If a, b, and c are positive integers, what is the range of the five nu

If a, b, c are positive integers what is the range of the five numbers 0, 10, a, b, c

(1) a+b+c< 13
(2) a<b<c<13

There are 3 variables (a,b and c) in the original condition. In order to match the number of variables to the number of equations, we need 3 equations. Since the condition 1) and the condition 2) each has 1 equation, we need 1 more equation. Therefore, there is high chance that E is the correct answer. If we use the condition 1) and the condition 2) at the same time, we get a=1, b=2 and c=9. The numbers are unique and the range becomes 10-0=10. Hence, the conditions are sufficient. So the answer is C. However, this is one of the key questions involving integer and statistics. Hence, we have to apply the Common Mistake Type 4(A).
Looking at the condition 1) and 2) separately:

For the condition 1), even if a=b=1 and c=10, the range is still 10-0=10. So, the answer is unique and the condition is sufficient.
For the condition 2), when a=1, b=2 and c=9, range is 10-0=10. When a=1, b=2 and c=12, the range becomes 12-0=12. The answer is not unique and the condition is not sufficient. So the correct answer is A.
l For cases where we need 3 more equations, such as original conditions with “3 variables”, or “4 variables and 1 equation”, or “5 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 80% chance that E is the answer (especially about 90% of 2 by 2 questions where there are more than 3 variables), while C has 15% chance. These two are the majority. In case of common mistake type 3,4, the answer may be from A, B or D but there is only 5% chance. Since E is most likely to be the answer using 1) and 2) separately according to DS definition (It saves us time). Obviously there may be cases where the answer is A, B, C or D.

Hi Karishma,

Can you please clarify why have we taken the sum as 12 in first case. I know that is the maximum that we can consider but the question is simply asking for a range and not the maximum range. If we take a+b+c = 11 then the values will differ.

We need the range of 0, 10, a, b, c
The smallest number here is 0. a, b and c are positive integers so they are at least 1. Hence the minimum value in this set will remain 0 only.
The largest number right now is 10. If one of a, b and c is greater than 10, it will increase the range. So we need to find the maximum value that any one of these can take. To take the maximum value, the sum of all 3 will also be maximum. The maximum sum is 12 and that is why we try to find the maximum value of a, b or c when the sum is 12.

Hi Karishma,

Can you please clarify why have we taken the sum as 12 in first case. I know that is the maximum that we can consider but the question is simply asking for a range and not the maximum range. If we take a+b+c = 11 then the values will differ.[/quote]

We need the range of 0, 10, a, b, c
The smallest number here is 0. a, b and c are positive integers so they are at least 1. Hence the minimum value in this set will remain 0 only.
The largest number right now is 10. If one of a, b and c is greater than 10, it will increase the range. So we need to find the maximum value that any one of these can take. To take the maximum value, the sum of all 3 will also be maximum. The maximum sum is 12 and that is why we try to find the maximum value of a, b or c when the sum is 12.[/quote]

Thanks for your response but I think I didn't frame my question correctly so let me try again.

If a question asks us for Range do we automatically try for the possible maximum value one of them can ascertain instead of the actual value. For eg if a+b+c<13 so I calculate value based on the assumption that it will be max and hence a+b+c = 12 and not a+b+c=11 or 10.

Since we need to determine if A is sufficient to get the answer , we are taking all the possibilities.

Let's see it like this

If you take a+b+c= 12 and we know that minimum value of any of these three variables is 1. => Say a=b=1 => c=10. Thus Range would be 10-0=10.
Now, say a+b+c=10(as you mentioned). in that case, say a=b=1 => c=8. but we already have 10 and 0 in the original list. => Range will still be 10-0=10, no matter what the value of a,b and c be.

Thus, Statement 1 is sufficient.

Where Statement 2 says a<b<c<13. In this case we can have c=12. It would change the range of the list to 12 instead of 10. Thus, it is insufficient.

Thanks for your response but I think I didn't frame my question correctly so let me try again.

If a question asks us for Range do we automatically try for the possible maximum value one of them can ascertain instead of the actual value. For eg if a+b+c<13 so I calculate value based on the assumption that it will be max and hence a+b+c = 12 and not a+b+c=11 or 10.[/quote]

If we can get the actual value - great! But in this question, all we know is a + b + c < 13. We don't know the actual values of a, b and c. All we are trying to establish is that no matter what the actual values of these variables, under the given constraints, the range will stay at 10 only.

Hi Karishma,

A small query, in GMAT we can take same value (1 in this case) for different variables (in this case a,b) ? Got this confusion while doing this question.

Yes, unless it is explicitly stated otherwise, different variables CAN represent the same number.

Hi,

My query is regarding Statement 1. Why we cant take a=b=0 and c=12, then a+b+c=12 which changes range to 12-0=12. Please help.

You've gotten caught in a classic trap here — go back and reread the question stem:

"If a, b, c are positive integers what is the range of the five numbers 0, 10, a, b, c"

0 isn't positive, so the smallest value a and b can equal is 1.

It's really easy to forget little pieces of information from the question stem, so I highly recommend 1) slowing down your reading of the question stem and 2) jotting down things you're likely to miss later. It takes a little more time up front, but it'll save you time, pain, and points down the line.

Bunuel

I have a general question pertaining to values of variables. In this question, though this concept did not matter, I want to know when we are given that a, b, and c are variables, then can the variables have same value i.e. a=b?

Unless it is stated otherwise, different variables CAN represent the same number.

i Just want to know why cant a,b be 0. If a+b= 0 then according to the first piece of information given c could be anything less then 13. if c = 11 or 12 then we could have different values for range couldnt we?

It seems you missed the highlighted part in the stem:

If a, b, and c are positive integers, what is the range of the five numbers 0, 10, a, b, and c ?

i think this is high quality question. i always forget 0 is not a positive integer.

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