Post your response to the following: How are the concepts of the Greatest Common Factors (GCF), divisibility, and Lowest Common Multiple (LCM) used when computing fractions? Provide an example and demonstrate how the concepts are used.
Example of application to fractions:
5/6+ 7/15
= 25/30 + 14/30 ..... (1)
= 39/30
= 13/10 .... (2)
= 1 3/10
(1) LCM of denominators 6 and 15 is 30
(2) To simplify a fraction to simplest terms, both numberator and denominator are divided by the GCF.
In order to find the GCF or the LCM, divisibility is needed to factorize the numbers.
However, divisibility is not required to find LCM or GCF if the extended Euclidean algorithm is used.
http://en.wikipedia.org/wiki/Extended_Euclidean_algorithm
When computing fractions, the concepts of Greatest Common Factor (GCF), divisibility, and Lowest Common Multiple (LCM) play an important role. Let's break down each concept and see how they are used in fraction calculations.
1. Greatest Common Factor (GCF): The GCF of two or more numbers is the largest number that divides evenly into each of them. When working with fractions, finding the GCF helps simplify the fractions by dividing both the numerator and the denominator by their GCF.
For example, let's say we have the fraction 16/24. To simplify this fraction, we find the GCF of 16 and 24, which is 8. Then, we divide both the numerator (16) and the denominator (24) by the GCF (8):
16 ÷ 8 = 2 (numerator after division)
24 ÷ 8 = 3 (denominator after division)
So, the simplified form of the fraction 16/24 is 2/3.
2. Divisibility: Divisibility refers to whether one number divides evenly into another without leaving a remainder. In fraction calculations, determining divisibility is helpful when trying to simplify or compare fractions.
For example, let's say we have the fractions 5/15 and 9/18, and we want to compare them. We can simplify both fractions by finding their GCF first. The GCF of 5 and 15 is 5, and the GCF of 9 and 18 is 9.
Now, let's divide both the numerator and the denominator of each fraction by their respective GCF:
5 ÷ 5 = 1 (numerator of 5/15 after division)
15 ÷ 5 = 3 (denominator of 5/15 after division)
9 ÷ 9 = 1 (numerator of 9/18 after division)
18 ÷ 9 = 2 (denominator of 9/18 after division)
So, the simplified forms of the fractions 5/15 and 9/18 are 1/3 and 1/2, respectively. Now we can compare them easily and see that 1/3 is less than 1/2.
3. Lowest Common Multiple (LCM): The LCM of two or more numbers is the smallest multiple that is divisible by all the given numbers. In fraction calculations, finding the LCM is essential when adding or subtracting fractions with different denominators.
For example, let's say we want to add the fractions 1/4 and 3/8. To do this, we need to find a common denominator. The LCM of 4 and 8 is 8. So, we convert both fractions to have a denominator of 8:
1/4 = 2/8 (multiply numerator and denominator of 1/4 by 2)
3/8 = 3/8 (already has a denominator of 8)
Now, we can add the fractions:
2/8 + 3/8 = 5/8
So, the sum of 1/4 and 3/8 is 5/8.
In summary, the concepts of GCF, divisibility, and LCM are crucial when computing fractions. GCF helps simplify fractions, divisibility aids in comparing fractions, and LCM is needed when adding or subtracting fractions with different denominators.