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# There are some students in the two examination halls A and B. To make the number of students equal in each hall, 10 students are sent from $ A $ to $ B $. But if 20 students are sent from $ \mathrm{B} $ to $ \mathrm{A} $, the number of students in $ \mathrm{A} $ becomes double the number of students in B. Find the number of students in the two halls.

Given:

There are some students in the two examination halls A and B. To make the number of students equal in each hall, 10 students are sent from \( A \) to \( B \). But if 20 students are sent from \( \mathrm{B} \) to \( \mathrm{A} \), the number of students in \( \mathrm{A} \) becomes double the number of students in B.

To do:

We have to find the number of students in the two halls.

Solution:Let the number of students in hall A and the number of students in hall B be $x$ and $y$ respectively.

If 10 candidates are sent from A to B, the number of students in each hall is same.

This implies,

$x-10=y+10$

$x=y+10+10$

$x=y+20$.....(i)

If 20 candidates are sent from B to A, the number of students in A is double the number of students in B.

$x + 20 = 2(y-20)$

$x + 20 = 2y-40$

$y+20+20=2y-40$ (From (i))

$2y-y=40+40$

$y=80$

Substituting $y=80$ in equation (i), we get,

$x=80+20$

$x=100$

Therefore, the number of students in hall A is 100 and the number of students in hall B is 80.

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