tens digit of Z)
Not enough information to answer either yes or no ---> the statement is insufficient.
Therefore, the answer is A.
Does that make sense? Please let me know if there are any questions.
The original question:
Integers x and y have three digits and z is the sum of x and y. Is the tens digit of z equal to the sum of the tens digits of x and y?
1) Both x and y have units digits greater than 6
2) The sum of tens digit of x and y is 7
Remember that the fundamental task on DS is not to answer the question, but rather to determine whether you have enough information to answer the question. If you can give the original question either a definite yes answer or a definite no answer, that's sufficient. Insufficient means no definitive answer is possible.
Statement #1: Both x and y have units digits greater than 6
If the units digits of x and of y are both more than 6, then their sum will be more than 12. Therefore, a digit from the one's place will carry into the tens place, and the tens digit of Z cannot possibly equal the tens digit of X plus the tens digit of Y. This determines that the answer to the question is: No. Thus, we have determined a definitive answer to the question, and that is sufficient.
Statement #2: The sum of tens digit of x and y is 7
Well, the fact that the sum of tens digit of x and y is a single digit means nothing will carry from the tens to the hundreds place, so that means it's possible that tens digit of Z equals the tens digit of X plus the tens digit of Y. Without information about the one's digits, though, and whether something in the one's digit would carry, we cannot answer this question with certainty. Numerical examples
143 + 134 = 247 (tens digit of X + tens digit of Y = tens digit of Z)
128 + 159 = 287 (tens digit of X + tens digit of Y =/ tens digit of Z)
Not enough information to answer either yes or no ---> the statement is insufficient.
Therefore, the answer is A.
Does that make sense? Please let me know if there are any questions.
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pemdas
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@mike, of course we can answer Yes/No for this doesn't seek value
now the question arise, whether in your accounting for the ten's digits of x and y you *assume* the numbers being positive or negative? Because the original question doesn't specify it.
With the ten's digit of positive x (x=108) equal to 0 and the ten's digit of negative y (y=-107) equal to 0, their sum is not 1 which would have carried out 1 into the tens, BUT 0 for the ten's digit of z, as perform addition x+y such as 108+(-107)=1 (you don't have a three-digit number any longer, and this is correct).
The question asked in the original entry requires us to answer if the ten's digit of z equals to the sum of tens' digits of x and y. That is if (0+0)=0. The answer is Yes, as we have no ten's digit.
Another example, x=257 and y=-157, x+y=257+(-157)=100
Or x=617 and y=-507, x+y=617+(-507)=110, here obviously (1+0)=1 and you answer Yes!
care to reconsider?
now the question arise, whether in your accounting for the ten's digits of x and y you *assume* the numbers being positive or negative? Because the original question doesn't specify it.
With the ten's digit of positive x (x=108) equal to 0 and the ten's digit of negative y (y=-107) equal to 0, their sum is not 1 which would have carried out 1 into the tens, BUT 0 for the ten's digit of z, as perform addition x+y such as 108+(-107)=1 (you don't have a three-digit number any longer, and this is correct).
The question asked in the original entry requires us to answer if the ten's digit of z equals to the sum of tens' digits of x and y. That is if (0+0)=0. The answer is Yes, as we have no ten's digit.
Another example, x=257 and y=-157, x+y=257+(-157)=100
Or x=617 and y=-507, x+y=617+(-507)=110, here obviously (1+0)=1 and you answer Yes!
care to reconsider?
mikemcgarry wrote: Remember that the fundamental task on DS is not to answer the question, but rather to determine whether you have enough information to answer the question. If you can give the original question either a definite yes answer or a definite no answer, that's sufficient. Insufficient means no definitive answer is possible.
Statement #1: Both x and y have units digits greater than 6
If the units digits of x and of y are both more than 6, then their sum will be more than 12. Therefore, a digit from the one's place will carry into the tens place, and the tens digit of Z cannot possibly equal the tens digit of X plus the tens digit of Y. This determines that the answer to the question is: No. Thus, we have determined a definitive answer to the question, and that is sufficient.
...
Therefore, the answer is A.
Does that make sense? Please let me know if there are any questions.
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pemdas
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@@sud21, please post the source of this question?
i believe your source shows as OA the choice A, and I stress that choice A is not correct, as demonstrated in my previous post. Please reveal the source.
i believe your source shows as OA the choice A, and I stress that choice A is not correct, as demonstrated in my previous post. Please reveal the source.
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user123321
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nice one...maybe the question shud have added two +ve integers.
Thanks
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Mike@Magoosh
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In response to pemdas, I will say: when the GMAT talks about "digits", the assumption is that they are always talking about positive digits.
One runs into various paradoxes when one starts talking about "negative digits" --- for example, does the number -147 have three negative digits? If so, the one's digit, -7, would be less than 6, and thus not satisfy Statement #1. In that reading, a negative 3-digit number could't possibly have a one's digit greater than 6, because all its digits would be negative. If you had some digits positive and some digits negative, wouldn't you have to subtract them? So that a negative hundred's digit of 2 plus a positive one's digit of 3 would be -197? That raises the troubling prospect that the "digits" of a negative number could differ from the digits that typographically appear when the number is written. It would also mean that the digits of any given number are no longer unique.
This bucket gets more and more disturbing, the more you think about it.
To avoid that entire kettle of fish, the GMAT makes the blanket assumption: when they are talking about a 2- or 3-digit number, and about the digits of said number, they are always, 100% of the time, talking about a positive number with positive digits.
