Question

The frequency distribution of the age of students in a class of 40 students is given below:

\(Age\)	15	16	17	18	19	20
No. of Students	5	8	5	12	x	y

If the mean deviation about the median is 1.25, then \(4x + 5y\) is equal to:

• A. 43
• B. 44
• C. 47
• D. 46

Answer 1

The Correct Option is B

The problem is to find the value of \(4x + 5y\) using the given frequency distribution and information about the mean deviation around the median.

The age distribution given in the question is:

\(Age\)	15	16	17	18	19	20
No. of Students	5	8	5	12	x	y

The total number of students is:

\(5 + 8 + 5 + 12 + x + y = 40\)

So, we have:

\(x + y = 10\)

Next, we need to find the median age. The median is the middle value in a frequency distribution when the students are arranged in ascending order. To find the median, we need the cumulative frequency distribution:

\(Age\)	15	16	17	18	19	20
Cumulative Frequency	5	13	18	30	30+x	40

The median corresponds to the 20th value since the total number of students is 40. From the cumulative distribution, we can see that the median age is 18.

Now, the mean deviation about the median is given by:

\(\frac{1}{N}\sum |x_i - Median|\)

With a mean deviation of 1.25, we have:

\(\frac{1}{40}[5|15-18| + 8|16-18| + 5|17-18| + 12|18-18| + x|19-18| + y|20-18|] = 1.25\)

Simplifying within the modulus:

\(\frac{1}{40}(15 + 16 + 5 + 0 + x + 2y) = 1.25\)

Simplifying further:

\(36 + x + 2y = 50\)

Which simplifies to:

\(x + 2y = 14\)

We now have two equations:

• \(x + y = 10\)
• \(x + 2y = 14\)

Subtract the first equation from the second:

\((x + 2y) - (x + y) = 14 - 10\)

Solving gives:

\(y = 4\)

Substitute \(y = 4\) in \(x + y = 10\):

\(x + 4 = 10\)

\(x = 6\)

Therefore, \(4x + 5y\) becomes:

\(4(6) + 5(4) = 24 + 20 = 44\)

Thus, the value of \(4x + 5y\) is 44.

Answer 2

We are given:
x + y = 10 $\quad \cdots \text{(1)}$

The median is:
M = 18

The formula for the Mean Deviation (M.D.) is:

$\text{M.D.} = \frac{\sum f_i |x_i - M|}{\sum f_i}$

Substituting the given values:

1.25 = $\frac{36 + x + 2y}{40}$

Simplifying:

x + 2y = 14 $\quad \cdots \text{(2)}$

From equations (1) and (2), solving simultaneously:

x + y = 10

x + 2y = 14

Subtracting (1) from (2):

y = 4

Substituting y = 4 into (1):

x = 6

Now, substituting x = 6 and y = 4 into 4x + 5y:

4x + 5y = 4(6) + 5(4) = 24 + 20 = 44

Final Answer: 44

The table values are as follows:
 

Age \(x_i\)	f	\(|x_i - M|\)	\(f_i|x_i - M|\)
15	3	3	15
16	8	2	16
17	5	1	5
18	12	0	0
19	x	1	x
20	y	2	2y