Question

How do you isolate c in the equation $ a = b\left( {\dfrac{1}{c} - \dfrac{1}{d}} \right) $ ?

Answer 1

Hint: An equation consisting of variables a, b, c and d is given to us. In order to isolate c, we have to solve this equation such that the variable is on one side of the equation and all the other variables on the other side. In order to do this we have to follow a series of steps.

Complete step-by-step answer:
Given to us is an equation
 $ a = b\left( {\dfrac{1}{c} - \dfrac{1}{d}} \right) $ and this equation consists of four variables a, b, c and d. In order to solve this equation to isolate c, we have to follow a series of steps.
Firstly, let us divide both the sides of the equation by the variable b. This gives
 $ \dfrac{a}{b} = \dfrac{b}{b}\left( {\dfrac{1}{c} - \dfrac{1}{d}} \right) $
On solving, we get
 $ \dfrac{a}{b} = \left( {\dfrac{1}{c} - \dfrac{1}{d}} \right) $
This way we have eliminated the variable b from one side. Similarly, let us bring the term $ \dfrac{1}{d} $ to the left side. The negative sign on this term becomes positive when it is shifted to the other side of the equation.
Hence we can write the equation as
 $ \dfrac{a}{b} + \dfrac{1}{d} = \dfrac{1}{c} $
Now, let us take L.C.M on the left side to combine the two terms. Now the equation becomes $ \dfrac{{ad + b}}{{bd}} = \dfrac{1}{c} $
Now we have the variable c on one side and rest of them on the other but c is in the reciprocal form. We can now eliminate this and write the equation as
 $ \dfrac{{bd}}{{ad + b}} = c $
Hence the final equation is $ c = \dfrac{{bd}}{{ad + b}} $
So, the correct answer is “ $ c = \dfrac{{bd}}{{ad + b}} $ ”.

Note: One might go wrong in changing the signs when terms are shifted across the equation. One can also solve this equation by adding the term $ \dfrac{1}{d} $ on both sides because this term will be eliminated on the right side of the equation.