# Incorrect in solving a/b−a/c=1 for c

Incorrect in solving $\frac{a}{b}-\frac{a}{c}=1$ for c
I have this:
$\frac{a}{b}-\frac{a}{c}=1$
Solve for c. Then,
$\frac{a}{b}-\frac{a}{c}=1\cdot bc$
$=ac-ab=bc$
$=a\left(c-b\right)=bc$
$c=\frac{bc}{a}+b$
This is my final result.
But the correct result is:
$c=\frac{ab}{a-b}$
What I development wrong in this equation ?
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Andrejkoxg
To isolate c we can proceed as follow
$\frac{a}{b}-\frac{a}{c}=1\phantom{\rule{thickmathspace}{0ex}}⟺\phantom{\rule{thickmathspace}{0ex}}\frac{a}{c}=\frac{a}{b}-1=\frac{a-b}{b}\phantom{\rule{thickmathspace}{0ex}}⟺\phantom{\rule{thickmathspace}{0ex}}\frac{c}{a}=\frac{b}{a-b}\phantom{\rule{thickmathspace}{0ex}}⟺\phantom{\rule{thickmathspace}{0ex}}c=\frac{ab}{a-b}$

Paul Reilly
You have expressed the solution to c in terms of c, where you need to have it in terms of a and b. This is what gives you a different(although correct) answer.
From $ac-ab=bc$, you should have taken the variables with $c$ to one side, like so:
$ac-bc=ab\phantom{\rule{thickmathspace}{0ex}}⟹\phantom{\rule{thickmathspace}{0ex}}c\left(a-b\right)=ab$
Can you continue from here?