Integers $m$ and $n$ are each greater than 100 . If $m+n=300$, then $m: n$ could be equal to
(A) $9: 1$
(B) $17: 8$
(C) $5: 3$
(D) $4: 1$
(E) $3: 2$
We try each of the five possibilities.

If $m: n=9: 1$, then we set $m=9 x$ and $n=x$, so $9 x+x=300$ or $10 x=300$ or $x=30$, so $m=9(30)=270$ and $n=30$.

If $m: n=17: 8$, then we set $m=17 x$ and $n=8 x$, so $17 x+8 x=300$ or $25 x=300$ or $x=12$, so $m=17(12)=204$ and $n=8(12)=96$.

If $m: n=5: 3$, then we set $m=5 x$ and $n=3 x$, so $5 x+3 x=300$ or $8 x=300$ or $x=\frac{75}{2}$, so $m=5\left(\frac{75}{2}\right)=\frac{375}{2}$ and $n=3\left(\frac{75}{2}\right)=\frac{225}{2}$.

If $m: n=4: 1$, then we set $m=4 x$ and $n=x$, so $4 x+x=300$ or $5 x=300$ or $x=60$, so $m=4(60)=240$ and $n=60$.

If $m: n=3: 2$, then we set $m=3 x$ and $n=2 x$, so $3 x+2 x=300$ or $5 x=300$ or $x=60$, so $m=3(60)=180$ and $n=2(60)=120$.

The only one of the possibilities for which $m$ and $n$ are integers, each greater than 100 , is $m: n=3: 2$.

ANSWER: (E)