The question asks to find the value of k, given a dataset where the mean is also k. The data provided is "Value32k45Frequencyk2k3k4k5k". We need to interpret this string to establish the values and their corresponding frequencies.

Interpreting the Data

The string "Value32k45Frequencyk2k3k4k5k" can be parsed as follows:

• Values ($x_i$): 3, 2, k, 4, 5
• Frequencies ($f_i$): k, 2k, 3k, 4k, 5k

This interpretation aligns the number of values with the number of frequencies, creating pairs for calculation.

Data Table

Calculating the Mean

The formula for the mean ($\bar{x}$) of a frequency distribution is:

$ \bar{x} = \frac{\sum (f_i \times x_i)}{\sum f_i} $

In this problem, the mean ($\bar{x}$) is given as k.

Sum of (Frequency × Value)

Calculate the sum of the product of each frequency and its corresponding value:

$ \sum (f_i \times x_i) = (k \times 3) + (2k \times 2) + (3k \times k) + (4k \times 4) + (5k \times 5) $

$ \sum (f_i \times x_i) = 3k + 4k + 3k^2 + 16k + 25k $

$ \sum (f_i \times x_i) = 3k^2 + (3 + 4 + 16 + 25)k $

$ \sum (f_i \times x_i) = 3k^2 + 48k $

Sum of Frequencies

Calculate the sum of all frequencies:

$ \sum f_i = k + 2k + 3k + 4k + 5k $

$ \sum f_i = (1 + 2 + 3 + 4 + 5)k $

$ \sum f_i = 15k $

Solving for k

Now, substitute these sums into the mean formula, setting the mean equal to k:

$ k = \frac{3k^2 + 48k}{15k} $

Assuming $k \neq 0$ (since frequencies like 'k', '2k', etc., must be positive), we can multiply both sides by $15k$:

$ k \times (15k) = 3k^2 + 48k $

$ 15k^2 = 3k^2 + 48k $

Rearrange the equation to solve for k:

$ 15k^2 - 3k^2 - 48k = 0 $

$ 12k^2 - 48k = 0 $

Factor out the common term $12k$:

$ 12k(k - 4) = 0 $

This equation yields two possible solutions for k:

• $12k = 0 \implies k = 0$
• $k - 4 = 0 \implies k = 4$

Since the frequencies are given as k, 2k, 3k, 4k, and 5k, the value of k must be positive. Therefore, $k=0$ is not a valid solution in this context.

The only valid solution is k = 4.