For a set of 3 numbers, assuming there is only one mode, does the mode

Platinum GMAT Official Solution:

The mode is the number that appears the most times in the set, and the median is the middle number when the set is sorted from least to greatest. The range is the largest value in the set minus the smallest value.

Given that there is only one mode, at least two of the numbers must be equal. In particular, either all the numbers are equal, or two of the numbers are equal and the third is different.

Evaluate Statement (1) alone.

First, assume that all three of the numbers in the set are equal. Represent this set by {A, A, A}. The range is equal to A - A = 0, because the highest number and the lowest number in the set are the same.

By Statement (1), the range is equal to the median. Since the range is 0, the median is 0. Now the set can be represented by {A, 0, A}, since the median is the middle number. But all the numbers in the set are the same, so A = 0 and the set can be represented by {0, 0, 0}. The mode is 0 because this is the number that appears the most times.

Therefore, when all three numbers in the set are equal, the mode is equal to the range.

Now, assume that only two of the numbers in the set are equal. The set can be represented as {A, A, B}, where A is not the same as B. When sorted from least to greatest, the set become either {A, A, B} or {B, A, A}. The median of both of these sets is A. By Statement (1), the range is A because the range is equal to the median. The mode is also A, because it appears in the set more times than B.

Therefore, when only two numbers in the set are equal, the mode is equal to the range.

Whether three numbers in the set are equal, or only two, the mode is always equal to the range.

Statement (1) is SUFFICIENT.

Evaluate Statement (2) alone.

First, assume that all three of the numbers in the set are equal. Represent this set by {A, A, A}. Statement (2) says that A = 2A, because A is both the largest number and the smallest number in the set. The only way A = 2A is if A = 0.

When A = 0, the set becomes {0, 0, 0}. The range is 0 - 0 = 0, and the mode is 0. Thus, the mode equals the range when all three numbers are equal.

Now, assume that only two of the numbers in the set are equal. The set can be represented as {A, A, B}, where A is not the same as B. When sorted from least to greatest, the set become either {A, A, B} or {B, A, A}. Statement (2) says that A = 2B, or B = 2A, depending on whether A or B is larger.

Assuming A is the larger number, A = 2B. For example, A = 8 and B = 4. Then the sorted set becomes {4, 8, 8}. In this case, the range is 8 - 4 = 4. The mode is 8, because it appears the most times in the set. The mode is NOT equal to the range.

Now, assume B is the larger number, B = 2A. For example, A = 3 and B = 6. Then the sorted set becomes {3, 3, 6}. In this case, the range is 6 - 3 = 3. The mode is 3. The mode is equal to the range in this case.

Whether or not the mode is equal to the range depends on whether A or B is larger. Therefore the answer cannot be determined from Statement (2) alone.

Statement (2) is NOT SUFFICIENT.

Since Statement (1) alone is SUFFICIENT and Statement (2) alone is NOT SUFFICIENT, answer A is correct.

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