When 70% of a number x is added to another number y, the sum becomes 165% of the value of y. When 60% of the number x is added to another number z, then the sum becomes 165% of the value of z. Which one of the following is correct?

The correct option is z < x < y.

To understand why this relationship holds true, let us break down the problem into mathematical equations step-by-step. Let the numbers $x$, $y$, and $z$ be positive real numbers.

Step 1: Analyze the relationship between x and y
The problem states that when 70% of $x$ is added to $y$, the sum becomes 165% of $y$. We can write this mathematically as:

$$y+0.70x=1.65y$$

Subtract $y$ from both sides of the equation:

$$0.70x=1.65y-y$$

$$0.70x=0.65y$$

Now, find the ratio of $x$ to $y$:

$$\frac{x}{y}=\frac{0.65}{0.70}=\frac{13}{14}$$

This can be rewritten to express $y$ in terms of $x$:

$$y=\frac{14}{13}x$$

Step 2: Analyze the relationship between x and z
The problem also states that when 60% of $x$ is added to $z$, the sum becomes 165% of $z$. Writing this as an equation:

$$z+0.60x=1.65z$$

Subtract $z$ from both sides of the equation:

$$0.60x=1.65z-z$$

$$0.60x=0.65z$$

Now, find the ratio of $x$ to $z$:

$$\frac{x}{z}=\frac{0.65}{0.60}=\frac{13}{12}$$

This can be rewritten to express $z$ in terms of $x$:

$$z=\frac{12}{13}x$$

Step 3: Compare x, y, and z
Now we can write all three variables in terms of $x$:
$z=\frac{12}{13}x$
$x=\frac{13}{13}x$
$y=\frac{14}{13}x$

Comparing the coefficients of $x$:

$$\frac{12}{13}<\frac{13}{13}<\frac{14}{13}$$

Therefore, we get the following inequality:
$z<x<y$