Are there definition of percent? In a school I was taught that percent is the same as 1/100. But I think that definition is not rigorous enough because that would imply for example that $5+4\%=5+4/100=5.04$ but this seems weird. So how one can define the percent rigorously?
 A: Typically, when we speak of something like $4\%$, the immediate question that should arise is 

"Four percent of what?"  



*

*"of" usually translates to multiplication.


We can call that "what" $x$, for now.
So if we speak of $4\%$ of $x$, this is equivalent to $0.04\times x = \frac {4}{100}\times x$...
We rarely see $5 + 4\%$. It would be more likely we encounter $5 + 4\%$ of $x$. 
For example, if we invest $\$5$, and we earn $4\%$ interest on our investment by year's end, we have $$5 + 4\%\times 5 = 5 + 0.04\times 5 = 5\times (1+0.04) = \$5.20$$ by years end, with a net earning of $\$.20$.
In your calculation, the only way to conclude that $5 + 4\% = 5.04$ is by making the assumption that $4\%$ means $4\%$ of $1$. It is very likely that what was intended was something along the lines of the example I give immediately above. Another example: 
If we know only that we invested $x$ dollars, and learn that we earned a $4\%$ profit, what that means, implicitly is that we earned a profit of $4\%\times x = 0.04\times x$, where $x$ is the invested quantity.
A: The definition is entirely correct. $1\% = \frac1{100}$.
This means that $5=500\%$, and $5 + 4\% = 500\% + 4\% = 504\% = 5.04$.

Usually, percents are used to denote the amount "of the whole" of something, so, for example, $4\%$ of one hundred dollars equals four dollars, and four percent of one dollar is four cents.
In your case, $5+4\%$ of something simply means taking $5$ whole parts of that something and adding another four percent of the same thing, so $(5+4\%)$ of one dollar is equal to $5$ dollars and $4$ cents.
