Question

If the median of the observations \(x,(x / 5),(x / 2),(x / 3),(x / 7)\), \((x / 4),(x / 8)(x>0)\) is 10 value of \(x\) is (a) 30 (b) 20 (c) 50 (d) 40

Short answer

The value of x in the given exercise is 40. We found this by first arranging the given observations in ascending order, identifying the median as \(\frac{x}{4}\), and then solving for x when the median is given as 10.

Step-by-step solution

Arrange observations in ascending order

Let's arrange the given observations in ascending order. \(\dfrac{x}{8}, \dfrac{x}{7}, \dfrac{x}{5}, \dfrac{x}{4}, \dfrac{x}{3}, \dfrac{x}{2}, x\)

Determine the median from the ordered sequence

As there are 7 observations in the sequence, the median will be the 4th value (middle value). Thus, the median is \(\frac{x}{4}\). We are given the median value as 10. So: \(\frac{x}{4} = 10\)

Solve for x

To find the value of x, we will now solve the equation from Step 2: \(\frac{x}{4} = 10\) To do this, multiply both sides of the equation by 4: \(x = 10 \times 4\) Simplifying the equation, we get: \(x = 40\) The value of x in the given exercise is 40, which corresponds to the option (d) 40.

Key concepts

Understanding Statistics

Statistics is a branch of mathematics that deals with the collection, analysis, interpretation, and presentation of numerical data. In statistics, the median is a measure of central tendency which separates a higher half from a lower half in a dataset. To calculate the median in a sequence of numbers, you need to follow a set of steps. First, arrange the numbers in ascending order; this is the order from the smallest to the largest value. Then, if the total number of observations is odd, the median is the middle number. If the number is even, the median is the average of the two middle numbers.

In the exercise provided, we see an application of these principles. The median value plays a crucial role in understanding the distribution of the dataset comprised of the terms involving variable 'x'. The process of calculating the value of 'x' from the given median is a fundamental exercise that showcases how statistics is applied in solving algebraic equations.

The Importance of Ascending Order

Arranging data in ascending order is a preliminary yet essential step in various statistical calculations, particularly when determining measures like the median. By ordering sequences from the smallest to the largest value, ascending order provides an organized framework that allows for simpler identification of the central position in a dataset.

When dealing with a sequence that includes variables, as in the given exercise, placing the sequence in ascending order relative to variable 'x' helps in revealing the median's position. This is critical because an unordered list could lead to incorrect interpretations and results. The process of sorting ensures that each element is sequenced correctly, which is inherently linked to the accurate determination of the median and, ultimately, to the solution of any associated algebraic equation.

Arithmetic Sequence Characteristics

An arithmetic sequence is a sequence of numbers in which each term after the first is found by adding a constant called the common difference to the previous term. This sequence can be represented in the form of 'a, a+d, a+2d, a+3d,...' where 'a' is the first term, and 'd' is the common difference.

In the context of the given exercise, though the terms form a sequence depending on 'x', it is not strictly an arithmetic sequence due to the varying denominators. However, understanding arithmetic sequences can prove beneficial as they provide a foundational understanding of how regular patterns in numerical sequences operate. Furthermore, when 'x' is known, the sequence of observations can then be treated similarly to arithmetic sequences for further analysis, such as finding the sum of terms or identifying other statistical measures.