Given:
a:b = (1/a + 1/c):(1/b + 1/c),a ≠ b,
Required:
Express c on terms of a and b.
Explanation:
The given expression is:
[tex]\frac{a}{b}=\frac{\frac{1}{a}+\frac{1}{c}}{\frac{1}{b}+\frac{1}{c}}[/tex]Solve it by cross multiplication as:
[tex]\begin{gathered} a(\frac{1}{b}+\frac{1}{c})=b(\frac{1}{a}+\frac{1}{c}) \\ \frac{a}{b}+\frac{a}{c}=\frac{b}{a}+\frac{b}{c} \\ \frac{a}{b}-\frac{b}{a}=\frac{b}{c}-\frac{a}{c} \end{gathered}[/tex]Solve by taking L.C.M.
[tex]\begin{gathered} \frac{a^2-b^2}{ab}=\frac{b-a}{c} \\ \frac{(a-b)(a+b)}{ab}=\frac{b-a}{c} \\ \frac{(a-b)(a+b)}{ab}=-\frac{(a-b)}{c} \\ \frac{a+b}{ab}=-\frac{1}{c} \\ c=-\frac{ab}{a+b} \end{gathered}[/tex]Final Answer:
Thus c in terms of a and b is:
[tex]c=-\frac{ab}{a+b}[/tex]