Understanding the Problem: Arithmetic Mean with Frequencies

The question asks us to find the value of 'x' given that the numbers 4 and 9 have specific frequencies, and their combined arithmetic mean is known. The frequencies are given in terms of 'x'.

Here's the information provided:

• Number 1: 4, with frequency 'x'.
• Number 2: 9, with frequency '(x - 1)'.
• Arithmetic Mean of these numbers: 6.

We need to use the formula for calculating the arithmetic mean for a set of data with frequencies.

Calculating Arithmetic Mean with Frequencies

When you have data points with associated frequencies, the arithmetic mean is calculated by summing the product of each data point and its frequency, and then dividing by the sum of the frequencies.

The formula is:

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

Where:

• \(\bar{x}\) is the arithmetic mean.
• \(v_i\) is the i-th data point (value).
• \(f_i\) is the frequency of the i-th data point.
• \(\sum (v_i \times f_i)\) is the sum of the products of values and their frequencies.
• \(\sum f_i\) is the sum of the frequencies.

Setting up the Equation to Find x

Using the given numbers and their frequencies, we can set up the equation based on the arithmetic mean formula:

Value 1: \(v_1 = 4\), Frequency 1: \(f_1 = x\)

Value 2: \(v_2 = 9\), Frequency 2: \(f_2 = (x - 1)\)

Arithmetic Mean: \(\bar{x} = 6\)

Sum of (Value \(\times\) Frequency):

\(\sum (v_i \times f_i) = (4 \times x) + (9 \times (x - 1))\)

\(= 4x + 9x - 9\)

\(= 13x - 9\)

Sum of Frequencies:

\(\sum f_i = x + (x - 1)\)

\(= 2x - 1\)

Now, substitute these into the arithmetic mean formula:

\(6 = \frac{13x - 9}{2x - 1}\)

Solving for the Value of x

Now we need to solve the equation for x:

\(6 = \frac{13x - 9}{2x - 1}\)

Multiply both sides by \((2x - 1)\) to remove the denominator:

\(6 \times (2x - 1) = 13x - 9\)

Distribute the 6 on the left side:

\(12x - 6 = 13x - 9\)

Now, gather the 'x' terms on one side and the constants on the other side. Subtract \(12x\) from both sides:

\(-6 = 13x - 12x - 9\)

\(-6 = x - 9\)

Add 9 to both sides:

\(-6 + 9 = x\)

\(3 = x\)

So, the value of x is 3.

Verification of the Value of x

Let's check if x = 3 yields an arithmetic mean of 6.

• Frequency of 4 = x = 3
• Frequency of 9 = x - 1 = 3 - 1 = 2
• Total frequency = 3 + 2 = 5
• Sum of (Value \(\times\) Frequency) = (4 \(\times\) 3) + (9 \(\times\) 2) = 12 + 18 = 30
• Arithmetic Mean = \(\frac{\text{Sum of (Value} \times \text{Frequency)}}{\text{Sum of Frequencies}} = \frac{30}{5} = 6\)

The calculated mean is 6, which matches the given arithmetic mean in the problem. Therefore, the value of x = 3 is correct.

Conclusion

The value of x that satisfies the given conditions is 3.

Revision Table: Key Concepts

Additional Information: Measures of Central Tendency

The arithmetic mean is one of several measures used to describe the center of a dataset. Other common measures include the median and the mode.

• Median: The middle value in a dataset that is ordered from least to greatest. If there's an even number of data points, the median is the average of the two middle values.
• Mode: The value that appears most frequently in a dataset. A dataset can have one mode (unimodal), more than one mode (multimodal), or no mode.

Understanding these measures helps provide a more complete picture of the distribution of data.