Which expression is equivalent to \( 30\left(\frac{1}{2} x-2\right)+40\left(\frac{3}{4} y-4\right) \) ? \( 45 x y-220 \) \( 15 x-30 y-220 \) \( 15 x+30 y-220 \) \( 15 x+30 y-64 \)

To simplify the expression \( 30\left(\frac{1}{2} x-2\right)+40\left(\frac{3}{4} y-4\right) \), we first distribute: 1. For the first term: \( 30 \times \frac{1}{2} x - 30 \times 2 = 15x - 60 \). 2. For the second term: \( 40 \times \frac{3}{4} y - 40 \times 4 = 30y - 160 \). Now, combine these results: \[ 15x - 60 + 30y - 160 = 15x + 30y - 220. \] Thus, the equivalent expression is \( 15x + 30y - 220 \). Thank you for your attention, and remember that sometimes math problems can feel like puzzles! It's all about breaking them down methodically. If you enjoyed this simplification, you might want to explore algebraic identities or even delve into the world of polynomial equations, where similar techniques can yield unimaginably beautiful results!