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.
_________________