The average score of $10$ students in a test is $25.$ The lowest score is $20.$ Then the highest score is:
$$A.100,$$ $$B.70,$$ $$C.30,$$ $$D.75$$
The answer key suggests option $B$ as the answer. I don't understand why this differs from my solution:
Let the marks obtained by each student be $x_1,x_2,…,x_{10}$ and $S=\sum_{i=1}^{10} x_i$. Then, we have $\frac{S}{10}=\frac{\sum_{i=1}^{10} x_i}{10}=20\implies S=200.$ If say, only one student obtained $25$, i.e let $x_k=25$, then, $$S-x_k=200-25=175.$$ Considering $S'=\sum_{i\neq k}x_i=175$. Now, if $\exists x_p=100$, then $100+(S'-x_p)=175\implies (S'-75)=75$, thus we are getting a situation, where it's possible for a student to score hundred, then, the rest students, will have to score, $75$ marks in total. This is the highest option given. Hence, the highest possible score can be $100.$ According to the information given, this can be a possible case, when 1 student scores 20, 8 students have a total score of 75 and one student scores 100, i.e then, average of this will still be 20. So, option $A$ is correct.
Best Answer
Ignoring the typos, your solution is not making a lot of sense, and the overuse of symbols makes it less readable than it could have been.
Note that the question is incorrectly phrased: it should be asking for the highest possible score, not the highest score, because there is insufficient information to deduce the latter.
This happens when everyone else has obtained the lowest score $20,$ in which case those nine students have scored a total of $9\times20=180$ marks, in which case the top scorer has scored $10\times25-180=70$ marks.
On the other hand, to determine possible highest score, we observe that the top score is minimum when $10\times25-20=230$ marks are equally distributed among the nine remaining students.
Hence, the top score must fall on the interval $$\left[25\frac59,70\right].$$