Trivial exam question - If a fraction's value increases by X, how does the denominator decrease? I'm doing some preparation work for a certain graduate-school entrance exam and this is one of the questions (paraphrased/rewritten):

Consider: $x = \frac{K}{y}$
If the value of X increases by 50%, the value of Y increases by what fraction? $x$, $y$ and $K$ are positive values. $K$ is a constant.

The possible answers are:


*

*$\frac14$

*$\frac13$

*$\frac12$

*$\frac23$

*$\frac34$


My problem was that I couldn't comprehend the question before I ran out of time, when I told it to reveal the answer it was $\frac13$. Even now, I don't understand the question, and I'm having a hard time visualising it.
Can you re-word and explain the question and the way that $X$ and $Y$ change such that 33% is the answer?
And how could I answer this question without spending too much time thinking about it? 
 A: Let's say you have $X_1 = \dfrac{k}{Y_1}$, the value before it increases, and $X_2 = \dfrac{k}{Y_2}$ the value after it decreases.
We want to know by which fraction the value of $Y$ has increased, so we want to know $a$, where $Y_2 - Y_1 = aY_1$
We know the value of $X$ has increased by $50\%$ so $X_2 = \dfrac{3}{2}X_1$. So we have $\dfrac{k}{Y_1} = \dfrac{3}{2}*\dfrac{k}{Y_2}$. Hence $Y_1 = \dfrac{2}{3}Y_2$.
Consequently, $Y_2 - Y_1 = \dfrac{1}{3}Y_1$ so the answer is $\dfrac{1}{3}$.
A: $X = \frac{k}{Y}$. So, $X(1+.5) = \frac{k}{Y} (1+.5) = \frac{k}{Y \cdot \frac{1}{1+.5}} = \frac{k}{\frac{2}{3}Y}$, so $Y$ becomes $\frac{2}{3}Y$, which decreases $Y$ by $\frac{1}{3}$.
A: This question (not yours, the exam question) doesn't make any sense at all. 
They say that the value k / Y increases by 50%. For that to happen, either k, or Y, or both, can change. If only k changes then it increases by 50% = 1/2. If only Y changes, then it decreases by 1/3. Dividing by a bigger number makes the result smaller, not bigger! 
But if you take X = 3/4 for example, increasing it by 50% gives 9/8, so k would be multiplied by 3 and Y multiplied by 2. X = 4/3 increased by 50% gives X = 2/1 so both k and Y shrink. Since nothing is said about how k and Y change, the question doesn't make any sense at all. 
A: Increasing by $50$% means taking what you have and adding half over again - or multiplying by $1+\frac12=\frac32$.
For example, increasing $\frac23$ by $50$% gives $\frac 11=\frac22$ when we keep the numerator the same - and $2$ is a decrease of $\frac13$ from $3$ (since $\frac13\cdot3=1 = 3-2$).
The point is that when we increase $X$ by $\frac32$, we get $$\begin{align}\frac32 X &= \frac32\cdot\frac kY\\&=\frac k{\frac23Y}\end{align}$$and multiplying $Y$ by $\frac23=1-\frac13$ is the same as removing $\frac13$ of its value.

Let's say we wanted $Y$ to decrease by $20$% instead. Then we for some value $p$, we want $$pX=\frac k{\frac45 Y}$$since a decrease of $20$% means removing $\frac15$ of its value - or multiplying by $\frac45$. Then we see that we would need $p=\frac54 = 1+\frac14$ - or an increase of $25$%.
