Toolkit 96Ratio Manipulationab=cd\frac{a}{b}=\frac{c}{d}ba=dca+bb=c+dd\frac{a+b}{b}=\frac{c+d}{d}ba+b=dc+daa+b=cc+d\frac{a}{a+b}=\frac{c}{c+d}a+ba=c+dcab−a=cd−c\frac{a}{b-a}=\frac{c}{d-c}b−aa=d−cca−ba+b=c−dc+d\frac{a-b}{a+b}=\frac{c-d}{c+d}a+ba−b=c+dc−da1b1=a2b2=⋯=anbn=k⟹λ1a1+λ2a2+⋯+λnanλ1b1+λ2b2+⋯+λnbn=k\frac{a_1}{b_1}=\frac{a_2}{b_2}=\cdots=\frac{a_n}{b_n}=k\quad\Longrightarrow\quad\frac{\lambda_1a_1+\lambda_2a_2+\cdots+\lambda_na_n}{\lambda_1b_1+\lambda_2b_2+\cdots+\lambda_nb_n}=kb1a1=b2a2=⋯=bnan=k⟹λ1b1+λ2b2+⋯+λnbnλ1a1+λ2a2+⋯+λnan=k