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Is the positive two-digit integer N less than 40 ?

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Is the positive two-digit integer N less than 40 ?

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I have a question about the explanation for the following problem.
Official Guide 13, 2015
Data Sufficiency, #75, p. 281
Is the positive two-digit integer N less than 40 ?
1) The units digit of N is 6 more than the tens digit
2) N is 4 less than 4 times the units digit


I have a question on the answer explanation for statement 2. The explanation says that 2 is sufficient because the largest that the units digit of N could be is 9. 4 less than 4 times 9 is 36 -4 = 32. If the maximum possible value of N is 32, then N is definitely less than 40.

I understand that this is sufficient by doing test cases, but I don't understand why using 9 as the units digit is a valid test case. Shouldn't 8 be the only units digit that makes this a valid test case?

If 9 is the units digit of N, and 9 x 4 - 4 = 36 - 4 = 32, then N is 32. But if N is 32, then the units digit of 32 is 2, not 9. How would this be a valid test case?

On the other hand, if 8 is the units digit of N, and 8 x 4 - 4 = 32 - 4 = 28, then N is 28. If N is 28, then the units digit of 28 is 8, which makes this a valid test case because we used 8 as the units digit of N to begin with. Shouldn't this be the only valid test case?

I understand that either way, we arrive at the same answer, which is that this statement is sufficient because N will be less than 40. But what I want to understand is why 9 (or other digits) are valid test cases as the units digit of N, if what you end up getting for N (after you do the multiplication and subtraction) is a 2 digit integer of which the units digit is not the same as the units digit that you began testing with. I want to understand why the answer explanation is different from my understanding and whether I am misinterpreting or misunderstanding the provided statement.

Please let me know if you need me to clarify. Thanks for your help!

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Quote:
Is the positive two-digit integer N less than 40 ?

(1) The units digit of N is 6 more than the tens digit
(2) N is 4 less than 4 times the units digit

Target question: Is N less than 40

Given: N is a positive two-digit integer

Statement 1: The units digit of N is 6 more than the tens digit
This statement is, essentially, restricting the value of the tens digit.
If the units digit is 6 more than the tens digit, then the tens digit cannot be very big.
For example, the tens digit cannot be 8, because the units digit would have to be 14, which is impossible.
Likewise, the tens digit cannot be 4, because the units digit would have to be 10, which is also impossible.
So, the greatest possible value of the tens digit of N is 3.
As such, N must be less than 40
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: N is 4 less than 4 times the units digit.
Well, 9 is the greatest possible value of any integer, and if the units digit were 9, then N would equal (4)(9) - 4, which is less than 40
So, no matter what value the units digit has, the resulting number (N), must be less than 40
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Answer = D

Cheers,
Brent

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You’re absolutely right that 9 can’t be the units digit of N for statement 2. What the (deeply, deeply flawed) explanation is trying to convey is that any units digit must be 9 or less, so given that even 9 would yield a number that is less than 40, even though 9 doesn’t technically work here, we know for sure that the numbers that do work will all be less than 40.

Takeaway: The Official Guide explanations are often not very good.

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paml wrote:
I have a question about the explanation for the following problem.
Official Guide 13, 2015
Data Sufficiency, #75, p. 281
Is the positive two-digit integer N less than 40 ?
1) The units digit of N is 6 more than the tens digit
2) N is 4 less than 4 times the units digit


I have a question on the answer explanation for statement 2. The explanation says that 2 is sufficient because the largest that the units digit of N could be is 9. 4 less than 4 times 9 is 36 -4 = 32. If the maximum possible value of N is 32, then N is definitely less than 40.

I understand that this is sufficient by doing test cases, but I don't understand why using 9 as the units digit is a valid test case. Shouldn't 8 be the only units digit that makes this a valid test case?

If 9 is the units digit of N, and 9 x 4 - 4 = 36 - 4 = 32, then N is 32. But if N is 32, then the units digit of 32 is 2, not 9. How would this be a valid test case?

On the other hand, if 8 is the units digit of N, and 8 x 4 - 4 = 32 - 4 = 28, then N is 28. If N is 28, then the units digit of 28 is 8, which makes this a valid test case because we used 8 as the units digit of N to begin with. Shouldn't this be the only valid test case?

I understand that either way, we arrive at the same answer, which is that this statement is sufficient because N will be less than 40. But what I want to understand is why 9 (or other digits) are valid test cases as the units digit of N, if what you end up getting for N (after you do the multiplication and subtraction) is a 2 digit integer of which the units digit is not the same as the units digit that you began testing with. I want to understand why the answer explanation is different from my understanding and whether I am misinterpreting or misunderstanding the provided statement.

Please let me know if you need me to clarify. Thanks for your help!
We're not actually saying that N could equal 32. All we're saying is that, even when we plug in the maximum value for units digit (9), (4)(units digit) - 4 < 40.
So, FOR ANY UNITS DIGIT, the resulting value of N must be less than 40

Cheers,
Brent

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Brent & Dave -

This was very helpful. Thank you for confirming my understanding.

Thanks!

