If a<b<c<d and a, b, c, d are non zero integers, the mean of...

If $a<b<c<d$ and $a, b, c, d$ are non zero integers, the mean of $a, b, c, d$ is 0 and the median of $a, b, c$, d is 0 , then which of the following is correct?
(1) $\mathrm{b}=-\mathrm{c}$
(2) $\mathrm{a}=-\mathrm{d}$
(3) Both (1) and (2)
(4) None of these

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Key Concepts

Arithmetic Mean

The arithmetic mean is the sum of a set of numbers divided by the count of those numbers. When provided with the mean in a problem, it allows one to create an equation representing the total sum. This equation can be combined with other conditions to derive relationships between the elements of the set.

Median in Even-Numbered Data Sets

For a set with an even number of elements, the median is defined as the average of the two central numbers after ordering the set. When the median is given, it imposes a specific relationship between these two middle numbers, leading to useful constraints in problem solving.

Ordered Sequences and Inequalities

The ordering of numbers (e.g., a < b < c < d) provides additional information that restricts the potential values each element can assume. This hierarchical structure, when combined with numerical conditions like the mean or median, is essential in deducing relationships and solving for the variables.

Symmetry in Equations

When the sum of two numbers yields zero, it indicates that the numbers are negatives of each other. Recognizing this symmetry simplifies the problem by directly relating pairs of variables. Such symmetric relationships often play a crucial role when the overall sum or central tendencies (like the median) are defined to be zero.

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