Lesson 4: Lesson Four--Finding Least Common Denominator, Comparing and Ordering Fractions
Lesson Four-Finding the common denominator or two or more fractions, comparing and ordering fractions.
Section One: Finding the Lowest Common Denominator (LCD) of two or more fractions
First you need to know something about the "Least Common Multiple" (LCM) of two or more numbers (in the case of fractions, two or more unlike denominators). Since this is important for finding the lowest common denominator of two or more fractions--which is necessary to add or subtract fractions--it is necessary to review this concept.
A "least common multiple" is the LOWEST number that both denominators go into evenly ( a multiple of a number is what you get when you multiply that number by another number. If you don't know the multiples of 1 through 12, go back and study your times tables!)
Let's take the two fractions, 2/3 and 3/4. You cannot add or subtract these because they don't have common denominators. You can say however, "Okay, multiply the first fraction by 4 and the second fraction by 3 (in other words, multiply numerator and denominator of one fraction by the denominator of the other fraction) to get a common denominator of 12, so that:
2x4/3x4 = 8/12, and 3x3/4x3 = 9/12, and to add, 8/12 + 9/12 = 17/12 = 1 5/12 after reducing (and the improper fraction 17/12 becomes the mixed number 1 5/12 since 12 divided into 17 is 1 with a remainder of 5/12).
And this can certainly be done with those two fractions, but what about this example?
3/4 + 7/8. Are you going to do the same thing as before and get a common denominator of 32? Look what happens if you do this...
3x8/4x8 + 7x4/8x4 = 24/32 + 28/32 = 52/32 = 1 20/32, which needs to be reduced by 4 to get 1 5/8.
But if you knew that the least common multiple of 4 and 8 was 8, thus the LOWEST common denominator was 8, you wouldn't even have to touch the second fraction which already has a denominator of 8! You would get...
3x2/4x2 + 7/8 = 6/8 + 7/8 = 13/8 = 1/ 5/8, meaning you could skip the reducing step because you couldn't reduce 1 5/8 any more.
Thus finding the LOWEST common denominator using the least common multiple is a way to add or subtract fractions quicker, saving an extra step in reducing fractions.
Very often, in fact, one of the denominators IS the lowest common denominator!
Let's do one more example and I hope you see why it's best to use the LCM to find the LCD:
5/9 + 11/15. You could say the common denominator is 9 x 15 = 135, but wouldn't you rather work with a smaller number?
Multiples of 9 : 9,18,27,36,45,54 etc. Multiples of 15: 15, 30, 45, 60 etc.
From this it is easy to see the LCM is 45, and thus the LCD is 45.
5/9 = 25/45 (multiply numerator and denominator by 5) and 11/15 = 33/45 (multiply numerator and denominator by 3), so that 25/45 + 33/45 = 58/45 = 1 13/45 (which cannot be reduced since 13 is prime).
In fact, the ONLY time you should use the "multiply fractions by the opposite denominator" is when the two denominators are relatively prime to each other!
