Let \(x\) and \(y\) be real numbers such that

\(2 < \frac{x - y}{x + y} < 5.\)

If \(\frac{x}{y}\) is an integer, what is its value?

Havingfun Jul 30, 2019

#1**+2 **

See this link: **https://www.wolframalpha.com/input/?i=Give+integer+solutions+++++%5B++2+%3C+%5Cfrac%7Bx+-+y%7D%7Bx+%2B+y%7D+%3C+5+%5D**

Guest Jul 30, 2019

#2**+2 **

\(\text{Let }\dfrac{x}{y}=z.\\ 2 < \dfrac{x - y}{x + y} < 5\\ 2 < \dfrac{z - 1}{z + 1} < 5\\ 2z + 2 \stackrel{\boxed{1}}{<} z - 1 \stackrel{\boxed{2}}{<} 5z + 5\\ \boxed{1}: 2z + 2 < z - 1\\ z < -3\\ \boxed{2}: z - 1 < 5z + 5\\ -6 < 4z\\ z > -\dfrac{3}{2}\\ \text{Combine }\boxed{1} \text{ and }\boxed{2}: -\dfrac{3}{2} < z < -3\\ \text{The only integer solution is } z = -2\\ \therefore \dfrac{x}{y} = -2\)

.MaxWong Jul 30, 2019

#4**+4 **

Note that x-y must be larger than x+y if the ratio is to be between 2 and 5. This will be the case if y is negative.

So we have (x-y)/(x+y) > 2, or. x-y > 2x+2y or x < -3y or x/y > -3 where the < becomes > because we are dividing by a negative number.

We also have (x-y)/(x+y)<5, or x-y < 5x+5y or 4x > -6y or x > -(3/2)y or x/y < -3/2 where > becomes < because we are dividing by a negative number.

Hence -3 < x/y < -3/2, giving x/y = -2

Max got the right answer with incorrect reasoning! It happens!

Alan Jul 30, 2019