Step 1 : Understanding the Question
This question is a statistical problem focused on finding the 'Median' of a specific numerical set. The median represents the exact middle value in an ordered list of data. In this scenario, the data set consists of the first six multiples of the number 5. Because the total number of observations (n = 6) is an even number, the calculation requires an additional step compared to a data set with an odd number of values.
Step 2 : Key Formulas and approach
The approach involves three specific stages: generating the data set, arranging it in ascending order, and applying the median formula for an even number of terms.
Key Formulas:
1. If $n$ is even, $\text{Median} = \frac{\left( \frac{n}{2} \right)^{th} \text{ term} + \left( \frac{n}{2} + 1 \right)^{th} \text{ term}}{2}$
2. $\text{Multiples of } x = \{x \cdot 1, x \cdot 2, x \cdot 3, \dots, x \cdot n\}$
Step 3 : Detailed Explanation
Generating the Set: The first six multiples of 5 are obtained by multiplying 5 by the natural numbers 1 through 6. This gives us the set: $5, 10, 15, 20, 25, 30$.
Analyzing the Count: We observe that there are $n = 6$ terms in the set. Since 6 is an even number, there is no single middle term that divides the set into two equal halves.
Locating Middle Positions: According to the formula for even data sets, we must identify the terms at the position $\frac{6}{2} = 3^{rd}$ and $\frac{6}{2} + 1 = 4^{th}$.
Identifying Terms: In our ordered sequence ($5, 10, 15, 20, 25, 30$), the value at the $3^{rd}$ position is 15, and the value at the $4^{th}$ position is 20.
Calculating the Mean of Middle Terms: The median is the arithmetic average of these two central values: $\text{Median} = \frac{15 + 20}{2} = \frac{35}{2}$. This calculation yields 17.5.
Step 4 : Final Answer
The median of the first six multiples of 5 is 17.5, which is option (B).