I was reading the following puzzle:
Look here, George, said his cousin Reginald: "by what fractional part does four-fourths exceed three-fourths?"
"By one-fourth!" shouted everybody at once.
"Try another one," George suggested.
"With pleasure, when you have answered that one correctly," was Reginald’s reply.
According to the solution the correct answer is $\frac{1}3$ because $3$ of anything if increased by $\frac{1}3$ become $4$.
I don't really get it though. I can think of having a set of $N$ objects and somehow consider that each object is $\frac{1}N$ to the set so to add one more object you consider it as equivalent to adding $\frac{1}N$ but that seems a very weird way to state this.
Can someone please explain what this means?
The amount added is $\frac44-\frac34=\frac14$, so the amount by which $\frac44$ exceeds $\frac34$ is $\frac14$. However, the question does not ask for the amount of the excess: it asks what fraction of the original amount of $\frac34$ has been added to make the new amount of $\frac44$. Thus, what’s wanted is
$$\frac{\text{amount added}}{\text{original amount}}=\frac{\frac14}{\frac34}=\frac13\,.$$