\(A=\frac{\frac{1}{1\cdot300}+\frac{1}{2\cdot301}+\frac{1}{3\cdot302}+...+\frac{1}{101\cdot400}}{\frac{1}{1\cdot102}+\frac{1}{2\cdot103}+\frac{1}{3\cdot104}+...+\frac{1}{299\cdot400}}\)
\(A=\frac{\frac{1}{299}\left(\frac{299}{1\cdot300}+\frac{299}{2\cdot301}+\frac{299}{3\cdot302}+...+\frac{299}{101\cdot400}\right)}{\frac{1}{101}\left(\frac{101}{1\cdot102}+\frac{101}{2\cdot103}+\frac{101}{3\cdot104}+...+\frac{299}{299\cdot400}\right)}\)
\(A=\frac{\frac{1}{299}\left(1-\frac{1}{300}+\frac{1}{2}-\frac{1}{301}+\frac{1}{3}-\frac{1}{302}+...+\frac{1}{101}-\frac{1}{400}\right)}{\frac{1}{101}\left(1-\frac{1}{102}+\frac{1}{2}-\frac{1}{103}+\frac{1}{3}-\frac{1}{104}+...+\frac{1}{299}-\frac{1}{400}\right)}\)
