Question

Assume that neither xx nor yy is equal to 0, to permit division by xx and by yy.

What is the ratio x : y : z ?
(1) x+ y = 2z
(2) 2x+ 3y = z

• A. $5:2:1$
• B. $3:-2:1$
• C. $4:2:1$
• D. $5:-3:1$

Answer 1

Answer: (D)

$5:-3:1$

For this problem, you do not necessarily need to know the value of $x,y$ or $z$. You simply need to know the ratio $x:y:z$ (in other words, the value of $\frac{x}{y}$ AND the value of $\frac{y}{z}$). You need to manipulate the information given to see whether you can determine this ratio.

(1) INSUFFICIENT: There is no way to manipulate this equation to solve for a ratio. If you simply solve for $\frac{x}{y}$, for example, you get a variable expression on the other side of the equation:
$x+y=2x$
$x=2z-y$
$\frac{x}{y}=\frac{2z-y}{y}=\frac{2x}{y}=-1$

2) INSUFFICIENT: As in the previous example, there is no way to manipulate this equation to solve for a ratio. If you simply solve for $\frac{x}{y}$, for example, you get a variable expression on the other side of they equation:
$2x+3y=z$
$2x=z-3y$
$\frac{x}{y}=\frac{z-3y}{2y}=\frac{z}{2y}-\frac{3}{2}$

(1) AND (2) SUFFICIENT: Use substitution to combine the equations:
$x+y=2z$
$2x+3y=z$
Since $z=2x+3y$, you can substitute:
$x+y=2(2x+3y)$
$x+y=4x+6y$

Therefore, you can arrive at a value for the ratio $x:y$:
$-3x=5y$
$\frac{-3x}{y}=\frac{5y}{y}$ Divide by y.
$\frac{-3x}{-3y}=\frac{5}{-3}$ Divide by -3
$\frac{x}{y}=\frac{5}{-3}$

You can also substitute for $x$ to get a value for the ratio $y:z$:
$x+y=2z$
$x=2z-Y$
$2x+3y=z$
$2(2z-y)+3y=z$
$4z-2y+3y=z$
$y=-3z$
$\frac{y}{z}=-3$

This tells you that $x:y=-5/3$, and $y:z=-3/1$. Both ratios contain a 3 for the y variable and both also contain a negative sign, so assign the value -3 to $y$. This means that x must be 5 and z must be 1. Therefore, the ratio $x:y:z=5:-3:1$.

You can test the result by choosing $x=5,y=-3$, and $z=1$,or $x=10,y=-6$, and $z=2$. In either case, the original equations hold up.
The correct answer is (C).