The problem provides two relationships between three numbers x, y, and z.

From the first statement: "When 70% of a number x is added to another number y, the sum becomes 165% of the value of y."

This can be expressed as an equation:

$0.7 x + y = 1.65 y$

Subtracting y from both sides yields:

$0.7 x = 0.65 y$

From this, we can express y in terms of x:

$y = \frac{0.7}{0.65} x = \frac{70}{65} x = \frac{14}{13} x$

Since $\frac{14}{13} > 1$ , it implies that $y > x$ .

From the second statement: "When 60% of the number x is added to another number Z, then the sum becomes 165% of the value of z."

This can be expressed as an equation:

$0.6 x + z = 1.65 z$

Subtracting z from both sides yields:

$0.6 x = 0.65 z$

From this, we can express z in terms of x:

$z = \frac{0.6}{0.65} x = \frac{60}{65} x = \frac{12}{13} x$

Since $\frac{12}{13} < 1$ , it implies that $z < x$ .

Now, comparing y and z relative to x:

We have $y = \frac{14}{13} x$ and $z = \frac{12}{13} x$ .

Since $\frac{14}{13} > \frac{12}{13}$ , it follows that $y > z$ .

Combining the inequalities $y > x$ , $z < x$ , and $y > z$ , the correct order is $z < x < y$ .

Options 2, 3, and 4 are incorrect as they do not align with the derived order of the numbers $z < x < y$ .