Question

The following table gives the distribution of 100 families according to their daily expenditure: 

Expenditure (in Rs)	0–10	10–20	20–30	30–40	40–50
Number of families	14	\(x\)	27	\(y\)	15

If the mode of the distribution is \(24\), find the missing frequencies \(x\) and \(y\).

• A. \(x=16, y=20\)
• B. \(x=18, y=21\)
• C. \(x=23, y=21\)
• D. \(x=21, y=23\)

Hint:

Always start by checking if the sum of the options \(x + y\) matches the total remaining frequency.
Here, \(100 - (14 + 27 + 15) = 44\).
For Option (C), \(23 + 21 = 44\), which is correct.
This helps quickly narrow down or verify the options.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The mode of a grouped frequency distribution is the value that occurs most frequently.
It is calculated from the modal class, which is the class interval with the highest frequency.

Step 2: Key Formula or Approach:
The formula to calculate the mode of grouped data is:
\[ \text{Mode} = L + \left(\frac{f_1 - f_0}{2f_1 - f_0 - f_2}\right) \times h \]
where:
- \(L\) is the lower limit of the modal class.
- \(f_1\) is the frequency of the modal class.
- \(f_0\) is the frequency of the class preceding the modal class.
- \(f_2\) is the frequency of the class succeeding the modal class.
- \(h\) is the width of the class interval.

Step 3: Detailed Explanation:
We are given that the total number of families is 100.
Therefore, the sum of all frequencies must equal 100:
\[ 14 + x + 27 + y + 15 = 100 \]
\[ x + y + 56 = 100 \implies x + y = 44 \quad \text{--- (Equation 1)} \]
The mode of the distribution is given as 24.
Since the value 24 lies within the class interval 20-30, this is our modal class.
From this modal class, we identify the following parameters:
- Lower limit of modal class, \(L = 20\)
- Frequency of modal class, \(f_1 = 27\)
- Frequency of preceding class, \(f_0 = x\)
- Frequency of succeeding class, \(f_2 = y\)
- Class width, \(h = 10\)
Substituting these values into the mode formula:
\[ 24 = 20 + \left(\frac{27 - x}{2(27) - x - y}\right) \times 10 \]
\[ 24 - 20 = \left(\frac{27 - x}{54 - (x + y)}\right) \times 10 \]
Substitute \(x + y = 44\) from Equation 1 into the denominator:
\[ 4 = \left(\frac{27 - x}{54 - 44}\right) \times 10 \]
\[ 4 = \left(\frac{27 - x}{10}\right) \times 10 \]
The factor of 10 cancels out:
\[ 4 = 27 - x \implies x = 27 - 4 = 23 \]
Now, substitute the value of \(x\) back into Equation 1 to find \(y\):
\[ 23 + y = 44 \implies y = 44 - 23 = 21 \]
Thus, the missing frequencies are \(x = 23\) and \(y = 21\).
This matches the third option.

Step 4: Final Answer:
Therefore, the correct option is (C).