2150.41 – Adding x to Numerator and Denominator

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

Solution

Cross multiply, then solve for x:

$\begin{aligned}\dfrac{a + x}{b + x} &= \dfrac{c}{d} \\[1em]d(a + x) &= c(b + x) \\[1em]a d + d x &= b c + c x \\[1em]d x - c x &= b c - a d \\[1em]x(d - c) &= bc - ad \\[1em]x &= \dfrac{bc - ad}{d - c}\end{aligned}$

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.