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Understanding the Context

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Key Insights

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\frac{36\pi x^3}{20\pi x^3} = \frac{36}{20} = \frac{9}{5}
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?
Solution: The ratio of inscribed to plain tablets is $5:3$, so the total number of parts is $5 + 3 = 8$. Since there are 48 tablets, each part represents $ \frac{48}{8} = 6 $ tablets. Thus, the number of inscribed tablets is $5 \times 6 = 30$. We are told that inscribed tablets must be displayed in groups of 7, so we seek the greatest multiple of 7 that is less than or equal to 30. The multiples of 7 below 30 are $7, 14, 21, 28$. The greatest is $28$. Therefore, the largest number of inscribed tablets that can be grouped in sevens is $28$.

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