The fraction $\frac{m}{n}$, where $m$ and $n$ are positive integers is given. If we increase the numerator of the fraction by 5%, then by what percent should we increase the denominator in order to decrease the fraction $\frac{m}{n}$ by 10%?
(answer is $50/3$)
I did this, $\frac{105}{n}=90$ seems wrong?
1. Let $x=\frac{m}{n}$ and $x'=\frac{m'}{n'}$.
$$x'=0.9x\implies\frac{m'}{n'}=0.9\frac{m}{n}.$$
You know how $m$ changes:
$$\frac{1.05m}{n'}=0.9\frac{m}{n}\implies n'=\frac{1.05}{0.9}n=\frac{7}{6}n=(1+\frac{1}{6})n.$$
That much.
2. Say $x=\frac{4}{9}$. $x'=\frac{4.2}{9k}$. We want $$\frac{x'-x}{x}=-0.1$$ $$x'=0.9x$$ $$\frac{4.2}{9k}=0.9(4/9)\implies k=\frac{4.2}{9\cdot0.9(4/9)}=\frac{7}{6}.$$
3. Note that $n'=(1+\frac{50}{3}\color{blue}{\%})n=(1+\frac{1}{6})n$. (Credit to Daniel)