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but you cannot use 0,1,2,3 correct? The result would a non-positive one digit integer or a one digit integer, which would violate "Given".

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I'm assuming you're asking this in the context of statement 2. If so, you're absolutely right.

Units 0 --> 0*4 - 4 = -4 No good
Units 1 --> 1*4 - 4 = 0 No good
Units 2 --> 2*4 - 4 = 4 No good
Units 3 --> 3*4 - 4 = 8 No good

And as Paml noted, many more scenarios won't work simply because the units digit won't match properly. For example, Units 4 --> 4*4 - 4 = 12. Well, the units can't be both 4 and 2, so this won't work either, etc. It's just faster to see that even 9 will produce an N under 40, so any units digit that works will produce an N that is less than 40.

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It's important to point out that, when solving Data Sufficiency questions, we must avoid doing more work than is necessary. Our SOLE goal here is to determine whether or not the statements are sufficient.

Consider the following example:
Quote:
Is integer K less than 60?

1) K is a two-digit number such that the tens digit is equal to the remainder when the square of the units digit is divided by 6.
Notice that the tens digit equals the remainder when some value is divided by 6.
When we divide by 6, the only possible remainders are 0, 1, 2, 3, 4, or 5.
So, the LARGEST possible value for the tens digit is 5, so we can be certain that K < 60.
We're done! Time to move on.

Does it make a difference that we haven't determined the value of K? No, the question doesn't ask us to find the value of K.
Does it make a difference that the tens digit of K cannot be 5 (the only possible values of K are 11, 15, 17, 33, 39, 42, 44 and 48)?
No, we've already done enough to determine that K is definitely less than 60.
Don't spend any more time lingering on this statement. KEEP MOVING.

The same applies with the given question:
Quote:
Is the positive two-digit integer N less than 40 ?

2) N is 4 less than 4 times the units digit
If we try to MAXIMIZE N, we need to think of a really big value for the units digit.
Let's try 9, the biggest digit there is.
(4)(9) - 4 = 32
Since 32 is less than 40, we know that what whatever value N has, it must be less than 40
Does it make a difference that N cannot equal 32?
No.
We just used this information to confirm that N must be less than 40
Since we can already be certain that N must be less than 40, we need not waste any time finding possible values of N.

Cheers,
Brent

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jak5189 wrote:
but you cannot use 0,1,2,3 correct? The result would a non-positive one digit integer or a one digit integer, which would violate "Given".
Nice point! This is important to remember on any DS question: if you're picking numbers, you can't pick any numbers that violate the statement(s) that you're plugging those numbers into. So since n is a two digit number that's also 4*(a single digit number) - 4, it must be at least 4*4 - 4 and at most 4*9 - 4.

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Hi All,

We're told that N is a positive two-digit integer. We're asked if N is less than 40. This is a YES/NO question. We can solve it by TESTing VALUES.

1) The UNITS digit of N is 6 MORE than the TENS digit

The information in Fact 1 severely limits the possible values of N (there are only 3 options)
IF....
N = 17, then the answer to the question is YES
N = 28, then the answer to the question is YES
N = 39, then the answer to the question is YES
The answer is ALWAYS YES
Fact 1 is SUFFICIENT

2) N is 4 less than 4 times the units digit

Since the largest possible units digit is 9, the largest possible value of N would have to be LESS than (4)(9) = 36. With that limitation, there's no need for any additional work - since N will always be LESS than 36, the answer to the question is ALWAYS YES.
Fact 2 is SUFFICIENT

Final Answer: D

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paml wrote:
I have a question about the explanation for the following problem.
Official Guide 13, 2015
Data Sufficiency, #75, p. 281
Is the positive two-digit integer N less than 40 ?
1) The units digit of N is 6 more than the tens digit
2) N is 4 less than 4 times the units digit
We need to determine whether the positive two-digit integer N is less than 40.

Statement One Alone:

The units digit of N is 6 more than the tens digit.

With the information in statement one, we know the units digit is the larger of the two digits in N. So let’s say the units digit is 9 (the largest digit possible); then the tens digit will be 3 (since 9 is 6 more than 3). Thus this makes N = 39, which is less than 40. Now let’s say the units digit is 8; then the tens digit will be 2 and thus N = 28, which is still less than 40. Let’s say the units digit is 7; then the tens digit will be 1 and thus N = 17, which is again less than 40. At this point, we can’t make the units digits any smaller; if we did, the tens digits would be 0 or negative, but we know N is a positive two-digit integer.

Statement one is sufficient to answer the question.

Statement Two Alone:

N is 4 less than 4 times the units digit.

Again, let’s test some possible numerical values for the units digit. Let’s start with 9 again since it’s the largest digit possible. If the units digit is 9, then N = 4(9) - 4 = 32, which is less than 40. If the units digit is any smaller, then N will be less than 32, which means N will always be less than 40. If you can’t see this, look at the following:

If the units digit is 8, then N = 4(8) - 4 = 28.

If the units digit is 7, then N = 4(7) - 4 = 24, etc.

Statement two is also sufficient to answer the question.

Answer: D

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