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1) The average (arithmetic mean) of the 10 numbers is 10 2) The greatest of the 10 numbers is 15

Target question:What is the range of 10 numbers?

Statement 1: The average (arithmetic mean) of the 10 numbers is 10 Let's TEST some values. There are several sets of numbers that satisfy statement 1. Here are two: Case a: the numbers are {10, 10, 10, 10, 10, 10, 10, 10, 10, 10}. In this case, the answer to the target question is the range is 0 Case b: the numbers are {0, 10, 10, 10, 10, 10, 10, 10, 10, 20}. In this case, the answer to the target question is the range is 20 Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: The greatest of the 10 numbers is 15 There are several sets of numbers that satisfy statement 2. Here are two: Case a: the numbers are {10, 10, 10, 10, 10, 10, 10, 10, 10, 15}. In this case, the answer to the target question is the range is 5 Case b: the numbers are {0, 10, 10, 10, 10, 10, 10, 10, 10, 15}. In this case, the answer to the target question is the range is 15 Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined There are still several sets of numbers that satisfy BOTH statements. Here are two: Case a: the numbers are {5, 10, 10, 10, 10, 10, 10, 10, 10, 15}. In this case, the answer to the target question is the range is 10 Case b: the numbers are {0, 10, 10, 10, 10, 10, 10, 10, 15, 15}. In this case, the answer to the target question is the range is 15 Since we cannot answer the target question with certainty, the combined statements are NOT SUFFICIENT

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

Since Range = Max – Min, we have 3 variables and 1 equation, so C is most likely to be the answer. There is no equation relating the range, the average and the maximum, so the answer is most likely to be E).

Conditions 1) & 2): If the data set is { 5, 5, 5, 5, 5, 15, 15, 15, 15, 15 }, then its range is 10. If the data set is { 4, 5, 5, 5, 6, 15, 15, 15, 15, 15 }, then its range is 11. Since we don’t have a unique solution, both conditions are not sufficient, when taken together.

Therefore, E is the answer. Answer: E

Normally, in problems which require 2 equations, such as those in which the original conditions include 2 variables, or 3 variables and 1 equation, or 4 variables and 2 equations, each of conditions 1) and 2) provide an additional equation. In these problems, the two key possibilities are that C is the answer (with probability 70%), and E is the answer (with probability 25%). Thus, there is only a 5% chance that A, B or D is the answer. This occurs in common mistake types 3 and 4. Since C (both conditions together are sufficient) is the most likely answer, we save time by first checking whether conditions 1) and 2) are sufficient, when taken together. Obviously, there may be cases in which the answer is A, B, D or E, but if conditions 1) and 2) are NOT sufficient when taken together, the answer must be E.
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