3 lcm and lcd, addition and subtraction of fractions

Definition of LCM
The least common multiple (LCM) of two or more numbers is
the least number that is the multiple of all of these numbers.
LCM and LCD

Definition of LCM
Example A. Find the LCM of 4 and 6.
LCM and LCD

Definition of LCM
The multiples of 4 are 4, 8, 12, 16, 20, 24, …
LCM and LCD

Definition of LCM
The multiples of 6 are 6, 12, 18, 24, 30,…
LCM and LCD

Definition of LCM
The smallest of the common multiples is 12,
LCM and LCD

Definition of LCM
The smallest of the common multiples is 12, so LCM{4, 6 } = 12.
LCM and LCD

Definition of LCM
LCM and LCD
We may improve the above listing-method for finding the LCM.

Definition of LCM
The smallest of the common multiples is 12, then LCM{4, 6 }=
12.
LCM and LCD
Given two or more numbers, we start with the largest number,

Definition of LCM
LCM and LCD
list its multiples in order until we find the one that can divide all
the other numbers.

Definition of LCM
LCM and LCD
Example B. Find the LCM of 8, 9, and 12.
the other numbers.

Definition of LCM
LCM and LCD
The largest number is 12 and the multiples of 12 are 12, 24,
36, 48, 60, 72, 84 …
the other numbers.

Definition of LCM
LCM and LCD
36, 48, 60, 72, 84 … The first number that is also a multiple
of 8 and 9 is 72.
the other numbers.

Definition of LCM
LCM and LCD
36, 48, 60, 72, 84 … The first number that is also a multiple
of 8 and 9 is 72. Hence LCM{8, 9, 12} = 72.
the other numbers.

But when the LCM is large, the listing method is cumbersome.
LCM and LCD

It's easier to find the LCM by constructing it instead.
LCM and LCD

To construct the LCM:
LCM and LCD

a. Factor each number completely
LCM and LCD

b. For each prime factor, take the highest power appearing in
the factorizations.
LCM and LCD

the factorizations. The LCM is their product.
LCM and LCD

Example C. Construct the LCM of {8, 15, 18}.
LCM and LCD

Factor each number completely,
LCM and LCD

8 = 23
LCM and LCD

8 = 23
15 = 3 * 5
LCM and LCD

8 = 23
15 = 3 * 5
18 = 2 * 32
LCM and LCD

8 = 23
15 = 3 * 5
18 = 2 * 32
From the factorization select the highest degree of each prime
factor:
LCM and LCD

8 = 23
15 = 3 * 5
18 = 2 * 32
factor
LCM and LCD

LCM and LCD
8 = 23
15 = 3 * 5
18 = 2 * 32
factor: 23, 32, 5,

LCM and LCD
8 = 23
15 = 3 * 5
18 = 2 * 32
factor: 23, 32, 5, then LCM{8, 15, 18} = 23*32*5 = 8*9*5 = 360.

LCM and LCD
The LCM of the denominators of a list of fractions is called the
least common denominator (LCD).
8 = 23
15 = 3 * 5
18 = 2 * 32
factor: 23, 32, 5, then LCM{8, 15, 18} = 23*32*5 = 8*9*5 = 360.

8 = 23
15 = 3 * 5
18 = 2 * 32
factor: 23, 32, 5, then LCM{8, 15, 18} = 23*32*5 = 8*9*5 = 360.
LCM and LCD
The LCM of the denominators of a list of fractions is called the
least common denominator (LCD). Following is an
application of the LCM.

LCM and LCD
Example D.
a. From one pizza, Joe wants 1/3, Mary wants 1/4 and Chuck
wants 1/6. How many equal slices should we cut the pizza into
and how many slices should each person take?

LCM and LCD
In picture:
Joe
Example D.

LCM and LCD
Mary
In picture:
Joe
Example D.

LCM and LCD
Mary Chuck
In picture:
Joe
Example D.

LCM and LCD
We find the LCM of 1/3, 1/4, 1/6 by searching.
Mary Chuck
In picture:
Joe
Example D.

LCM and LCD
We find the LCM of 1/3, 1/4, 1/6 by searching. The multiples of
6 are 6, 12, 18, 24, …
Mary Chuck
In picture:
Joe
Example D.

LCM and LCD
6 are 6, 12, 18, 24, … Since 12 is also the multiple of 3 and 4,
then 12 is the LCM.
Mary Chuck
In picture:
Joe
Example D.

LCM and LCD
then 12 is the LCM. Hence we should cut it into 12 slices
Mary Chuck
In picture:
Joe
Example D.

