minimizing this expression: $\frac{(x+y)(c+d)-(z+w)(a+b)}{(a+b)(c+d)}$ Jody and Shelli each receive a box of buttons on Saturday and on Sunday.  Each button is either red or white, and the number of buttons in each of the four boxes is from $1$ to $100$, inclusive.
On both days, the percentage of red buttons in Jody's box is greater than the percentage of red buttons in Shelli's box.  If $J \%$ of Jody's total number of buttons are red and $S \%$ of Shelli's total number of buttons are red, what is the least possible value of $J-S$?  Express your answer to the nearest integer.
Ans. -96  (Source:  2021 MathCounts Target Round, calculators allowed)
I had Jody receiving on Saturday and Sunday respectively $a$ and $b$ buttons of which $x$ and $y$ are red.  Shelli on Saturday and Sunday receive respectively $c$ and $d$ buttons, of which $z$ and $w$ are red.
We have $\frac{x}{a} > \frac{z}{c}$ and  $\frac{y}{b} > \frac{w}{d}$ and
$J-S=100 \cdot (\frac{x+y}{a+b} - \frac{z+w}{c+d})$.  So,
$J-S=100 \cdot \frac{(x+y)(c+d)-(z+w)(a+b)}{(a+b)(c+d)}$
I'm not sure how to minimize this expression.

Heuristic: use a graphical reasoning on the percentages transformed into slopes of the diagonals in these parallelograms:

(in fact, take a grid $200 \times 200$).
