# Thread: find the number of student

1. ## find the number of student

question.
More than half of a certain class got A in their first term examination. More than half of the pupils with A in their first term examination also got A in their second term examination. Altogether 10 pupils got A in both examinations. What is the largest possible number of pupils in the class?

2. Originally Posted by lekge
question.
More than half of a certain class got A in their first term examination. More than half of the pupils with A in their first term examination also got A in their second term examination. Altogether 10 pupils got A in both examinations. What is the largest possible number of pupils in the class?
Let the total number of pupils be $\displaystyle N$; let $\displaystyle x$ be the percentage of the class who got a A in the first exam (from given we know $\displaystyle x>50\%$); also let $\displaystyle y$ be the percentage of the pupils with A in their first exam also got A in their second exam (from given we know $\displaystyle y>50\%$). Now I am going to break the number of pupils in this class into four groups: (1) The pupils who did not get A in either of the two exams; (2) the pupils who got A ONLY in the first exam; (3) the pupils who got A in BOTH exams; (4) the pupils who got A ONLY in the second exam.

Using the variables we assigned earlier and the given information, we can expressed the number of the four groups as:

Group (1): $\displaystyle (1-x)N$
Group (2): $\displaystyle xN - 10$
Group (3): $\displaystyle 10$
Group (4): $\displaystyle yxN-10$

We know that the sum of the numbers for these four groups should equal to the total number of pupils in the class, hence we have:

$\displaystyle (1-x)N+(xN-10)+10+(yxN-10)=N$

which simplifies to $\displaystyle yxN=10$. Now we need to do a little analysis, in order to make $\displaystyle N$ to be as large as possible, we need the product $\displaystyle yx$ to be as small as possible. We know $\displaystyle x>50\%$ and $\displaystyle y>50\%$, it is hard to just randomly pick the percentages $\displaystyle x$ and $\displaystyle y$ here, so lets assume both $\displaystyle x=y=50\%$, we got $\displaystyle N=40$. However this can not be true, since both percentages need to be strictly greater than $\displaystyle 50\%$, so the largest possible number of pupils in the class will be $\displaystyle 39$.

Roy