\frac{\Delta A}{A_1} \times 100 = \frac{\frac{3\sqrt{3}}{16} s^2}{\frac{\sqrt{3}}{4} s^2} \times 100 = \frac{3}{4} \times 100 = 75\%

["Understanding the Area Increase: A Step-by-Step Breakdown", "When analyzing geometric shapes, understanding how changes in dimensions affect surface or area measurements is crucial in fields like architecture, engineering, and design. One elegant example involves calculating the percentage increase in area when a shape’s new area is compared to its original area using a precise ratio of squared side lengths.", "### The Formula: Relative Area Change", "Consider the equation:\n[\n\frac{\Delta A}{A_1} \ imes 100 = \frac{\frac{3\sqrt{3}}{16} s^2}{\frac{\sqrt{3}}{4} s^2} \ imes 100\n]\nHere, ( \Delta A ) represents the change in area, and ( A_1 ) is the original area. This expression tells us what fraction of the original area the change represents—expressed as a percentage.", "### Simplifying the Ratio of Areas", "Start with the ratio:\n[\n\frac{\frac{3\sqrt{3}}{16} s^2}{\frac{\sqrt{3}}{4} s^2}\n]\nSince ( s^2 ) appears in both numerator and denominator, it cancels out:\n[\n\frac{3\sqrt{3}/16}{\sqrt{3}/4}\n]", "Recall that dividing by ( \frac{\sqrt{3}}{4} ) is the same as multiplying by its reciprocal:\n[\n\left( \frac{3\sqrt{3}}{16} \right) \ imes \left( \frac{4}{\sqrt{3}} \right)\n]", "Now simplify:\n- The ( \sqrt{3} ) terms cancel:\n[\n\frac{3}{16} \ imes 4 = \frac{12}{16}\n]\n- Simplify ( \frac{12}{16} ) to ( \frac{3}{4} ).", "### Calculating the Percentage Increase", "Now plug this simplified ratio back into the original expression:\n[\n\frac{3}{4} \ imes 100 = 75%\n]", "Thus, a change in area represented by ( \frac{3\sqrt{3}}{16} s^2 ) compared to ( \frac{\sqrt{3}}{4} s^2 ) corresponds to a 75% increase in the original area ( A_1 ).", "This formula highlights how relative area change depends not just on raw numbers, but on squaring side dimensions—making it an essential tool for scaling and comparative analysis in geometry and design.", "### Why This Matters", "Whether designing structures, fabricating models, or calculating surface coverage, knowing the exact proportional change helps ensure accuracy and efficiency. Understanding how area ratios relate to side length ratios enhances problem-solving in both academic and professional settings.", "---", "Key Takeaways:\n- Area change is calculated relative to the original area.\n- Ratios of squared quantities (like side lengths) affect area.\n- Simplifying algebraic expressions accurately reveals meaningful percentages.\n- This method provides clear insight into proportional growth in two-dimensional spaces.", "---", "By mastering such calculations, you empower yourself to tackle geometry-driven challenges with confidence and precision."]