Question

If $x\%$ of $y$ is equal to $1\%$ of $z$, $y\%$ of $z$ is equal to $1\%$ of $x$ and $z\%$ of $x$ is equal to $1\%$ of $y$, then the value of $xy+yz+zx$ is.

• A. $1$
• B. $2$
• C. $3$
• D. $4$

Answer 1

The correct option is C

$3$

Solution:

Step 1: Forming the equation

$x\%$ of $y$ is equal to $1\%$ of $z$:

$\Rightarrow y\times \frac{x}{100}=z\times \frac{1}{100}$

$\Rightarrow$$xy=z.....\left(1\right)$

$y\%$ of $z$ is equal to $1\%$ of $x$:

$$\begin{gathered} \Rightarrow z\times \frac{y}{100}=x\times \frac{1}{100} \\ \Rightarrow yz=x....\left(2\right) \end{gathered}$$

$z\%$ of $x$ is equal to $1\%$ of $y$:

$$\begin{gathered} \Rightarrow x\times \frac{z}{100}=y\times \frac{1}{100} \\ \Rightarrow zx=y........\left(3\right) \end{gathered}$$

Adding (1), (2), (3) we get,

$xy+yz+zx=z+x+y$

Step 2: Finding the values of $x,y,z$:

From (1):

$xy=z$

$\Rightarrow x=\frac{z}{y}......\left(4\right)$

From (3) and (4) we get,

$\Rightarrow x=\frac{z}{zx}$

$\Rightarrow x^2=1$

$\Rightarrow x=1$

Similarly, $y=z=1$

Step 3: Finding the value of required expression:

$$\begin{gathered} xy+yz+zx=z+x+y \\ \Rightarrow xy+yz+zx=1+1+1 \\ \Rightarrow xy+yz+zx=3 \end{gathered}$$

Final answer: Hence, option(C) is the correct one.