Question

In the first four papers of \[100\] marks each, Rishi got $95,72,73,83$ marks respectively. If he wants an average of greater than or equal to $75$ and less than $80$ marks, find the range of marks he should score in the fifth paper.
a. $52 \le x < 77$
b. $25 \le x < 75$
c. $75 \le x < 80$
d. $73 \le x < 100$

Answer 1

Hint: Here in this question average concept and inequality concept will get used. They are as follows which are mentioned below.
Average = Sum of observations$/$ number of observation
$ \le $ Less than equal to
$ \ge $ Greater than equal to
$ > $ Greater than
$ < $ Less than

Complete step-by-step answer:
Marks obtained by Rishi in first paper $ = 95$
Marks obtained by Rishi in second paper $ = 72$
Marks obtained by Rishi in third paper $ = 73$
Marks obtained by Rishi in fourth paper $ = 83$
So, we have to find marks of Rishi in the fifth paper, for this we will apply the average concept.
Average = Sum of observations$/$ number of observation
Let marks obtained by Rishi in fifth paper be $x$
$75 \le $Average $ < 80$
(As given in the question average should be greater than equal to$75$ and less than$80$)
Average$ = \dfrac{{95 + 72 + 73 + 83 + x}}{5}$
$ \Rightarrow 75 \le \dfrac{{95 + 72 + 73 + 83 + x}}{5} < 80$ (Putting the value of average)
$ \Rightarrow 375 \le 95 + 72 + 73 + 83 + x < 400$ (Multiplying $5$ to the whole equation)
$ \Rightarrow 375 \le 323 + x < 400$
Now by subtracting $323$ from whole equation we will get,
$ \Rightarrow 52 \le x < 77$
Hence option A is the correct answer.

Note:
While solving inequality we must pay attention to the direction of the inequality.
In which direction an arrow should point plays a vital role. Closed side is for lesser quantity while the open side is for greater quantity.

Things which do not affect the inequality:-
1. Add or subtract a number from both the sides.
2. Multiply or divide both sides from a positive number.
3. Simplification can be done within sides.
Things which do not affect the inequality:-
1. Multiply or divide both sides from a negative number.
2. Swapping left or right hand sides.