Doubt when something is some fraction less than some quantity.

Question

Why is $\frac{1}{4}$ less than X equals $\left(\text{X}-\frac{\text{X}}{4}\right)$ and not equals $\left(\text{X}-\frac{{1}}{4}\right)$ ?

For example: I was given the following problem to solve:

Q) One fourth less than 50% of 120 equals to ?

My solution:

50% of 120 = 60

then I did $60-\frac{1}{4}= \frac{239}{4}$

Answer 1

The phrase "one fourth less than" is a little bit ambiguous. It might mean "subtract $1/4$" or "find $75\%$ of".

You read it the first way.

The problem intended you to read it the second way. That is in fact the much more sensible way to read it, since the context tells you that this is a problem about percentages. Those involve multiplication and division, not addition and subtraction.

In this case there's another reason to guess that the second reading is correct. If these were real numbers that meant something (like dollars or pounds of potatoes) then the $1/4$ would be negligible in comparison to the other round numbers. Only "one fourth of the amount" would make sense.

That said, the question is very badly written. The author expects you to read his mind, since the words themselves are not clear

Answer 2

\begin{align*} \dfrac{x}{4} &\lt x-\dfrac{x}{4}\Large{?}\\ \\ x-\dfrac{x}{4} &=\dfrac{4x-x}{4}=\frac{3x}{4}\\ \\ \dfrac{x}{4} &\lt \frac{3x}{4}\\ \\ \therefore\space \dfrac{x}{4} &\lt x-\dfrac{x}{4} \end{align*}