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Understanding mean and median is fundamental in statistics and often in solving mathematical problems. The mean, or average, of a set of numbers is calculated by adding up all the numbers and then dividing by the total count of numbers. It gives us a central value that represents the entire set. In mathematical terms, for a set of numbers \(x_1, x_2,..., x_n\), the mean \(\overline{x}\) is given by \(\overline{x} = \frac{\sum_{i=1}^{n} x_i}{n}\).

The median, on the other hand, is the middle number in a sorted list of numbers. If there is an odd number of terms, the median is simply the middle term. If there is an even number of terms, the median is the average of the two middle numbers. The median divides the dataset into two equal halves, and in the case of our example with three numbers \(a, b, 2a\), the median is naturally \(b\) since it's the middle value when the numbers are ordered.
To clearly elucidate this with our problem, the mean of \(a, b, 2a\) was calculated by \(\frac{a + b + 2a}{3} = 15\), and since \(b\) is the median, its value is directly obtained as 12 when ordered correctly.

Inequalities are a part of mathematics that deal with the relative size or order of two values. They are represented by symbols like \(>\), \(<\), \(\geq\), and \(\leq\), which denote 'greater than', 'less than', 'greater than or equal to', and 'less than or equal to' respectively. The core concept involves understanding that the inequality \(a < b\) means \(a\) is strictly less than \(b\)

When solving problems that involve inequalities, such as the given problem where \(a < b < 2a\), it is essential to maintain the order of the numbers since inequalities affect the steps involved in arriving at a solution. Correctly interpreting and maintaining inequalities is key to solving variables within such constraints. In this problem, identifying \(b\) as the median, given the constraints, was crucial for solving for \(a\) effectively, as it provided a straightforward value that could be used in subsequent calculations.

Solving for variables is a core skill in algebra that involves finding the values of unknown quantities. The goal is to isolate the variable of interest on one side of an equation or inequality. For linear equations, this often involves a series of operations such as addition, subtraction, multiplication, or division applied to both sides of the equation to maintain equality.

In the context of our example, after calculating the mean and determining the median, we solved for \(a\) by substituting the known value of \(b\) and simplifying the equation. This moved us from a general mean equation \(\frac{a + b + 2a}{3} = 15\) to a specific linear equation \(3a + 12 = 45\), which we then solved by isolating \(a\) and dividing both sides by 3 to get \(a = 11\). Such methods of isolating the variable form the bedrock of algebra and are widely applicable across many domains of mathematics.