Question

The average of $25$ results is $18$. The average of the first twelve of them is $14$ and of the last twelve is $17$. Find the thirteenth result.
A). $80$
B). $79$
C). $78$
D). $77$

Answer 1

Hint: The given question is based on the concept of average. Average is defined as the mean value which is equal to the ratio of the sum of a number of a given set of values to the total number of values present in the set. It can be expressed as: Average = Sum of values/Total number of values.
We are given:
1. The average of $25$ results is $18$, from here we can calculate the total sum of $25$ results.
2. The average of the first twelve results is $14$, from here we can calculate the sum of the first twelve results.
3. The average of last twelve results, i.e., average of fourteenth result from the beginning to twenty fifth results is $17$, from here we can calculate the sum of last twelve results. ( $25 - 12 + 1 = 14th$ term from beginning is $12th$ from the last)
Formula Used: $Average = \dfrac{{{x_1} + {x_2} + {x_3} + ...... + {x_n}}}{n}$ or
$Average = \dfrac{{\sum\limits_{i = 1}^n {{x_i}} }}{n}$ Here, $\sum\limits_{i = 1}^n {{x_i}} $ symbol denotes the sum ${x_1} + {x_2} + {x_3} + ..... + {x_n}$ .

Complete step-by-step solution:
Let those results be ${x_1},{x_2},...,{x_{25}}$
The average of the first $25$ results is $18$.
$ \Rightarrow \dfrac{{\sum\limits_{i = 1}^{25} {{x_i}} }}{{25}} = 18$
After cross multiplication, we get
$ \Rightarrow \sum\limits_{i = 1}^{25} {{x_i}} = 18 \times 25$
$ \Rightarrow \sum\limits_{i = 1}^{25} {{x_i}} = 450$
So, the total sum of first $25$results is $450$
The average of the first twelve results is $14$
$ \Rightarrow \dfrac{{\sum\limits_{i = 1}^{12} {{x_i}} }}{{12}} = 14$
After cross multiplication, we get
$ \Rightarrow \sum\limits_{i = 1}^{12} {{x_i}} = 14 \times 12$
$ \Rightarrow \sum\limits_{i = 1}^{12} {{x_i}} = 168$
So, the sum of the first twelve results is $168$
The average of the last twelve results is $17$
$ \Rightarrow \dfrac{{\sum\limits_{i = 14}^{25} {{x_i}} }}{{12}} = 17$
After cross multiplication, we get
$ \Rightarrow \sum\limits_{i = 14}^{25} {{x_i}} = 17 \times 12$
$ \Rightarrow \sum\limits_{i = 14}^{25} {{x_i}} = 204$
So, the sum of the last twelve results is $204$
But we have to find the thirteenth result i.e., ${x_{13}}$
The total sum of $25$ results = Sum of twelve results from beginning + thirteenth result + Sum of last twelve results.
$ \Rightarrow \sum\limits_{i = 1}^{25} {{x_i}} = \sum\limits_{i = 1}^{12} {{x_i}} + {x_{13}} + \sum\limits_{i = 14}^{25} {{x_i}} $
$ \Rightarrow 450 = 168 + {x_{13}} + 204$
Shift numbers to one side
$ \Rightarrow {x_{13}} = 450 - 168 - 204$
On simplifying, we get
$ \Rightarrow {x_{13}} = 450 - 372$
$ \Rightarrow {x_{13}} = 78$
Therefore, the thirteenth result is $78$.

Note: If any term is given in the form of- from the last, convert the position of the term to, from the beginning, and don’t forget to add $1$. We added $1$ here so that the position of the term does not get repeated. We should take care of the calculations so as to be sure of our final answer.