LOWEST COMMON MULTIPLE
HOW TO COMPARE FRACTIONS
In this Lesson, we will answer the following:
WE WILL SEE that to add fractions, or to compare fractions that have different denominators, we must construct a common denominator. What denominator should we choose? We should choose the lowest common multiple of the original denominators. The student therefore must be clear as to what that means.
Here are the first few multiples of 6:
6, 12, 18, 24, 30.
And here are the first few multiples of 8:
8, 16, 24, 32, 40.
24 is a common multiple of 6 and 8. It is their lowest common multiple, which we abbreviate as the LCM.
The LCM is the first time that the multiples of 6 meet the multiples of 8.
Example 1. Find the LCM of 9 and 12.
Solution. Go through the multiples of 12 until you come to a multiple of 9.
12, 24, 36.
36 is the first multiple of 12 that is also a multiple of 9. 36 is their LCM.
Example 2. Find the LCM of 2 and 8.
8 itself is their LCM.
When the larger number is itself a multiple of the smaller number, then the larger number itself is their LCM.
Example 3. Find the LCM of 5 and 20.
Solution. 20 is their LCM.
Now the product of two numbers will always be a common multiple. The product of 6 and 4, for example, is 24, and 24 is a common multiple -- but it is not their lowest common multiple. Their lowest common multiple is 12.
Compare Lesson 22, Question 4.
Example 4. What is the LCM of 10 and 27?
Answer. 10 and 27 have no common divisors except 1. Therefore their LCM is 10 × 27 = 270.
(1 is a common divisor of every pair of numbers, but some pairs have 1 as their only common divisor. 10 and 27 are such a pair.)
Example 5. What is the LCM of 8 and 12?
Answer. 24. Their LCM is not 8 × 12, because 8 and 12 have common divisors besides 1; for example, 4.
(To find the LCM from prime factors, see Lesson 33.)
Answer. Make a common denominator. Choose the LCM of 2 and 8 -- which is 8 itself. Example 2, above.
Answer. Again, we will make the denominators the same, and then compare the numerators. As a common denominator, we will choose the LCM of 4 and 32, which is 32 itself.
terms by 8,
Answer. As a common denominator, choose the LCM of 6 and 9.
Answer. Choose 18.
multiplied both terms by 2.
We choose a common multiple of the denominators because we change denominators by multiplying them
Adding fractions (as we will see in Lesson 25) involves the same technique as comparing them, because the denominators -- the units -- must be the same. For example,
In the next Section, Question 4, we will see how to compare fractions by cross-multiplying.
At this point, please "turn" the page and do some Problems.
Continue on to the next Section.
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