2150.41 – Adding x to Numerator and Denominator

Let a quantity $x$ be added to both the numerator and the denominator of a fraction $a/b$ , where neither $a$ nor $b$ is $0$ . Let the new value of the fraction be $c/d$ . Find the value of $x$ in terms of $a, b, c,$ and $d$ .

Solution

Cross multiply, then solve for x:

\begin{aligned} \frac{a + x}{b + x} &=& \frac{c}{d}, \\ d(a + x) &=& c(b + x), \\ a d + d x &=& b c + c x \\ d x - c x &=& b c - a d, \\ x(d – c) &=& bc – ad, \\ x &=& \frac{bc – ad}{d – c}. \end{aligned}

Afterthought: the denominator of that fraction might be zero. Do we need to specify something to keep that from happening? Poke around and see what you find.