Example D.
LCM and LCD
Mary Chuck
In picture:
Joe
Mary ChuckJoe

Example D. From one pizza, Joe wants 1/3, Mary wants 1/4
and Chuck wants 1/6. How many equal slices should we cut
the pizza into and how many slices should each person take?
What is the fractional amount of the pizza they want in total?
LCM and LCD
Mary Chuck
In picture:
Joe
Mary ChuckJoe

LCM and LCD
Mary ChuckJoe
Joe: of the 12
1
3 * 12 = 4 slices
1
3
slices
Mary: of the 121
4
1
4
* 12 = 3 slices
Chuck: of the 121
6
1
6 * 12 = 2 slices
b. What is the fractional amount of the pizza they want in total?
The number of slices taken is 4+3+2=9, out of total of 12 slices.
Hence the fractions amount that’s taken is 9/12 or
3
4
1
6
.

Your Turn: From one pizza, Joe wants 3/8 of it, Mary wants 1/6
of it and Chuck wants 5/12 of it. How many equal slices should
we cut the pizza and how many slices should each person
take?
LCM and LCD

take?
LCM and LCD
In the above example, we found that is the same .
1
3
4
12

take?
LCM and LCD
1
3
4
12
The following theorem tells us how to convert the denominator
of a fraction to a fraction with a different denominator.

take?
LCM and LCD
1
3
4
12
Multiplier Theorem:

take?
LCM and LCD
1
3
4
12
Multiplier Theorem:
To convert the fraction into a fraction with denominator d,
the new numerator is * d.
a
b
a
b

take?
LCM and LCD
1
3
4
12
Multiplier Theorem:
a
b
a
b
Example E. Convert to a fraction with denominator 48.
9
16

take?
LCM and LCD
1
3
4
12
Multiplier Theorem:
a
b
a
b
The new denominator is 48,
9
16

take?
LCM and LCD
1
3
4
12
Multiplier Theorem:
a
b
a
b
The new denominator is 48, then the new numerator is
48*
9
16
9
16

take?
LCM and LCD
1
3
4
12
Multiplier Theorem:
a
b
a
b
48*
9
16
9
16
3

take?
LCM and LCD
1
3
4
12
Multiplier Theorem:
a
b
a
b
48* = 27.
9
16
9
16
3

take?
LCM and LCD
1
3
4
12
Multiplier Theorem:
a
b
a
b
48* = 27.
9
16
9
16
3 9
16
Hence =
27
48 .

Addition and Subtraction of Fractions

Suppose a pizza is cut into 4 equal slices and Joe takes
one slice or ¼ of the pizza, Mary takes two slices or 2/4 of
the pizza,
1
4
2
4

the pizza, altogether they take
+
1
4
2
4
=
3
4
of the entire pizza.
1
4
2
4

+
1
4
2
4
=
3
4
of the entire pizza. In picture:
+ =
1
4
2
4
3
4

+
Addition and Subtraction of Fractions With the Same
Denominator
1
4
2
4
=
3
4
+ =
1
4
2
4
3
4

+
Denominator
To add or subtract fractions of the same denominator, keep
the same denominator, add or subtract the numerators
1
4
2
4
=
3
4
+ =
1
4
2
4
3
4

+
Denominator
1
4
2
4
=
3
4
±
a
d
b
d = a ± b
d
+ =
1
4
2
4
3
4

+
Denominator
,then simplify the result.
1
4
2
4
=
3
4
±
a
d
b
d = a ± b
d
+ =
1
4
2
4
3
4

Example F.
a. 7
12
+
11
12

Example F.
a. 7
12
+ =
7 + 11
12
11
12 =

Example F.
a. 7
12
+ =
7 + 11
12
18
12
11
12 =

Example F.
a. 7
12
+ =
7 + 11
12
18
12
= 18/6
12/6
=
11
12 =

Example F.
a. 7
12
+ =
7 + 11
12
18
12
= 3
2
18/6
12/6
=
11
12 =

Example F.
a. 7
12
+ =
7 + 11
12
18
12
= 3
2
18/6
12/6
=
+ =b.
8
15
4
15
–
2
15
11
12 =

Example F.
a. 7
12
+ =
7 + 11
12
18
12
= 3
2
18/6
12/6
=
+ =
8 + 4 – 2
15
=b.
8
15
4
15
–
2
15
11
12 =

Example F.
a. 7
12
+ =
7 + 11
12
18
12
= 3
2
18/6
12/6
=
+ =
8 + 4 – 2
15
=b.
8
15
4
15
–
2
15
10
15
11
12 =

Example F.
a. 7
12
+ =
7 + 11
12
18
12
= 3
2
18/6
12/6
=
+ =
8 + 4 – 2
15
=
2
3
=b.
8
15
4
15
–
2
15
10
15
11
12 =

Example F.
a. 7
12
+ =
7 + 11
12
18
12
= 3
2
18/6
12/6
=
+ =
8 + 4 – 2
15
=
2
3
=b.
8
15
4
15
–
2
15
10
15
11
12 =
Fractions with different denominators can’t be added directly
since the “size” of the fractions don’t match.

