The age distribution of 400 persons in a colony having median age 32 is given below:
`{:("Age (in years) :", 20-25,25-30,30-35,35-40,40-45,45-50),("Frequency :", 110,x,75,55,y,30):}`
Then, x - y is

Options

A. 10
B. 20
C. -10
D. -20

Explanation

To solve the problem, we need to find the values of \( x \) and \( y \) from the given age distribution and then calculate \( x - y \).

### Step 1: Identify the Median Class
The median age is given as 32, which lies in the age group 30-35. Therefore, the median class is 30-35.

### Step 2: Write Down the Given Frequencies
The frequency distribution is as follows:
- Age (in years): 20-25, 25-30, 30-35, 35-40, 40-45, 45-50
- Frequency: 110, \( x \), 75, 55, \( y \), 30

### Step 3: Calculate Cumulative Frequencies
We will calculate the cumulative frequencies for each age group:
- For 20-25: 110
- For 25-30: \( 110 + x \)
- For 30-35: \( 110 + x + 75 = 185 + x \)
- For 35-40: \( 185 + x + 55 = 240 + x \)
- For 40-45: \( 240 + x + y \)
- For 45-50: \( 240 + x + y + 30 = 270 + x + y \)

### Step 4: Use the Median Formula
The formula for median is given by:
\[ \text{Median} = L + \left( \frac{N/2 - F}{f} \right) \times H \]
Where:
- \( L \) = lower limit of the median class = 30
- \( N \) = total frequency = 400
- \( F \) = cumulative frequency of the class preceding the median class = \( 110 \) (for 20-25)
- \( f \) = frequency of the median class = 75
- \( H \) = class width = \( 35 - 30 = 5 \)

Substituting the values into the formula:
\[ 32 = 30 + \left( \frac{200 - (110)}{75} \right) \times 5 \]
\[ 32 - 30 = \left( \frac{90 - x}{75} \right) \times 5 \]
\[ 2 = \left( \frac{90 - x}{75} \right) \times 5 \]
\[ 2 = \frac{90 - x}{15} \]
Multiplying both sides by 15:
\[ 30 = 90 - x \]
Rearranging gives:
\[ x = 90 - 30 = 60 \]

### Step 5: Find the Value of \( y \)
The total frequency must equal 400:
\[ 110 + x + 75 + 55 + y + 30 = 400 \]
Substituting \( x = 60 \):
\[ 110 + 60 + 75 + 55 + y + 30 = 400 \]
Calculating the left side:
\[ 330 + y = 400 \]
Solving for \( y \):
\[ y = 400 - 330 = 70 \]

### Step 6: Calculate \( x - y \)
Now we can find \( x - y \):
\[ x - y = 60 - 70 = -10 \]

### Final Answer
The value of \( x - y \) is \(-10\).

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