Question

If $x,y$ and $z$ are positive numbers such that $5x=\frac{y}{4},\frac{y}{4}=\frac{z}{5}$ and $y+z=\frac{x}{k}$, what is the value of $k$?

• A. $\frac{1}{2}$
• B. $\frac{1}{25}$
• C. $\frac{1}{5}$
• D. $\frac{1}{45}$
• E. $\frac{5}{4}$

Answer 1

Answer: (D)

$\frac{1}{45}$

Equation $5x=\frac{y}{4}$ can be written as:

$y=20x$

Equation $\frac{y}{4}=\frac{z}{5}$ may be written as

$z=\left(\frac{5}{4}\right)y$

Substitute $y$ by $20x$ to get

$z=\left(\frac{5}{4}\right)20x=25x$

Substitute $y$ by $20x$ and $z$ by $25x$ in equation $y+z=\frac{x}{k}$, divide all terms by $x$ and solve for $k$:

$\Rightarrow 20x+25x=\frac{x}{k}$

$\Rightarrow 20+25=\frac{1}{k}$
$\Rightarrow k=\frac{1}{45}$