Question

The mean of fifteen different natural numbers is 13. The maximum value for the second largest number is
A) 46
B) 51
C) 52
D) 53

Answer 1

Hint: First assume the second largest and largest natural number. For the maximum value of the second largest number, assume the least different natural numbers for the first 13 numbers. After that simplify the equation and get the desired result.

Complete step-by-step answer:
Let the second largest and largest natural number be $x$ and $y$.
For the maximum value of the second largest number, assume the least different natural numbers for the first 13 numbers.
As we know,
$\bar x = \dfrac{{\sum {{x_i}} }}{n}$
Substitute the values,
$ \Rightarrow 13 = \dfrac{{1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + x + y}}{{15}}$
Add the terms in the numerator,
$ \Rightarrow 13 = \dfrac{{91 + x + y}}{{15}}$
Cross-multiply the terms,
$ \Rightarrow 91 + x + y = 195$
Move the constant term on the right side and subtract,
$ \Rightarrow x + y = 104$
Since the sum is an even number. So, for maximum value, both numbers must be odd. So, $y = x + 2$ will give the maximum value of the second largest number.
Substitute the value in the above equation,
$ \Rightarrow x + \left( {x + 2} \right) = 104$
Simplify the terms,
$ \Rightarrow 2x + 2 = 104$
Move the constant term on the right side and subtract,
$ \Rightarrow 2x = 102$
Divide both sides by 2,
$\therefore x = 51$
Thus, the maximum value for the second-largest natural number is 51.

Hence, option (A) is correct.

Note: Arithmetic Mean is the most common measurement of central tendency. According to the layman, the mean of data represents an average of the given collection of the data. It is equivalent to the sum of all the observations of a given data divided by the total number of observations.
The mean of data for n values in a set of data namely ${x_1},{x_2},{x_3}, \ldots ,{x_n}$ is given by,
$\bar x = \dfrac{{{x_1} + {x_2} + {x_3} + \ldots + {x_n}}}{n}$
For calculating the arithmetic when the number of observations along with the frequency of observation is given such that ${x_1},{x_2},{x_3}, \ldots ,{x_n}$ are the recorded observation and ${f_1},{f_2},{f_3}, \ldots ,{f_n}$ are the corresponding frequencies of the observation,
$\bar x = \dfrac{{{f_1}{x_1} + {f_2}{x_2} + {f_3}{x_3} + \ldots + {f_n}{x_n}}}{{{f_1} + {f_2} + {f_3} + \ldots + {f_n}}}$