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Lesson 23

LOWEST COMMON MULTIPLE

HOW TO COMPARE FRACTIONS


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.

LCM

The LCM is the first time that the multiples of 6 meet the multiples of 8.


 1.   How can we find the lowest common multiple of two numbers?
 
  Go through the multiples of the larger number
until you come to a multiple of the smaller number.
 

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.

LCM

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.


 2.   When is the LCM of two numbers simply their product?
 
  When they have no common divisors except 1.
 

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.)

Comparing fractions

In Lesson 20 we saw how to compare fractions that have equal numerators or equal denominators.  We will now see how to compare any two fractions.


 3.   How can we compare fractions when the numerators and denominators are different?
LCM
  Change them to equivalent fractions that will have equal denominators. As the common denominator, choose the LCM of the original denominators.
Then the larger the numerator, the larger the fraction.
 
  Example 6.   Which is larger,   1
2
 or  3
8
?

Answer.  Make a common denominator.  Choose the LCM of 2 and 8 -- which is 8 itself.  Example 2, above.

We will change  1
2
 to an equivalent fraction with denominator 8:
1
2
 =  4
8
.
The denominators are now the same, and we can compare  4
8
 with  3
8
.

We see:

4
8
 is larger than  3
8
.

That is,

1
2
 is larger than  3
8
.

  Example 7.   Which is larger,   3
4
 or  25
32
?

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.

  We will express  3
4
 with denominator 32.   On multiplying both

terms by 8,

3
4
 =  24
32
.
We are now comparing   3
4
 , or
24
32
   with    25
32
.
  25 is larger than 24; therefore  25
32
 is larger than  3
4
.

  Example 8.   Which is larger,   5
6
 or  7
9
?

Answer.  As a common denominator, choose the LCM of 6 and 9.

Answer.  Choose 18.

5
6
 =  15
18
,    7
9
 =  14
18
.
  To change  5
6
 , we multiplied both terms by 3.  To change  7
9
 , we

multiplied both terms by 2.

We choose a common multiple of the denominators, because we change denominators by multiplying themexclamation

Now, 15 is larger than 14. Therefore,  5
6
 is larger than  7
9
.

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,

5
6
 +  7
9
 = 15
18
 +  14
18
 
       =  29
18
.

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.

or

Continue on to the next Section.

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