1. **State the problem:** The value $Y$ varies inversely with $x$, and we know $y=12$ when $x=5$. We need to find $y$ when $x=4$. 2. **Formula for inverse variation:** When two variables vary inversely, their product is constant. This means: $$ y \times x = k $$ where $k$ is a constant. 3. **Find the constant $k$:** Using the given values $y=12$ and $x=5$: $$ 12 \times 5 = k $$ $$ k = 60 $$ 4. **Use $k$ to find $y$ when $x=4$:** $$ y \times 4 = 60 $$ 5. **Solve for $y$:** $$ y = \frac{60}{4} $$ 6. **Simplify the fraction:** $$ y = \frac{\cancel{60}}{\cancel{4}} = 15 $$ **Final answer:** When $x=4$, $y=15$.