Example F.
a. 7
12
+ =
7 + 11
12
18
12
= 3
2
18/6
12/6
=
+ =
8 + 4 – 2
15
=
2
3
=b.
8
15
4
15
–
2
15
10
15
11
12 =
since the “size” of the fractions don’t match. For example
1
2

Example F.
a. 7
12
+ =
7 + 11
12
18
12
= 3
2
18/6
12/6
=
+ =
8 + 4 – 2
15
=
2
3
=b.
8
15
4
15
–
2
15
10
15
11
12 =
+
1
2
1
3

Example F.
a. 7
12
+ =
7 + 11
12
18
12
= 3
2
18/6
12/6
=
+ =
8 + 4 – 2
15
=
2
3
=b.
8
15
4
15
–
2
15
10
15
11
12 =
+
1
2
1
3
=
?
?
Fractions with different denominators can’t be added
directly since the “size” of the fractions don’t match.
For example

Example F.
a. 7
12
+ =
7 + 11
12
18
12
= 3
2
18/6
12/6
=
+ =
8 + 4 – 2
15
=
2
3
=b.
8
15
4
15
–
2
15
10
15
11
12 =
To add them, first find the LCD of ½ and 1/3, which is 6.
+
1
2
1
3
=
?
?

Example F.
a. 7
12
+ =
7 + 11
12
18
12
= 3
2
18/6
12/6
=
+ =
8 + 4 – 2
15
=
2
3
=b.
8
15
4
15
–
2
15
10
15
11
12 =
We then cut each pizza into 6 slices.
+
1
2
1
3
=
?
?

Example F.
a. 7
12
+ =
7 + 11
12
18
12
= 3
2
18/6
12/6
=
+ =
8 + 4 – 2
15
=
2
3
=b.
8
15
4
15
–
2
15
10
15
11
12 =
We then cut each pizza into 6 slices. Both fractions may be
converted to have the denominator 6.
+
1
2
1
3
=
?
?

Example F.
a. 7
12
+ =
7 + 11
12
18
12
= 3
2
18/6
12/6
=
+ =
8 + 4 – 2
15
=
2
3
=b.
8
15
4
15
–
2
15
10
15
11
12 =
converted to have the denominator 6. Specifically,
1
2
= 3
6
1
3
= 2
6
+
1
2
1
3
=
?
?

Example F.
a. 7
12
+ =
7 + 11
12
18
12
= 3
2
18/6
12/6
=
+ =
8 + 4 – 2
15
=
2
3
=b.
8
15
4
15
–
2
15
10
15
11
12 =
+
1
2
= 3
6
1
3
= 2
6
3
6
1
3
=
?
?

Example F.
a. 7
12
+ =
7 + 11
12
18
12
= 3
2
18/6
12/6
=
+ =
8 + 4 – 2
15
=
2
3
=b.
8
15
4
15
–
2
15
10
15
11
12 =
+
1
2
= 3
6
1
3
= 2
6
3
6
2
6
=
?
?

Example F.
a. 7
12
+ =
7 + 11
12
18
12
= 3
2
18/6
12/6
=
+ =
8 + 4 – 2
15
=
2
3
=b.
8
15
4
15
–
2
15
10
15
11
12 =
+
3
6
2
6
=
1
2
= 3
6
1
3
= 2
6
Hence, 1
2
+ 1
3
= 3
6
+ 2
6
= 5
6
5
6

These steps may be condensed using the following method.
To add or subtract fractions with different denominators,
1. find their LCD,
2. convert all the different-denominator-fractions to the have
the LCD as the denominator,
3. add and subtract the adjusted fractions then simplify.
We list the steps of the above traditional method here.
(Multiplier Method) To add or subtract fractions
1. multiply the problem by the number 1 as “LCD/LCD”,
2. expand then reduce the answer.
1
2
+
1
3
Example G. Add or subtract using the multiplier–method.
The LCD is 6,
so multiply the problem by 6/6 (=1),
expand the multiplication,
organize the answer.
( )6/6
1
2 +
1
3
a.
=

1. find their LCD,

1. find their LCD,
1
2 +
1
3
a.

