If $x = y - \frac{y}{10}$, where $y$ is a positive integer which increases in value, then $x$

### Concept & Logic This problem tests the understanding of proportional linear equations. When a variable ($x$) is defined as a constant positive fraction of another variable ($y$), their growth directions will identical. ### Step-by-Step Solution * **Simplify the given equation:** The equation is given as $x = y - \frac{y}{10}$. We can factor out $y$: $$x = y \left(1 - \frac{1}{10}\right)$$ $$x = y \left(\frac{9}{10}\right)$$ $$x = 0.9y$$ * **Analyze the relationship:** The equation $x = 0.9y$ represents a direct, positive linear relationship between $x$ and $y$. The coefficient (0.9) is a positive constant. * **Determine the behavior:** The problem states that $y$ is a positive integer which *increases* in value. Since $x$ is simply $90\%$ of $y$, as $y$ grows larger, $x$ must proportionally grow larger as well. There are no negative signs or exponential constraints that would cause $x$ to reverse direction or decrease. ### Exam Strategy & Shortcut Plug in simple test values to immediately see the trend. Let $y = 10 \Rightarrow x = 10 - 1 = 9$ Let $y = 20 \Rightarrow x = 20 - 2 = 18$ Let $y = 30 \Rightarrow x = 30 - 3 = 27$ The values of $x$ (9, 18, 27) are strictly increasing. ### Common Pitfall Some students overthink the subtraction sign, assuming that subtracting a component ($\frac{y}{10}$) that is also growing will eventually cause the total value to shrink. They fail to realize that the base value ($y$) is growing 10 times faster than the subtracted component. ### Final Answer **Therefore, the correct answer is increases in value.**