Thus, the ratio is $ \boxed{\dfrac{9}{5}} $.Question: A museum curator is cataloging a collection of 48 ancient tablets. If the ratio of inscribed tablets to plain tablets is $5:3$, and all inscribed tablets must be displayed in groups of 7, what is the greatest number of inscribed tablets that can be grouped without leaving any out?

["Thus, the ratio is $ \boxed{\dfrac{9}{5}} $: Displaying Inscribed Tablets in Perfect Groups at the Museum", "Curators and historians often face fascinating challenges when organizing ancient artifacts—balancing preservation, visibility, and thematic display. In the recent cataloging of a rare collection of 48 ancient tablets, the museum’s curator uncovered a precise 5:3 ratio between inscribed and plain tablets, offering a compelling mathematical puzzle alongside historical significance.", "### Understanding the Ratio and Tablet Distribution", "The problem states the ratio of inscribed tablets to plain tablets is $ \boxed{\dfrac{9}{5}} $—though in reality, this ratio simplifies directly from the 5:3 proportion across the total of 48 tablets. Let’s first determine how many inscribed tablets exist in the collection.", "Let the number of inscribed tablets be $ 5x $ and plain tablets $ 3x $. Then:", "[\n5x + 3x = 48 \Rightarrow 8x = 48 \Rightarrow x = 6\n]", "Thus:\n- Number of inscribed tablets = $ 5x = 5 \cdot 6 = 30 $\n- Number of plain tablets = $ 3x = 18 $", "With exactly 30 inscribed tablets, the curator can now organization with confidence: all 30 inscribed tablets must be grouped into rows of 7, as required for a ceremonial display highlighting precision and pattern.", "### Groups of Inscribed Tablets: Maximum Number Without Leftover", "Since each group must contain exactly 7 inscribed tablets, we calculate how many full groups fit:", "[\n\left\lfloor \frac{30}{7} \right\rfloor = 4 \quad \ ext{with a remainder of } 30 - (7 \cdot 4) = 2\n]", "Thus, the greatest number of inscribed tablets that can be grouped without leaving any out is 28 (4 full groups of 7), leaving 2 tablets ungrouped.", "This restriction challenges curators to balance mathematical elegance—like the 9:5 ratio derived from distribution—with practical display constraints.", "### Why This Matters", "Such ratio-based organization ensures thematic consistency, enhances visitor engagement through intentional structure, and preserves historical integrity. In this case, the $ \boxed{\dfrac{9}{5}} $ ratio inspired the original proportion but the real test was grouping 30 tablets efficiently into groups of 7—answering the curator’s goal: display every inscribed tablet meaningfully, completely and cleanly.", "---", "Bottom Line:\nOut of 48 ancient tablets with a 5:3 inscribed-to-plain ratio (30 inscribed), the greatest number that can be grouped in sets of 7 without leaving any is 28. This blend of ratio logic and practical grouping exemplifies how numbers bring ancient collections to life—harmonizing history with exhibition wisdom."]