11) For which positive integers $a, b, c, d$ can we have $\frac{a}{b}+\frac{c}{d}=\frac{a+c}{b+d}$?
(A) When $\frac{a}{b}=\frac{c}{d}$
(B) When $a d^{2}=c b^{2}$
(C) When $b \cdot d=b+d$
(D) Never
(E) Always.
11) The answer is (D). Indeed $\frac{a}{b}+\frac{c}{d}=\frac{a d+b c}{b d}$. It is therefore necessary that $\frac{a d+b c}{b d}=\frac{a+c}{b+d}$, that is, $(a+c) b d=$ $(b+d)(a d+b c)$, from which $a d^{2}+c b^{2}=0$, which is never satisfied if $a, b, c, d$ are positive integers.