Question

If \(a

Short answer

Based on the given conditions and information about the mean and median of the numbers in the set, we determined that the smallest number is \(a\), the middle number is \(b\), and the largest number is \(2a\). We were given that the mean of the set is 15 and the median is 12, which allowed us to set up equations to solve for \(a\). By substituting the value of \(b\) in the mean equation and solving, we found that the value of \(a\) is 11.

Step-by-step solution

Understand the order of the numbers

Since we know that \(a<b<2a\), it is evident that the middle number in the set of 3 numbers is \(b\) and \(a\) should be the smallest number and \(2a\) should be the largest number. So, we have the sequence \((a, b, 2a)\).

Calculate the mean

According to the given problem, the mean of \(a\), \(b\), and \(2a\) is 15. The mean is the sum of the numbers divided by the number of terms. In this case, we have three terms, so the mean equation is: \(\frac{a + b + 2a}{3} = 15\)

Calculate the median

The median is the "middle" number, which is also given in the problem as 12. Since we know the order of the numbers, we can directly say that \(b = 12\).

Solve for \(a\)

Substitute the value of \(b\) in the mean equation found in step 2: \(\frac{a + 12 + 2a}{3} = 15\) Now solve for \(a\): \(3a + 12 = 45\) \(3a = 33\) \(a = 11\) Therefore, the value of \(a\) is 11, which corresponds to option (2).

Key concepts

Mean and Median Calculation

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 in Mathematics

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

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.