Does that make sense?
Mike
One runs into various paradoxes when one starts talking about "negative digits" --- for example, does the number -147 have three negative digits? If so, the one's digit, -7, would be less than 6, and thus not satisfy Statement #1. In that reading, a negative 3-digit number could't possibly have a one's digit greater than 6, because all its digits would be negative. If you had some digits positive and some digits negative, wouldn't you have to subtract them? So that a negative hundred's digit of 2 plus a positive one's digit of 3 would be -197? That raises the troubling prospect that the "digits" of a negative number could differ from the digits that typographically appear when the number is written. It would also mean that the digits of any given number are no longer unique.
This bucket gets more and more disturbing, the more you think about it. To avoid that entire kettle of fish, the GMAT makes the blanket assumption: when they are talking about a 2- or 3-digit number, and about the digits of said number, they are always, 100% of the time, talking about a positive number with positive digits.
Does that make sense?
Mike
Last edited by Mike@Magoosh on Fri Jan 13, 2012 11:14 am, edited 1 time in total.
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pemdas
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but the question starts from introducing integers x and y. The digits are rather a form of the number expression. It's clear that there are only ten digits 0-9 and no one says that we need to apply -ve digit definition here.mikemcgarry wrote:In response to pemdas, I will say: when the GMAT talks about "digits", the assumption is that they are always talking about positive digits.
Since this question is not involving any specific type of number property relationship(s), such as prime number(s) or remainder(s), for which we would be allowed to make *blanket assumption* of +ve numbers, we need to read whether integers x and y are +ve or -ve. A *default assumption* (not *blanket*sud21 wrote:Integers x and y have three digits and z is the sum of x and y. Is the tens digit of z equal to the sum of the tens digits of x and y?
1). Both x and y have units digits greater than 6
2). The sum of tens digit of x and y is 7
) would be to consider all integers, i.e. positive and negative ones.Success doesn't come overnight!
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Mike@Magoosh
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This is another reply to user pemdas
I love the fact that you think so carefully and so precisely about math. It seems to me you have an exceptionally strong background in math. You certainly have my respect.
With respect, I will disagree with you on the matter of digits. Again, as soon as you open the conversation of the value of the digits of negative numbers, the paradoxes are myriad. For example, you asserted that there are just ten digits, 0-9, no need to consider negative digits. That's one interpretation. Well, then, that would mean that 356 and -356 are two numbers that have all the same digits in the same places but are not equal(!!!) You could add two numbers, with the same non-zero digits, and get zero (!!!) I am aware of no mathematical conventions in place to negotiate these paradoxes. That is why I am inclined to think that the very word "digits" itself, like the word "prime", must mean: consider positive integers only. I readily confess, though, avoidance is not the only way to handle mathematical paradoxes, and there seems no generally recognized set conventions, so perhaps this is, as it were, a matter of mathematical taste. De gustibus non est disputandum.
All of this may well be moot. I don't know the source of the original question posted by sud21, but to the best of my knowledge, the OG (12e) doesn't include any questions at all about digits, nor does any of the Manhattan GMAT books. Nor have I see a question on digits on any GMAT I've taken or reviewed. While it's certainly plausible that the GMAT could ask about digits at some future date, it doesn't seem that it has been a topic in their purview. If the official GMAT were to ask about digits, I would expect that, true to form, they would be hyper-explicit about whether or not the numbers concerned could be negative.
Thus, it may be, pemdas, that our disagreement here is both (a) a matter of mathematical taste, and (b) wholly irrelevant to GMAT prep.
With respect,
Mike
I love the fact that you think so carefully and so precisely about math. It seems to me you have an exceptionally strong background in math. You certainly have my respect.
With respect, I will disagree with you on the matter of digits. Again, as soon as you open the conversation of the value of the digits of negative numbers, the paradoxes are myriad. For example, you asserted that there are just ten digits, 0-9, no need to consider negative digits. That's one interpretation. Well, then, that would mean that 356 and -356 are two numbers that have all the same digits in the same places but are not equal(!!!) You could add two numbers, with the same non-zero digits, and get zero (!!!) I am aware of no mathematical conventions in place to negotiate these paradoxes. That is why I am inclined to think that the very word "digits" itself, like the word "prime", must mean: consider positive integers only. I readily confess, though, avoidance is not the only way to handle mathematical paradoxes, and there seems no generally recognized set conventions, so perhaps this is, as it were, a matter of mathematical taste. De gustibus non est disputandum.
All of this may well be moot. I don't know the source of the original question posted by sud21, but to the best of my knowledge, the OG (12e) doesn't include any questions at all about digits, nor does any of the Manhattan GMAT books. Nor have I see a question on digits on any GMAT I've taken or reviewed. While it's certainly plausible that the GMAT could ask about digits at some future date, it doesn't seem that it has been a topic in their purview. If the official GMAT were to ask about digits, I would expect that, true to form, they would be hyper-explicit about whether or not the numbers concerned could be negative.
Thus, it may be, pemdas, that our disagreement here is both (a) a matter of mathematical taste, and (b) wholly irrelevant to GMAT prep.
With respect,
Mike
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Mike@Magoosh
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pemdas,
Thank you very much for your kind words.
If there's anything I can do to support you, please do not hesitate to ask.
Mike
Thank you very much for your kind words.
If there's anything I can do to support you, please do not hesitate to ask.
Mike
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