A number equal to 4 times of the mean and a frequency equal to k is inserted in the data of n observations. If the new mean is `(7)/(5)` times the old mean, then `(n)/(k)` is equal to

Answer

To solve the problem step by step, we can follow these steps:

### Step 1: Define the Old Mean
Let the old mean of the n observations be denoted as \( a \). Therefore, the sum of the old observations can be expressed as:
\[ \Sigma x_i = a \cdot n \]

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

### Step 2: Insert the New Data
We insert a new number equal to \( 4a \) with a frequency of \( k \). The total contribution of the new data to the sum of observations is:
\[ \text{New Contribution} = 4a \cdot k \]

**Hint:** When adding multiple identical values, you can multiply the value by the frequency.

### Step 3: Calculate the New Sum of Observations
The new sum of observations becomes:
\[ \Sigma x_i^{\text{new}} = \Sigma x_i + \text{New Contribution} = a \cdot n + 4a \cdot k \]
Factoring out \( a \):
\[ \Sigma x_i^{\text{new}} = a(n + 4k) \]

**Hint:** Always look for common factors when simplifying expressions.

### Step 4: Calculate the New Number of Observations
The new number of observations is:
\[ n + k \]

**Hint:** When new data is added, the total count increases by the frequency of the new data.

### Step 5: Calculate the New Mean
The new mean can be expressed as:
\[ \text{New Mean} = \frac{\Sigma x_i^{\text{new}}}{n + k} = \frac{a(n + 4k)}{n + k} \]

**Hint:** The mean is calculated by dividing the total sum by the total count.

### Step 6: Set Up the Equation
According to the problem, the new mean is \( \frac{7}{5} \) times the old mean:
\[ \frac{a(n + 4k)}{n + k} = \frac{7}{5} a \]
We can cancel \( a \) from both sides (assuming \( a \neq 0 \)):
\[ \frac{n + 4k}{n + k} = \frac{7}{5} \]

**Hint:** Canceling common factors simplifies the equation.

### Step 7: Cross-Multiply to Solve for n and k
Cross-multiplying gives:
\[ 5(n + 4k) = 7(n + k) \]
Expanding both sides:
\[ 5n + 20k = 7n + 7k \]

**Hint:** Distributing terms helps to rearrange the equation.

### Step 8: Rearranging the Equation
Rearranging the equation yields:
\[ 20k - 7k = 7n - 5n \]
This simplifies to:
\[ 13k = 2n \]

**Hint:** Grouping like terms can help isolate variables.

### Step 9: Solve for \( \frac{n}{k} \)
Now, we can express \( \frac{n}{k} \):
\[ \frac{n}{k} = \frac{13}{2} \]

**Hint:** To find the ratio, divide both sides by \( k \) and rearrange.

### Final Answer
Thus, the value of \( \frac{n}{k} \) is:
\[ \frac{n}{k} = \frac{13}{2} \]

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