A number equal to 2 times the mean and with a frequency equal to k is inserted in a data having n observation. If the new mean is `(4)/(3)` times the old mean, then the value of `(k)/(n)` is

Answer

To solve the problem step by step, let's denote the old mean as \( \alpha \).

### Step 1: Define the Old Mean
The old mean \( \alpha \) is given by the formula:
\[ \alpha = \frac{\sum x_i}{n} \]
From this, we can express the sum of the observations:
\[ \sum x_i = n \alpha \quad \text{(Equation 1)} \]

**Hint:** The mean is the total sum of observations divided by the number of observations.

### Step 2: Insert the New Number
We are inserting a number equal to \( 2\alpha \) with a frequency of \( k \). The new sum of observations becomes:
\[ \sum x_i + k(2\alpha) = n\alpha + k(2\alpha) \]
The new number of observations is:
\[ n + k \]

### Step 3: Calculate the New Mean
The new mean can be expressed as:
\[ \text{New Mean} = \frac{n\alpha + k(2\alpha)}{n + k} \]
This new mean is given to be \( \frac{4}{3} \alpha \). Therefore, we can set up the equation:
\[ \frac{n\alpha + k(2\alpha)}{n + k} = \frac{4}{3} \alpha \]

**Hint:** The new mean is the total sum of the new observations divided by the new total number of observations.

### Step 4: Simplify the Equation
We can multiply both sides by \( n + k \) to eliminate the denominator:
\[ n\alpha + k(2\alpha) = \frac{4}{3} \alpha (n + k) \]
Now, simplifying the left side:
\[ n\alpha + 2k\alpha = \frac{4}{3} \alpha n + \frac{4}{3} \alpha k \]

### Step 5: Factor Out \( \alpha \)
Assuming \( \alpha \neq 0 \), we can divide through by \( \alpha \):
\[ n + 2k = \frac{4}{3} n + \frac{4}{3} k \]

### Step 6: Clear the Fractions
To eliminate the fraction, multiply the entire equation by 3:
\[ 3n + 6k = 4n + 4k \]

### Step 7: Rearrange the Equation
Rearranging gives:
\[ 6k - 4k = 4n - 3n \]
This simplifies to:
\[ 2k = n \]

### Step 8: Find \( \frac{k}{n} \)
From \( 2k = n \), we can express \( k \) in terms of \( n \):
\[ k = \frac{n}{2} \]
Thus, we find:
\[ \frac{k}{n} = \frac{1}{2} \]

### Final Answer
The value of \( \frac{k}{n} \) is:
\[ \frac{k}{n} = \frac{1}{2} \]

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