Question

If a:b = (1/a + 1/c):(1/b + 1/c), and if a ≠ b, express c on terms of a and b

Answer 1

Given:

a:b = (1/a + 1/c):(1/b + 1/c),a ≠ b,

Required:

Express c on terms of a and b​.

Step-by-step explanation:

The given expression is:

`(a)/(b)=((1)/(a)+(1)/(c))/((1)/(b)+(1)/(c))`

Solve it by cross multiplication as:

`\begin{gathered} a((1)/(b)+(1)/(c))=b((1)/(a)+(1)/(c)) \\ (a)/(b)+(a)/(c)=(b)/(a)+(b)/(c) \\ (a)/(b)-(b)/(a)=(b)/(c)-(a)/(c) \end{gathered}`

Solve by taking L.C.M.

`\begin{gathered} (a^2-b^2)/(ab)=(b-a)/(c) \\ ((a-b)(a+b))/(ab)=(b-a)/(c) \\ ((a-b)(a+b))/(ab)=-((a-b))/(c) \\ (a+b)/(ab)=-(1)/(c) \\ c=-(ab)/(a+b) \end{gathered}`

Final Answer:

Thus c in terms of a and b is:

`c=-(ab)/(a+b)`