1992 AMC 12 第 13 题

Multiply the numerator and denominator by $ab$

After simplifying, write $a$ as a multiple of $b$ and apply the bound on $a+b$

Multiplying the numerator and denominator by $ab$ gives $$\frac{a^2b+a}{b+ab^2} =\frac{a(ab+1)}{b(ab+1)} =\frac ab.$$ Thus $a=13b.$ The condition $a+b\le100$ becomes $14b\le100,$ so $b$ may be any integer from $1$ through $7.$ There are $7$ pairs.

Thus the correct answer is C.