If the mean of the following frequency distribution is 5, then b =
`{:(x_(i) " :",3,5,7,4),(f_(i)" :",2,a,5,b):}`

Options

A. 10
B. 6
C. 8
D. none of these

Explanation

To find the value of \( b \) in the given frequency distribution, we will follow these steps:

### Step 1: Set up the frequency distribution
We are given the following frequency distribution:

\[ \begin{array}{|c|c|} \hline x_i & f_i \\ \hline 3 & 2 \\ 5 & a \\ 7 & 5 \\ 4 & b \\ \hline \end{array} \]

### Step 2: Calculate \( x_i f_i \)
We will calculate \( x_i f_i \) for each \( x_i \) and \( f_i \):

- For \( x_1 = 3 \) and \( f_1 = 2 \):
\[ x_1 f_1 = 3 \times 2 = 6 \]

- For \( x_2 = 5 \) and \( f_2 = a \):
\[ x_2 f_2 = 5 \times a = 5a \]

- For \( x_3 = 7 \) and \( f_3 = 5 \):
\[ x_3 f_3 = 7 \times 5 = 35 \]

- For \( x_4 = 4 \) and \( f_4 = b \):
\[ x_4 f_4 = 4 \times b = 4b \]

Now, we can summarize the \( x_i f_i \) values:

\[ \begin{array}{|c|c|} \hline x_i & f_i & x_i f_i \\ \hline 3 & 2 & 6 \\ 5 & a & 5a \\ 7 & 5 & 35 \\ 4 & b & 4b \\ \hline \end{array} \]

### Step 3: Calculate the total of \( x_i f_i \) and \( f_i \)
Now we will find the total of \( x_i f_i \) and \( f_i \):

\[ \text{Total of } x_i f_i = 6 + 5a + 35 + 4b = 41 + 5a + 4b \]

\[ \text{Total of } f_i = 2 + a + 5 + b = 7 + a + b \]

### Step 4: Set up the mean equation
We know that the mean \( \bar{x} \) is given as 5. The formula for mean is:

\[ \bar{x} = \frac{\sum x_i f_i}{\sum f_i} \]

Substituting the totals we found:

\[ 5 = \frac{41 + 5a + 4b}{7 + a + b} \]

### Step 5: Cross-multiply to eliminate the fraction
Cross-multiplying gives us:

\[ 5(7 + a + b) = 41 + 5a + 4b \]

Expanding both sides:

\[ 35 + 5a + 5b = 41 + 5a + 4b \]

### Step 6: Simplify the equation
Now, we can simplify the equation:

Subtract \( 5a \) from both sides:

\[ 35 + 5b = 41 + 4b \]

Subtract \( 4b \) from both sides:

\[ 35 + b = 41 \]

### Step 7: Solve for \( b \)
Now, isolate \( b \):

\[ b = 41 - 35 \]

\[ b = 6 \]

### Conclusion
Thus, the value of \( b \) is \( 6 \).

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