Question

If the median of 20, 25, 28, x, 33, 34 is 29, then what is the maximum possible value of x.
(a) 30
(b) 31
(c) 29
(d) 32

Answer 1

To solve this problem we need to know how we can calculate the median or even number of terms. Median of an odd number of terms is directly given by the middle term of the total numbers where terms are in ascending order, and if there are an even number of terms then median is the mean of the middle two terms of the total given numbers. In this question we already know that median is 29 so we will take the mean of the middle two terms i.e. x and 28 and equate it with 29 and solve the expression to get the value of x.

Complete step by step answer:
We are given the terms,
20, 25, 28, x, 33, 34
It is also given that the median of the above terms is 29.
So to solve this we should know what do we mean by median.
According to the definition of median, it is the “middle” term of a sorted list of numbers.
But suppose if we are given total odd number of terms and number of terms is n then the middle term will be directly $\dfrac{n+1}{2}$, but if there are total even number of terms and number of terms is n then there will be two middle terms i.e. $\dfrac{n}{2},\dfrac{n}{2}+1$ hence we cannot find its median directly.
So median of the even number of terms is given by the average or mean of the middle two terms,
But before finding the median we have to make sure that all the terms are in sorted order.
In our given question terms are,
20, 25, 28, x, 33, 34
These all are already in sorted order.
And there are total 6 terms which is even so its median will be given by the mean of $\dfrac{6}{2},\dfrac{6}{2}+1={{3}^{rd}}\,and\,{{4}^{th}}$ terms i.e. 28 and x, and the given median is equal to 29, so we get
$\dfrac{28+x}{2}=29$
Cross multiplying we get,
$\begin{aligned} & 28+x=2\times 29 \\\ & x=30 \\\ \end{aligned}$
Hence we get the correct option as (A).

Note:
In the question it is mentioned to find the maximum possible value of x but there is only one specific value of x so it is given just to confuse the student so be careful about that. Also go through the definitions of mean, median and mode to solve more problems of this type. And remember how to find the median for even number of terms for future references.