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Question: In the first four papers of \[100\] marks each, Rishi got \(95,72,73,83\) marks respectively. If he ...

In the first four papers of 100100 marks each, Rishi got 95,72,73,8395,72,73,83 marks respectively. If he wants an average of greater than or equal to 7575 and less than 8080 marks, find the range of marks he should score in the fifth paper.
a. 52≤x<7752 \le x < 77
b. 25≤x<7525 \le x < 75
c. 75≤x<8075 \le x < 80
d. 73≤x<10073 \le x < 100

Explanation

Solution

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 = 95
Marks obtained by Rishi in second paper =72 = 72
Marks obtained by Rishi in third paper =73 = 73
Marks obtained by Rishi in fourth paper =83 = 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 xx
75≤75 \le Average <80 < 80
(As given in the question average should be greater than equal to7575 and less than8080)
Average=95+72+73+83+x5 = \dfrac{{95 + 72 + 73 + 83 + x}}{5}
⇒75≤95+72+73+83+x5<80\Rightarrow 75 \le \dfrac{{95 + 72 + 73 + 83 + x}}{5} < 80 (Putting the value of average)
⇒375≤95+72+73+83+x<400\Rightarrow 375 \le 95 + 72 + 73 + 83 + x < 400 (Multiplying 55 to the whole equation)
⇒375≤323+x<400\Rightarrow 375 \le 323 + x < 400
Now by subtracting 323323 from whole equation we will get,
⇒52≤x<77\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.