find counterexample
the quotient of two proper fractions is a proper fraction
For a proper fraction, the absolute value of the numerator is less than the absolute value of the denominator.
So,
1/3 is a proper fraction
2/3 is a proper fraction
Is (2/3)/(1/3) a proper fraction?
To find a counterexample for the statement "the quotient of two proper fractions is a proper fraction," we need to find an example where this statement is false.
Let's consider the fractions 3/4 and 1/2. Both of these fractions are proper fractions because the numerator is less than the denominator. The quotient of these two fractions would be (3/4) / (1/2), which can be simplified as follows:
(3/4) / (1/2) = (3/4) * (2/1) = 6/4
The fraction 6/4 is not a proper fraction because the numerator is greater than or equal to the denominator. In fact, 6/4 can be simplified further to 3/2, which is an improper fraction.
So, the example of 3/4 ÷ 1/2 provides a counterexample to the statement "the quotient of two proper fractions is a proper fraction."
To find a counterexample, we need to find an example where the statement "the quotient of two proper fractions is a proper fraction" is false.
First, let's understand what a proper fraction is. A proper fraction is a fraction where the numerator (the top number) is smaller than the denominator (the bottom number). For example, 1/2, 3/4, and 5/8 are all proper fractions.
Now, considering the statement, we want to find two proper fractions whose quotient is not a proper fraction. Let's take the fractions 3/4 and 2/3 as an example.
To find the quotient of 3/4 ÷ 2/3, we can multiply the first fraction (3/4) by the reciprocal of the second fraction (2/3). The reciprocal of a fraction is obtained by swapping the numerator and the denominator.
(3/4) ÷ (2/3) = (3/4) * (3/2)
Now, let's multiply the two fractions:
(3/4) * (3/2) = (9/8)
The quotient of 3/4 and 2/3 is 9/8.
However, 9/8 is not a proper fraction because the numerator 9 is greater than the denominator 8.
Therefore, the fractions 3/4 and 2/3 provide a counterexample to the statement "the quotient of two proper fractions is a proper fraction."