# Lowest score that two students could have received

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?

• It means, choose any $2$ out of $20$ students (you have $\binom{20}{2}=190$ ways to do it), sum up their scores, and tell us what is the lowest value that you got. – barak manos Aug 21 '16 at 7:23
• BTW, if anything, I'd ask what does the wording "At least one student got every possible score" mean. How can a student take a single test, and get more than one score for that test??? – barak manos Aug 21 '16 at 7:25

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".

I think here is some misleading interpretation:

First of all, "The lowest score that two students could have received" this means Two among all the 20 score which is equal and two distinct students got that score.

The scores we can get are like this:

0,1,2,3,4,5,6,7,8,9,10 (11 distinct score),

10,10,10,10,10,10,10,10(8 of the remaining 9 student may got this to score 80 of 85 remaining point)

and 5 (the last student have got to make 85 points).

Now, look into the above list of scores. We see 2 students have got 5 among the lowest scores. Here is our answer.

Secondly, "At least one student got every possible score" this statement still seams misleading to me.

I think that "the lowest score that two students could have received" is $m$ means that there are at least TWO students with vote $m$, and if two students get the same vote than it is $\geq m$.

In your case there are $11$ students with votes $0,1,2,\dots,10$ and the remaining ones get votes $x_{12},\dots,x_{20}$ with $x_i\in \{0,1,\dots,10\}$.

Since the average is $7$ then $$140=7\cdot 20=\sum_{k=1}^{11}k+ \sum_{k=12}^{20}x_k$$ which implies that $$\sum_{k=12}^{20}x_k=140-55=85.$$ Let $m$ be minimum value of $x_{12},\dots,x_{20}$ then $$85=\sum_{k=12}^{20}x_k\leq 10\cdot 8+m= 80+m\quad\Rightarrow \quad m\geq 5.$$ $m=5$ can be obtained when $x_{12}=\cdots=x_{19}=10$ and $x_{20}=5$.