Maximizing Algebraic Fraction Value

Let $S = a + b + c$. To maximize $\frac{S + d}{S - d}$, we need $S - d$ to be as small as possible (minimum 1) and $S + d$ as large as possible. Set $d = 25$. To make denominator $S - d = 1$, we need $S = 26 \implies a + b + c = 26$. Then numerator $= S + d = 26 + 25 = 51$. Denominator $= 26 - 25 = 1$. Maximum value $= 51/1 = 51$.