In a class of 20 students, a test was administered, scored only in whole numbers from 0 to 10. At least one student got every possible score, and the average was 7. Compare quantity A with quantity B (i.e. Quantity A is greater, less or equal to quantity B) given below:
- Quantity A: $4$
- Quantity B: The lowest score that two students could have received
Question: What does the wording "The lowest score that two students could have received" mean?
I solved this one as follows:
Total score of $20$ students $= 20*7= 140$
As at least one student got every possible score, so $11$ students get 55 in total. And rest of the students can get $140-55= 85$ in total.
Now, to minimize the score of $9th$ student, I have to maximize $8$ students score, which is $10$. so, their total = $8*10=80$, and the $9th$ student can get $5$.
So, I get lowest 2 scores of $2$ students are $0$ and $1$. In total $2$.
But, right answer I find in book is $5$. What am I missing?
Best Answer
We can have a lot of fun with the ambiguity of the English language, but when this ambiguity crops up in a math problem it is not so much fun.
I would interpret a "score that two students could have received" as an integer $n$ such that there exists a list of $20$ scores satisfying all the requirements of the problem, and in that list of scores the integer $n$ appears at least twice.
I would interpret "the lowest score that two students could have received" to mean the minimum value of $n$ over all values $n$ could take in the previous paragraph over all lists of scores that satisfy the requirements of the problem. This value is $5$.
One of the ambiguities in the problem statement is that a score received by two students could (in some circumstances) mean the sum of the two students' scores. In that case the lowest score received by two students is $0+1=1$. (It not only "could be" $1$, it must be $1$.)
The reason to choose the interpretation leading to the answer $5$ is that this is the more usual interpretation in problems like this. If the sum of scores were intended, I would expect some indication such as "combined score".