A class had 48 students but only half of them had uniforms. During inspection, they form a 6×8 rectangle, and it was just enough to conceal in its interior everyone without a uniform. Later, some new students joined the class, but again only half of them had uniforms. During the next inspection, they used a different rectangular formation, again just enough to conceal in its interior everyone without a uniform. How many new students joined the class?
I understood this questions as, with adding more students, now two students will be with uniform and two without; in the rectangle. So, (6+2)x(8+2)=8x10=80 total students. So, 80-48=32students. So, Answer=32 new students. Is this correct?
Hint 1.
People 'inside' the rectangle form another rectangle, but with sides shorter by 2. We also know, that people without an uniform are half of all people, thus the bigger rectangle is two times bigger.
You have to solve then the equation: $$ab=2(a-2)(b-2)$$ for positive integer $a$ and $b$
Also notice, that pair $(8,10)$ (or $(10,8)$) doesn't satisfy this equation - forming a rectangle $8\times10$ makes, that on the circumference there are $32$ people, not $40(=80/2)$.
Hint 2.
To show the uniqueness of the solution you should take a look at the graph of the function $b(a)$