Question: "What is the percentage change in the result when we add $50$ to a certain number $x$, instead of subtracting $50$ from the same number $x$?"
Doubt: In the previous problem that I solved, the same question was modified as multiplied by $10$ instead of being divided, and we genuinely arrived at the correct answer. Here, the answer key reads "cannot be determined". If so, where did I go wrong? Any form of assistance is most welcome.
The the percentage change (in decimal form) from starting with $x$ and ending with $y$ is
$$\frac {y-x}x$$
So, for example, the percentage change from $10$ to $15$ is
$$\frac {15-10}{10}=0.5=50\text{%}$$
Now, the question is what is the percentage change if we added $50$ to some number $x$, instead of subtracting $50$ from $x$. We are starting with $x-50$ and ending with $x+50$. So, our percentage change is
$$\frac {(x+50)-(x-50)}{x-50}=\frac {100}{x-50}$$
Notice that the answer depends on what $x$ is, so there is no single answer. For example, if $x=100$, then the percentage change would be $100/50=2=200\text{%}$. If $x=150$, then the percentage change would be $100/100=1=100\text{%}$.