Consider the following pattern involving fractions: 15/20=3/4, 15-3/20-4=12/16=3/4
18/15=6/5, 18-6/15-5=12/10=6/5
1.) Make a conjecture about the pattern. Show that the conjecture is true for another example.
2.) Explain how inductive reasoning can be used to support the conjecture.
3.) Find a counter example to the conjecture. What does the counter example tell about the conjecture?
OR
If a counter example cannot be found, describe what it would need to look like. What does not finding a counter example tell about the conjecture?
I need help with these questions. Thanks in advance.
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30'
1.) Conjecture: When fractions have a common numerator and their denominators decrease by 1, subtracting the numerators and subtracting the denominators will yield equivalent fractions.
To show that the conjecture is true for another example:
Let's take the fractions 9/12 and 9/8.
Subtracting the numerators: 9 - 9 = 0
Subtracting the denominators: 12 - 8 = 4
The result is 0/4, which can be simplified to 0/1 or simply 0.
Now, let's check if the original fractions are equivalent:
9/12 = 3/4
9/8 = 9/8
Since 0/4 is equivalent to 3/4, we can conclude that the conjecture holds for this example.
2.) Inductive reasoning can be used to support the conjecture by observing patterns and using previous examples to make generalizations. In this case, we have identified a pattern where, when fractions have a common numerator and their denominators decrease by 1, subtracting the numerators and subtracting the denominators results in equivalent fractions. By applying this pattern repeatedly to different examples, we can build confidence in the conjecture's validity.
3.) A counterexample that would disprove the conjecture would involve fractions with a common numerator and denominators decreasing by 1, but subtracting the numerators and subtracting the denominators does not yield equivalent fractions. For example, if we consider the fractions 4/5 and 4/3, and apply the same pattern:
Subtracting the numerators: 4 - 4 = 0
Subtracting the denominators: 5 - 3 = 2
The resulting fraction is 0/2 which simplifies to 0/1 or simply 0. However, the original fractions are not equivalent:
4/5 ≠ 4/3
This counterexample shows that the conjecture is not always true.
The fact that we found a counterexample indicates that the conjecture is not universally true. It means that the pattern identified in the conjecture does not hold for all cases. It is crucial to consider counterexamples to ensure the validity of any conjecture or generalization.