Can someone help me with this problem? A) Separate the fractions 2/6, 2/5, 6/13, 1/25, 7/8, and 9/29 into two categories: those that can be written as a terminating decimal and those of non-terminating. Write an explanation of how you made your decision. B) Form a conjecture about which fraction can be expressed as terminating decimals. C) Test your conjecture on the following fractions: 6/12, 7/15, 28/140, and 0/7. D) Use the ideas of equivalent fractions and common multiples to verify your conjecture. Here is how I answered A, B, and C: A) terminating decimals: 2/5, 1/25, 7/8; non-terminating decimals: 2/6, 6/13, 9/29 B) Factor the denominator of the fraction after it is reduced to the lowest terms into a product of powers of prime numbers. If the prime factors are all 2 and 5 then the decimal will terminate. C) All form terminating decimal category except for the last one because zero divided by anything is zero. D) This one I can not figure out. So, is A, B and C correct?
Let's go through each part of your question and check if your answers are correct:
A) To determine whether a fraction can be written as a terminating decimal or a non-terminating decimal, we need to examine the denominator. If the denominator of a fraction, when reduced to its lowest terms, only contains prime factors of 2 and/or 5, then the fraction can be written as a terminating decimal. If the denominator has any other prime factors, then the fraction will result in a non-terminating decimal.
Now let's check your categorization:
2/6: The denominator is 6, which prime factorizes to 2 * 3. Since it has prime factor 3, it cannot be written as a terminating decimal. So, it belongs in the non-terminating decimals category.
2/5: The denominator is 5, which only has prime factor 5. Therefore, it can be written as a terminating decimal. So, it belongs in the terminating decimals category.
6/13: The denominator is 13, which is a prime number. Therefore, it cannot be written as a terminating decimal. So, it belongs in the non-terminating decimals category.
1/25: The denominator is 25, which only has prime factor 5. So, it can be written as a terminating decimal. Thus, it belongs in the terminating decimals category.
7/8: The denominator is 8, which prime factorizes to 2^3. Since it only contains prime factor 2, it can be written as a terminating decimal. Thus, it belongs in the terminating decimals category.
9/29: The denominator is 29, a prime number. Consequently, it cannot be expressed as a terminating decimal. So, it belongs in the non-terminating decimals category.
Your categorization is correct. Well done!
B) Your conjecture states that if the prime factors of the denominator, in its lowest terms, are only 2 and 5, then the fraction can be written as a terminating decimal. This generalizes the criteria we used in part A.
C) Let's test your conjecture on the fractions you mentioned:
6/12: 6/12 reduces to 1/2, and the denominator, 2, only has prime factor 2, so it can be written as a terminating decimal. Your conjecture holds.
7/15: This fraction does not reduce, and its denominator, 15, has prime factors 3 and 5. Since it has prime factor 3, it cannot be written as a terminating decimal. Your conjecture holds.
28/140: When reduced, it becomes 1/5, and the denominator, 5, has prime factor 5 only, so it can be expressed as a terminating decimal. Your conjecture holds.
0/7: This fraction represents division by zero, which is undefined. Therefore, it doesn't belong to either category. It's not a terminating decimal because it's not a valid fraction.
D) To use the ideas of equivalent fractions and common multiples to verify your conjecture, you can select more fractions and check if they fit the criteria of terminating or non-terminating decimals. For example, choose fractions with denominators that have prime factors beyond 2 and 5, such as 3, 7, 11, etc. If those fractions result in non-terminating decimals, it serves as additional evidence supporting your conjecture.
Overall, your answers for parts A, B, and C are correct. For part D, you can further test your conjecture by selecting more fractions and applying the criteria mentioned above.