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 objective is to determine the value of \(4x + 5y\) utilizing a provided frequency distribution and data on the mean deviation around the median.

The age distribution is as follows:

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

The total student count is given by:

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

This simplifies to:

\(x + y = 10\)

The median age is then calculated. The median is the central value in an ordered frequency distribution. The cumulative frequency distribution is required to find the median:

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

With a total of 40 students, the median corresponds to the 20th value. The cumulative distribution indicates that the median age is 18.

The mean deviation about the median is calculated using the formula:

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

Given a mean deviation of 1.25, the equation is:

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

After simplifying the absolute values:

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

Further simplification yields:

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

This reduces to:

\(x + 2y = 14\)

Two simultaneous equations are now established:

• \(x + y = 10\)
• \(x + 2y = 14\)

Subtracting the first equation from the second provides:

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

Solving this equation yields:

\(y = 4\)

Substituting \(y = 4\) into the equation \(x + y = 10\):

\(x + 4 = 10\)

\(x = 6\)

The expression \(4x + 5y\) is then evaluated as:

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

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