Question

This question has two parts. First, answer Part A. Then, answer Part B.

Part A: A statement about rational numbers is shown.
The product of two negative rational numbers is greater than either factor. Is the statement always true, sometimes true, or never true? Explain your answer. Provide at least two examples to support your answer.

Part B: A different statement about rational numbers is shown. The product of two positive rational numbers is greater than either factor. Provide at least two examples to show that this statement is only sometimes true.

Answer 1

Part A: The statement is sometimes true. For example, if we take -1/2 and -1/3, their product is 1/6 which is greater than either factor. Another example would be taking -2/3 and -3/4, their product is 1/2 which is also greater than either factor. However, if we take -1/2 and -1/4, their product is 1/8 which is less than -1/2, making the statement false.

Part B: The statement is only sometimes true. For example, if we take 1/2 and 1/3, their product is 1/6 which is less than either factor. Another example would be taking 2/3 and 3/4, their product is 1/2 which is equal to the first factor. However, if we take 3/4 and 4/5, their product is 12/20 which is less than the second factor.