Question about Inequalities/Ratios at the Middle School Level

Question

This was given on a middle school math test:

$y<0$ and $4x>y$ such that $x$ and $y$ are two integers. What is the ratio $x:y$?

Could anyone help me out?

Thanks

Answer 1

Since $y<0$, we obtain $$\frac{x}{y}<\frac{1}{4}.$$

Answer 2

You have to use the rules of algebra and the equations that are given: $$ 4x > y \Rightarrow (\text{divide both sides by }4) \Rightarrow x > \frac{1}{4} y $$ $$ \Rightarrow (\text{divide both sides by } y) \Rightarrow \frac{x}{y} < \frac{1}{4} $$ In the last step, we have to keep in mind the rule $y<0$, which both enables dividing by $y$ ($y\neq 0$) and the fact that $y$ is negative then flips the inequality sign. So the result is $$ \frac{x}{y} < \frac{1}{4} $$ In other words, the ratio x:y can have any negative value, or a value smaller than $\frac{1}{4}$ when it comes to the positive values. Both positive and negative values are allowed, because the sign of $x$ is not specified.