Question

The numbers $13,15,17,18$ and $n$ are arranged in ascending order. If the mean is equal to the median, find the value of $n$.

Answer 1

Hint: To solve the problem we need to have the knowledge of median and mean. We know that median is basically the midterm among the given numbers which are arranged in an order. To solve we will firstly find the mean and the median of the numbers in series. The second step is to equate the mean to median so as to find the value for $n$.

Complete step by step answer:
The question asks us to find the fifth number $''n''$ when arranged in ascending order consisting of numbers$13,15,17,18$ . The first step is to find the mean of the numbers. Mean, as we know, refers to the ratio of total sum of numbers to total numbers in the series. On writing it mathematically we get:
$\Rightarrow \dfrac{13+15+17+18+n}{5}$
The total numbers given to us including $n$ is $5$. So the denominator is $5$. On calculating it further we get:
$\Rightarrow \dfrac{63+n}{5}$
The next step is to find the median. The formula for median differs for both odd and even numbers. Since in the question we are given with $5$ numbers which is odd. So the formula for median is:
$\text{Median = }{{\left( \dfrac{\text{n+1}}{\text{2}} \right)}^{\text{th}}}\text{term}$
On applying the above formula for the numbers given to us, we get:
$\Rightarrow \text{Median = }{{\left( \dfrac{\text{5+1}}{\text{2}} \right)}^{\text{th}}}\text{term}$
$\Rightarrow \text{Median = }{{\text{3}}^{rd}}\text{term}$
On analysing the series the third term we get is $17$. So the median becomes $17$.
The last step is to equate mean and median with each other. On doing this we get:
$\Rightarrow \dfrac{63+n}{5}=17$
On cross multiplying we get:
$\Rightarrow 63+n=17\times 5$
$\Rightarrow n=85-63$
On further calculation we get:
$\Rightarrow n=22$
$\therefore $ The value of $n$ in numbers $13,15,17,18$ and $n$ is $22$.

Note: Always remember for the calculation of median for any number of terms, the terms should be arranged either in ascending or descending order. In the question above we are already given the terms in ascending order. So in this question we are not required to change the terms in order.