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 that cannot. Write an explanation of how you made your decision. b. orm a conjecture about which ractions 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 idea of equivalent fractions and common multiples to verify your conjecture.
By dividing>>
2/6 = 0.333333
2/5 = 0.40
6/13 = ?
1/25 = ?
7/8 = ?
9/29 = ?
If you get stuck on the rest of this assignment, please repost and explain what stumps you.
I got part a. I do not understand the rest. b. form a conjecture about which fractions 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 idea of equivalent fractions and common multiples to verify your conjecture.
b. What do think the fractions that are expressed as terminating decimals have in common? Think about equivalent fractions and common multiples.
c. Do these fractions follow the same pattern as what you decided about the first set of fractions?
d. Why or why not?
Note: The most important part of this assignment is what you think. The right answer is not as important as how you analyze this problem.
a. To determine whether a fraction can be written as a terminating decimal or not, you need to look at the denominator of each fraction.
A fraction can be expressed as a terminating decimal if its denominator can be factored into primes of only 2s and 5s. For example, if the denominator of a fraction is 2^a * 5^b, where a and b can be non-negative integers, then the fraction can be written as a terminating decimal.
On the other hand, if the denominator of a fraction has any prime factors other than 2s and 5s, then the fraction cannot be expressed as a terminating decimal.
b. Based on the explanation above, we can form a conjecture: Fractions that can only be expressed with prime factors of 2s and 5s in the denominator can be written as terminating decimals.
c. Let's test the conjecture on the given fractions: 6/12, 7/15, 28/140, and 0/7.
- For 6/12, the denominator is 12, which can be factored into 2 * 2 * 3. Since there is a factor of 3, it cannot be written as a terminating decimal.
- For 7/15, the denominator is 15, which can be factored into 3 * 5. Since there is no factor other than 2s and 5s, it can be written as a terminating decimal.
- For 28/140, the denominator is 140, which can be factored into 2 * 2 * 5 * 7. Since there is a factor of 7, it cannot be written as a terminating decimal.
- For 0/7, the denominator is 7, which is a prime number and cannot be factored further. Since there are no factors other than 2s and 5s, it can be written as a terminating decimal.
d. To verify our conjecture using equivalent fractions and common multiples, let's consider the fractions 2/3 and 2/6.
- The fraction 2/3 can be written as 2/3 * 2/2 = 4/6. As we can see, the denominator of 6 includes a factor of 3, which means it cannot be written as a terminating decimal.
- On the other hand, the fraction 2/6 can be simplified to 1/3. Since the denominator of 3 only includes prime factors of 2s and 5s, it can be written as a terminating decimal.
This verification process confirms that fractions with denominators having factors other than 2s and 5s cannot be written as terminating decimals, supporting our initial conjecture.