30. Higher Order Thinking Suppose $$ \frac{a}{b^{\prime}} \frac{c}{d^{\prime}} $$ and $$ \frac{f}{f} $$ represent three rational numbers. If $$ \frac{d}{b} $$ is less than $$ \frac{c}{d^{\prime}} $$ and $$ \frac{c}{d} $$ is less than $$ \frac{e}{f^{\prime}} $$ compare $$ \frac{a}{b} $$ and $$ \frac{e}{f} $$ Explain.

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To compare $$ \frac{a}{b} $$ and $$ \frac{e}{f} $$, we will analyze the given conditions step by step. ### Known Conditions: 1. $$ \frac{a}{b^{\prime}} $$ and $$ \frac{c}{d^{\prime}} $$ are rational numbers. 2. $$ \frac{f}{f} $$ is a rational number (which equals 1). 3. $$ \frac{d}{b} < \frac{c}{d^{\prime}} $$ 4. $$ \frac{c}{d} < \frac{e}{f^{\prime}} $$ ### Step-by-Step Analysis: 1. **Understanding the first inequality:** $$ \frac{d}{b} < \frac{c}{d^{\prime}} $$ This implies that $$ d \cdot d^{\prime} < c \cdot b $$. Rearranging gives us: $$ c \cdot b > d \cdot d^{\prime} $$ 2. **Understanding the second inequality:** $$ \frac{c}{d} < \frac{e}{f^{\prime}} $$ This implies that $$ c \cdot f^{\prime} < e \cdot d $$. Rearranging gives us: $$ e \cdot d > c \cdot f^{\prime} $$ ### Comparing $$ \frac{a}{b} $$ and $$ \frac{e}{f} $$: To compare $$ \frac{a}{b} $$ and $$ \frac{e}{f} $$, we need to relate $$ a $$ and $$ e $$ through the inequalities we have. From the first inequality, we have: $$ c \cdot b > d \cdot d^{\prime} $$ This suggests that $$ c $$ is relatively larger than $$ \frac{d \cdot d^{\prime}}{b} $$. From the second inequality, we have: $$ e \cdot d > c \cdot f^{\prime} $$ This suggests that $$ e $$ is relatively larger than $$ \frac{c \cdot f^{\prime}}{d} $$. ### Conclusion: To compare $$ \frac{a}{b} $$ and $$ \frac{e}{f} $$, we can infer that since $$ c $$ is larger than a certain ratio involving $$ d $$ and $$ d^{\prime} $$, and $$ e $$ is larger than a ratio involving $$ c $$ and $$ f^{\prime} $$, we can conclude that: - If $$ a $$ is related to $$ c $$ in a way that maintains the inequalities, we can say that $$ \frac{a}{b} $$ is likely to be less than $$ \frac{e}{f} $$ based on the transitive nature of inequalities. Thus, we can conclude that: $$ \frac{a}{b} < \frac{e}{f} $$ This conclusion is based on the relationships established by the inequalities provided. $$ \frac{a}{b} < \frac{e}{f} $$

Higher Order Thinking Suppose and
represent three rational numbers. If is less
than and is less than compare and
Explain.

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Higher Order Thinking Suppose and represent three rational numbers. If is less than and is less than compare and Explain.