2290.21 – Ratio-Difference-Sum-Ratio

If the ratio of 2x−y2 x - y to x+yx + y is 2/3, then what is the ratio of xx to yy? This sort of problem is known as "symbol pushing"--that is, simply doing algebraic manipulations for no reward other than just being able to do them.

  1. 15\frac{1}{5}
  2. 45\frac{4}{5}
  3. 11
  4. 65\frac{6}{5}
  5. 54\frac{5}{4}
Solution

Proceed as follows:

2x−yx+y=23,2(x+y)=3(2x−y),2x+2y=6x−3y,−4x=−5y,xy=54.\begin{aligned} \frac{2 x - y}{x + y} &=& \frac{2}{3}, \\ 2 (x + y) &=& 3 (2 x - y), \\ 2x + 2 y &=& 6 x - 3 y, \\ -4 x &=& -5 y, \\ \frac{x}{y} &=& \frac{5}{4}. \end{aligned}

The answer is e.