Solution: The volume of a sphere is $ \frac{4}{3}\pi y^3 $, and the volume of a hemisphere is $ \frac{2}{3}\pi (3y)^3 = \frac{2}{3}\pi \times 27y^3 = 18\pi y^3 $. The ratio is $ \frac{\frac{4}{3}\pi y^3}{18\pi y^3} = \frac{4}{54} = \frac{2}{27} $. \boxed{\dfrac{2}{27}}

["Title: Understanding the Volume Ratio: Sphere vs Hemisphere — A Clear Explanation with Formula Breakdown", "The volume of geometric shapes like spheres and hemispheres is a fundamental topic in mathematics and physics. Whether you’re studying geometry, engineering, or calculus, understanding volume ratios helps simplify complex calculations and clarify relationships between different forms. In this article, we explore a classic example that demonstrates how to calculate and compare the volumes of a sphere and a hemisphere using clear formulas and explanations.", "---", "### The Formula for the Volume of a Sphere", "The volume ( V ) of a sphere with radius ( r ) is given by the well-known formula:\n[\nV = \frac{4}{3}\pi r^3\n]", "This formula originates from advanced integration techniques but relates directly to three-dimensional space filled by the sphere.", "---", "### The Volume of a Hemisphere", "A hemisphere is exactly half of a sphere — specifically, one curved half with a flat circular base. If the radius of the hemisphere is ( r = 3y ), its volume is half the volume of a full sphere, adjusted for the geometric proportions:\n[\nV_{\ ext{hemisphere}} = \frac{1}{2} \left( \frac{4}{3}\pi (3y)^3 \right)\n]", "Let’s simplify this step-by-step:\n[\n(3y)^3 = 27y^3\n]\n[\n\frac{4}{3}\pi \ imes 27y^3 = 36\pi y^3\n]\n[\nV_{\ ext{hemisphere}} = \frac{1}{2} \ imes 36\pi y^3 = 18\pi y^3\n]", "So, the volume of the hemisphere with radius ( 3y ) is ( 18\pi y^3 ).", "---", "### Calculating the Volume Ratio", "Now, we compute the ratio of the sphere’s volume to the hemisphere’s volume. Assume the sphere has radius ( y ), so:\n[\nV_{\ ext{sphere}} = \frac{4}{3}\pi y^3\n]\n[\nV_{\ ext{hemisphere}} = 18\pi y^3\n]", "The volume ratio is:\n[\n\frac{V_{\ ext{sphere}}}{V_{\ ext{hemisphere}}} = \frac{\frac{4}{3}\pi y^3}{18\pi y^3}\n]", "Cancel ( \pi y^3 ) from numerator and denominator:\n[\n= \frac{\frac{4}{3}}{18} = \frac{4}{3 \ imes 18} = \frac{4}{54}\n]", "Simplify the fraction:\n[\n\frac{4}{54} = \frac{2}{27}\n]", "---", "### Final Answer", "[\n\boxed{\dfrac{2}{27}}\n]", "---", "### Why This Ratio Matters", "Understanding that the volume of a sphere of radius ( y ) is ( \frac{4}{3}\pi y^3 ) and the volume of a hemisphere of radius ( 3y ) is ( 18\pi y^3 ) gives us a clear insight: a hemisphere with three times the radius has 18 times the volume of a sphere with radius ( y ), but only 2 out of every 27 cubic units — indicating a precise geometric scaling relationship.", "This ratio appears in areas such as volume displacement, fluid dynamics, architecture, and physics, making it a valuable concept for students and professionals alike.", "---", "Keywords: sphere volume, hemisphere volume, volume ratio, geometric formulas, 4/3 π y³, 18π y³, volume calculation, math tutorial, geometry ratios", "By mastering these formulas and ratios, anyone working with spatial volumes can confidently solve complex problems with clarity and precision."]