Question

Range of the scores \(18,13,14,42,22,26\) and \(\mathrm{x}\) is \(44(\mathrm{x}>0)\). Find the sum of the digits of \(\mathrm{x}\). (1) 16 (2) 14 (3) 12 (4) Cannot be determined

Short answer

Answer: 12

Step-by-step solution

Calculate the current range of the given scores

Before adding x to the set, let's calculate the range: Range = Maximum Value - Minimum Value Using the given scores 18, 13, 14, 42, 22, 26, the current range would be: Range = 42 - 13 = 29 As of now, the range is 29.

Determine if x is the new maximum or minimum value

To achieve a range of 44, we need to determine whether x would be the new maximum value or minimum value. The value of x must be greater than 0. Since the desired range is larger than the current range, x will be the new maximum value.

Calculate the value of x

Since x is the new maximum value, we can calculate it using the desired range: x - Minimum Value = Desired Range x - 13 = 44 x = 44 + 13 x = 57 The value of x is 57.

Calculate the sum of the digits of x

Now, we find the sum of the digits of x: Sum = 5 + 7 Sum = 12 The sum of the digits of x is 12.

Choose the correct option

Finally, among the given options, we can choose the correct one. The sum of the digits is 12, which is: (3) 12

Key concepts

Minimum and Maximum Values

In statistics, the concepts of minimum and maximum values are fundamental. The **minimum value** is the smallest number in a set of data, while the **maximum value** is the largest. These two values help define the range of the data set.
To find the range of a dataset, simply subtract the minimum value from the maximum value.
For example, in the dataset given: 18, 13, 14, 42, 22, and 26, the minimum value is 13 and the maximum value is 42.
Thus, the initial range without considering any additional values is calculated as 42 - 13 = 29.
Knowing these values helps us understand the spread and variability of the data. Identifying a new minimum or maximum, as in the exercise's example where \(\mathrm{x}\) changes the current maximum, directly influences the resulting range.

Sum of Digits

The sum of digits is a simple mathematical concept, but it often comes in handy in problem-solving. To find the sum of the digits of a number, you add each individual digit together.
In the context of the example provided, once the value of \(\mathrm{x}\) was determined to be 57, calculating the sum of its digits involves adding 5 and 7.
Mathematically, this is represented as:

• 5 + 7 = 12

This result is not only applicable in statistics problems but also in other fields, including digit sum puzzles and encoding systems. Keeping track of such small details ensures accuracy in larger problem-solving contexts.

Problem Solving in Mathematics

Mathematical problem-solving involves a series of logical steps designed to find a solution to a given problem. It requires understanding each component and how they interact.
In our exercise, the goal was to find the sum of the digits of \(\mathrm{x}\) based on a known range. The process involves several steps:

• First, calculating the current range of the dataset.
• Next, determining whether \(\mathrm{x}\) influences the minimum or maximum value.
• Using these insights to calculate \(\mathrm{x}\).
• Finally, solving for the sum of the digits of \(\mathrm{x}\).

This logical sequence illustrates problem-solving by breaking down larger problems into manageable parts, making it easier to find a precise solution. Such structured approaches are crucial in mathematics and can be applied across various complex scenarios.