Question

On a recent chemistry test, the average (arithmetic mean) score among 5 students was 83, where the lowest and highest possible score were 0 and 100 respectively. If the teacher decides to increase each student's score by 2 points, and if none of the students originally scored more than 98 which of the following is true?
I. After the scores are increased, the average score is 85.
II. When the scores are increased, the difference between the highest and lowest score increases.
III. After the increase, all 5 scores are greater than or equal to 25.
A. 1 only.
B. 11 only.
C. 1 and 11 only.
D. 1 and 111 only.
E. None of these.

Answer 1

Hint: In this question, we are given average marks of 5 students and after that, it is given that 2 more marks were given to all students. We have to find the true statement among the given three statements. So, we will analyze every statement one by one and then find our required answer. The formula for finding mean is given by:
\[\text{Mean}=\dfrac{\text{Sum of all terms}}{\text{Number of terms}}\]

Complete step-by-step solution:
Here, we have to find the true statement from the statements given as:
I. After the scores are increased, the average score is 85.
II. When the scores are increased, the difference between the highest and lowest score increases.
III. After the increase, all 5 scores are greater than or equal to 25.
Let us analyze all statements one by one:
I. The original mean given was 83 and number of terms are 5. As we know mean is given by:
\[\text{Mean}=\dfrac{\text{Sum of all terms}}{\text{Number of terms}}\]
So, before increasing we get:
\[\begin{align}
  & 83=\dfrac{\text{Sum of all terms}}{\text{5}} \\
 & \Rightarrow \text{Sum of all terms}=83\times 5=415 \\
\end{align}\]
Hence, sum of all terms before increasing was 415. Now, two marks are rewarded to every student which means $2\times 5=10\text{ marks}$ are added to total marks of all students. So, new total sum of all terms become equal to 415+10 = 425.
\[\text{Mean}=\dfrac{\text{425}}{\text{5}}=85\]
Hence, new mean is equal to 85.
So, the option “I” is correct.
II. Since, it is given that the original marks of any student are not more than 98 so, they could also get increased to 100. Hence, all five students get 2 more marks. Since the same number of marks are increased for all students, so the difference between the highest and lowest scores will not increase and it will remain the same.
So, option II is incorrect.
III. For solving III, we have to suppose the marks for students as we can see that the total marks for all five students are equal to 425. Even if four of the students get 100 out of 100 marks, the fifth student will get 25 marks $\left( \text{425}-\left( \text{1}00+\text{1}00+\text{1}00+\text{1}00 \right)=\text{25} \right)$. So, minimum marks obtained by students after increasing are greater than or equal to 25.
So, option III is correct.
Hence, option D is the correct answer.

Note: Students should note that, when the same number of marks are increased for all students, the mean also simply gets increased by the same number. Hence, if two marks are increased for all students, the mean also gets increased by 2 marks (83+2 = 85). For the III part, students should use their general knowledge only. They can check this by taking an example.