Suppose [tex]$a, b, c$[/tex], and [tex]$d$[/tex] are four numbers which satisfy the conditions [tex]$a>b>c>d>0$[/tex]. Fill in the blank boxes using either [tex]$>, <$[/tex], and [tex]$=$[/tex] so as to make each expression a true statement.
Answer (2)
Since c > 0"> a > c > 0 , \frac{c}{a}"> c a > a c .
Since d > 0"> a > d > 0 and 0"> b > 0 , \frac{b}{a}"> d b > a b .
Since b > c > d > 0"> a > b > c > d > 0 , \frac{c}{a}"> d b > a c .
The final answer is: , >, >}"> > , > , > .
Explanation
Understanding the Problem We are given that b > c > d > 0"> a > b > c > d > 0 . We need to compare the given pairs of fractions.
Comparing c a and a c First, let's compare c a and a c . Since c > 0"> a > c > 0 , we know that 1"> c a > 1 and a c < 1 . Therefore, \frac{c}{a}"> c a > a c .
Comparing d b and a b Next, let's compare d b and a b . Since d > 0"> a > d > 0 and 0"> b > 0 , we have \frac{1}{a}"> d 1 > a 1 . Multiplying both sides of the inequality by b (which is positive) preserves the inequality, so \frac{b}{a}"> d b > a b .
Comparing d b and a c Finally, let's compare d b and a c . Since b > c > d > 0"> a > b > c > d > 0 , we have c"> b > c and d"> a > d . Thus, cd"> ab > c d . Dividing both sides by a d (which is positive), we get \frac{cd}{ad}"> a d ab > a d c d , which simplifies to \frac{c}{a}"> d b > a c .
Final Answer Therefore, the true statements are:
\frac{c}{a}"> c a > a c
\frac{b}{a}"> d b > a b
\frac{c}{a}"> d b > a c
Examples
Understanding inequalities between fractions is crucial in various real-life scenarios. For instance, when comparing the efficiency of two machines, you might have machine A completing a task in 'c' hours and machine B completing the same task in 'a' hours, where a > c > 0. The ratio a/c helps determine which machine is faster. Similarly, in financial investments, comparing returns (b/d vs. b/a) helps investors decide where to allocate their resources for optimal gains. These comparisons ensure informed decision-making in everyday tasks.
The relationships among the fractions derived from the inequalities b > c > d > 0"> a > b > c > d > 0 yield that \frac{c}{a}"> c a > a c , \frac{b}{a}"> d b > a b , and \frac{c}{a}"> d b > a c . Therefore, all comparisons result in the symbol '>', showing that the first quantity in each pair is greater than the second. Hence, the answer is , >, >"> > , > , > .
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