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.
You have to have brackets for your conjecture to be true
15/20 = 3/4 , then (15-3)/(20-4) = 12/16 = 3/4
18/15 = 6/5, then (18-6)/(15-5) = 12/10 = 6/5
suppose a fraction a/b is not in lowest terms and can be reduced to c/d
then a/b = c/d = k
is (a-c)/(b-d) = k ??
from c/d = k, and a/b = k
then c = dk and a = bk
(a-c)/(b-d)
= (bk - dk)/(b - d)
= k(b-d)/(b-d)
= k
so I have proven you conjecture to be true for all cases, except b = d, or else we would have divided by zero
1.) Conjecture: The pattern suggests that when you have a fraction A/B, where A and B are integers, if you subtract the numerator and denominator by the fraction itself, and simplify the resulting fraction, it will be equal to A/B in a simplified form.
To show this conjecture is true for another example, let's take the fraction 25/30:
25 - 25/30 - 30 = 20/5 = 4/1 = 4
So, 25/30 is indeed equal to 4/5, following the pattern.
2.) Inductive reasoning can be used to support the conjecture by examining multiple examples and looking for a repeating pattern. In this case, we have observed the pattern for two examples (15/20 and 18/15). By extrapolating from these examples, we can infer that the pattern holds for other fractions as well.
3.) Counter example: Let's consider the fraction 2/3:
2 - 2/3 - 3 = 1/1 = 1
So, in this case, the pattern does not hold true. The fraction 2/3 is not equal to 1/2 as the pattern would suggest.
The counter example tells us that the conjecture is not universally true. It suggests that there may be certain conditions or constraints for the pattern to hold, and it does not apply to all fractions.
If a counter example cannot be found, it would demonstrate that the conjecture holds true for all fractions meeting the specified conditions. It would show that the pattern is universally applicable and follows a consistent rule.