Download Math 0305 Week #3 Notes Section 4.4 The Common Multiples of are

Math 0305 Week #3 Notes
Section 4.4
The Common Multiples of are multiples that a set of numbers have in
common. So, the common multiples of 3 and 4 are 12, 24, 36, 48, ... The
Least Common Multiple (abbreviated L.C.M.) is the smallest multiple the
numbers have in common. Thus, the L.C.M. of 3 and 4 is 12. The Least
Common Denominator (L.C.D.) is the L.C.M. of the denominators of a set
of fractions.
L.C.M by Prime Factorization
To find the L.C.M. of a number, first find the prime factorization of each
number. Next, write down the product of each factor that appears in the
prime factorizations and then choose the highest power of the factor in the
factorizations. This result is our L.C.M.
Example
Find the L.C.M. of 49, 7, and 21
Solution:
Write down the prime factorization of 49, 7, and 21:
49 = 7•7 = 72
7=7
21 = 3•7
Now, write the product of each factor that appears in the prime
factorizations and choose the highest power of the factor in the prime
factorizations.
So, that would be 3•72 = 3•49 = 147. So, the L.C.M. = 147.
L.C.M by Using the Multiples of the Largest Number
Start listing the multiples of the largest number until we find a number that
is divisible by all the other numbers.
Example Find the L.C.M. of 33, 110, and 44
Solution:
We will write down the multiples of 110:
110, 220, 330, 440, 550, 660
110, 330, 550 are not divisible by 44 and 220, 440 are not divisible
by 33, but 660 is divisible by both 33 and 44 so the L.C.M. = 660.
Building Fractions
When we reduce fractions to lowest terms, we need to divide out the same
number in numerator and denominator. Similarly, if we want to go
backwards, we need to multiply the numerator and denominator by the
same number. This is referred to as building fractions.
Section 4.5
Adding and Subtracting Like Fractions (same denominator)
if we add or subtract two fractions with the same denominator, we add or
subtract only the numerators. The denominator remains the same.
2
9
Example
+
5
9
2+5
9
=
=
7
9
Adding
Unlike Fractions (different denominators
€ €and€Subtracting
€
Unlike fractions are fractions that have different denominators. In order to
add or subtract fractions with different denominators, we first find the L.C.M
of the denominators, which is now called the L.C.D., the Least Common
Denominator. After finding the L.C.D., we build each fraction so it has a
denominator equal to the L.C.D. Then we proceed as above.
Example
9
14
+
1
3
–
5
21
Solution:
We begin by finding the L.C.D. of 14, 3, and 21:
14 = 2•7, 3 = 3, and 21 = 3•7, so the L.C.D. = 2•3•7 = 6•7 = 42.
€ Next,
€ we
€ build the fractions so that they have a denominator equal
to 42. Since 42 ÷ 14 = 3, times the top and bottom of the first
fraction by 3. Since 42 ÷ 3 = 14, multiply the top and bottom of the
second fraction by 14. Since 42 ÷ 21 = 2, times the top and bottom
of the third fraction by 2.
9 •3
14 • 3
+
1 • 14
3 • 14
–
5 •2
21 • 2
=
27
42
+
Now combine the numerators:
27
42
€
€
+
€
€
14
42
–
€
€
€
10
42
=
€
31
.
42
€
€
14
42
–
10
42