Thus, the ratio of the volumes is \(\boxed{\frac{1}{125}}\).

["## Understanding Volume Ratios: Why the Ratio of Volumes is (\boxed{\frac{1}{125}})", "Volume is a fundamental concept in geometry and physics, playing a crucial role in fields ranging from engineering to chemistry. When comparing the volumes of two objects, especially those that are similar in shape, the ratio of their volumes often reveals critical scaling relationships—notably, when expressed as cubes, these ratios simplify to precise numerical values like (\boxed{\frac{1}{125}}).", "In this article, we explore why the ratio of certain volumes equals (\frac{1}{125}), how to determine such ratios systematically, and why understanding this principle is essential for solving complex problems in science and mathematics.", "---", "### Why Volume Ratios Matter", "The volume of a three-dimensional object scales with the cube of its linear dimensions. For example, if a cube’s side length is reduced to a fraction of the original, its volume decreases by the cube of that fraction. This cubic relationship enables rapid changes in volume, even with modest dimensional shifts—making volume ratios indispensable for scaling models, analyzing molecular structures, designing engineering prototypes, and more.", "One involving a volume ratio of (\frac{1}{125}) often arises when an object’s volume shrinks to ( \frac{1}{125} ) of its original size—equivalent to reducing its linear dimensions by a factor of ( \sqrt[3]{\frac{1}{125}} = \frac{1}{5} ). This straightforward cube root relationship underpins many practical applications across science and industry.", "---", "### How to Calculate Volume Ratios", "To determine the volume ratio of two similar objects, follow these key steps:", "1. Identify Similarity\n Confirm both shapes are similar, meaning corresponding linear dimensions (length, width, height) are proportional.", "2. Find the Linear Scaling Factor\n Determine the ratio of corresponding linear dimensions, ( \frac{L_2}{L_1} ), where ( L_1 ) and ( L_2 ) are the original and scaled dimensions, respectively.", "3. Apply the Cubic Relationship\n Since volume scales with the cube of linear dimensions:\n [\n \frac{V_2}{V_1} = \left(\frac{L_2}{L_1}\right)^3\n ]", "---", "### Example: Why (\frac{V_2}{V_1} = \frac{1}{125})", "Suppose we have two identical cubes initially with side length ( L ), giving an original volume:\n[\nV_1 = L^3\n]\nNow, scale each dimension by a factor of ( \frac{1}{5} ), yielding a smaller cube with side length ( \frac{L}{5} ). Its volume becomes:\n[\nV_2 = \left(\frac{L}{5}\right)^3 = \frac{L^3}{125}\n]", "Thus, the volume ratio is:\n[\n\frac{V_2}{V_1} = \frac{\frac{L^3}{125}}{L^3} = \frac{1}{125}\n]", "This illustrates how a length reduction by a factor of 5 results in a volume reduction by ( \frac{1}{125} ).", "---", "### Real-World Applications of the (\frac{1}{125}) Volume Ratio", "The ratio (\frac{1}{125}) appears in numerous practical situations:\n- Model Testing: Engineers reduce full-scale prototypes to small unmanned models; aerodynamic tests on scaled models rely on this ratio to simulate real-world forces.\n- Drug Dosage Scaling: When adjusting medication volumes based on body mass, volume reductions proportional to ( \frac{1}{125} ) (e.g., 1/5 size) inform pediatric dosing.\n- Architectural Models: Miniature architecture uses scaled-down units where volume ratios preserve visual and functional integrity.", "---", "### Key Takeaways", "- Volume ratios for similar objects are determined by cubing the linear scaling factor.\n- A volume ratio of (\frac{1}{125}) corresponds to a linear scale factor of ( \frac{1}{5} ).\n- Understanding these relationships enables accurate modeling, safe experimentation, and precise adjustments in science and design.", "Mastering volume ratios empowers learners and professionals alike to navigate spatial scaling confidently—boosting both accuracy and efficiency in real-world applications. Whether crafting models, analyzing data, or designing innovations, recognizing how scaling impacts volume remains a cornerstone skill in STEM fields.", "---", "Conclusion\nThe ratio of the volumes being (\boxed{\frac{1}{125}}) is a quintessential example of cube-based scaling, where a linear reduction by a factor of 5 produces a volume reduction to one-hundred twenty-fifths of the original. By grasping this fundamental principle, you unlock deeper insight into dimensional thinking—enabling smarter decisions across science, engineering, and beyond."]