Suppose $a, b, c$, and $d$ are four numbers which satisfy the conditions $a>b$. Fill in the blank boxes using either $>,<$, and $=$ so as to make each expression a true statement.
Answer (1)
The relationship between d c and 1 cannot be determined.
d d = 1 .
The relationship between d b and b b cannot be determined.
The only relationship that can be determined is d d = 1 , so the answer is = .
Explanation
Analyzing the Problem We are given that a , b , c , and d are four numbers with the condition b"> a > b . We need to fill in the blanks with , <"> > , < , or = to make the expressions true. Let's analyze each expression separately.
Analyzing the First Expression The first expression is d c □ 1 . We have no information about the relationship between c and d . Therefore, we cannot determine whether d c is greater than, less than, or equal to 1.
Analyzing the Second Expression The second expression is d d □ 1 . Assuming d is not zero, any non-zero number divided by itself is 1. Therefore, d d = 1 .
Analyzing the Third Expression The third expression is d b □ b b . Assuming b is not zero, b b = 1 . So we need to compare d b with 1. This is equivalent to comparing b and d . However, we have no information about the relationship between b and d . Therefore, we cannot determine whether d b is greater than, less than, or equal to 1.
Conclusion Based on the analysis, we can only determine the relationship for the second expression. The first and third expressions cannot be determined with the given information.
Examples
Imagine you're comparing fractions in cooking. If you know you have more apples than bananas ( b"> a > b ), but you don't know how much flour ( c ) you have compared to sugar ( d ), you can't say if the ratio of flour to sugar is more or less than 1. However, if you divide the amount of salt you have by itself, you always get 1, no matter the amount (as long as it's not zero). Similarly, if you don't know how the amount of bananas compares to the amount of sugar, you can't determine if the ratio of bananas to sugar is more or less than 1.