1. find their LCD,
1
2
+
1
3
The LCD is 6,
1
2 +
1
3
a.

1. find their LCD,
The LCD is 6,
1
2 +
1
3
a.

1. find their LCD,
1
2
+
1
3
The LCD is 6,
so multiply the problem by 6/6 (=1),( )6/6
1
2 +
1
3
a.
=

1. find their LCD,
1
2
+
1
3
The LCD is 6,
( )6 /6
1
2 +
1
3
a.

1. find their LCD,
1
2
+
1
3
The LCD is 6,
( )6 /6
1
2 +
1
3
a.
=
3 2
5/6

b.
5
12
1
6
+ –
16
9

( ) * 48 / 48
5
12
9
+ –
6
1
b.
5
12
1
6
+ –
16
9
16
The LCD is 48 so multiply the problem by 48/48,

( ) * 48 / 48
35
12
9
16
+ –
6
14 8
b.
5
12
1
6
+ –
16
9
expand the multiplication, then place the result over 48.

( ) * 48 / 48
= (4* 5 + 3*9 – 8) / 48
= (20 + 27 – 8) / 48
35
12
9
16
+ –
6
14 8
b.
5
12
1
6
+ –
16
9

( ) * 48 / 48
35
12
9
16
+ –
6
14 8
b.
5
12
1
6
+ –
16
9
48
39
=
16
13=
= (4* 5 + 3*9 – 8) / 48
= (20 + 27 – 8) / 48

( ) * 48 / 48
35
12
9
16
+ –
6
14 8
b.
5
12
1
6
+ –
16
9
48
39
This methods extend to the ± operations of the rational
(fractional) formulas. We will use this method extensively.
=
16
13=
= (4* 5 + 3*9 – 8) / 48
= (20 + 27 – 8) / 48

( ) * 48 / 48
35
12
9
16
+ –
6
14 8
b.
5
12
1
6
+ –
16
9
48
39
This methods extend to the ± operations of the rational
(fractional) formulas. We will use this method extensively.
When + or – two fractions, the cross–multiplying method
is an eyeballing trick that one may use to arrive at an answer.
Again, this answer needs to be simplified.
=
16
13=
= (4* 5 + 3*9 – 8) / 48
= (20 + 27 – 8) / 48

Cross Multiplication

A useful eyeballing-procedure with two fractions is the cross-
multiplication.
a
b
c
d

a
b
c
d
multiplication.
take their denominators and multiply them diagonally across.
To cross-multiply two given fractions as shown,

We obtain are two products.
a
b
c
d
multiplication.
a*d b*c

a
b
c
d
multiplication.
a*d b*c
! Make sure that the denominators cross up with the
numerators stay put.

a
b
c
d
multiplication.
a*d b*c
! Make sure that the denominators cross up with the
numerators stay put.
a
b
c
d
adbc
Do not cross downward as shown here.

Cross–Multiplication for Addition or Subtraction
We may cross multiply to add or subtract two fractions with
the product of the denominators as the common denominator.
a
b
c
d
±

a
b
c
d
± =
a*d ±b*c

a
b
c
d
± =
a*d ±b*c
bd

a
b
c
d
Afterwards we reduce if necessary for the simplified answer.
± =
a*d ±b*c
bd

a
b
c
d
Example D. Calculate
± =
a*d ±b*c
bd
3
5
5
6
–a.
5
12
5
9
–b.

a
b
c
d
± =
a*d ±b*c
bd
3
5
5
6
– =
5*5 – 6*3
6*5
a.
5
12
5
9
– =b.

a
b
c
d
± =
a*d ±b*c
bd
3
5
5
6
– =
5*5 – 6*3
6*5
7
30
=a.
5
12
5
9
– =b.

a
b
c
d
± =
a*d ±b*c
bd
3
5
5
6
– =
5*5 – 6*3
6*5
7
30
=a.
5
12
5
9
– =5*12 – 9*5
9*12
15
108
=b.

a
b
c
d
± =
a*d ±b*c
bd
3
5
5
6
– =
5*5 – 6*3
6*5
7
30
=a.
5
12
5
9
– =5*12 – 9*5
9*12
15
108
=b. 5
36=

LCM and LCD
Exercise A. Find the LCM.
1. a.{6, 8} b. {6, 9} c. {3, 4}
d. {4, 10}
2. a.{5, 6, 8} b. {4, 6, 9} c. {3, 4, 5}
d. {4, 6, 10}
3. a.{6, 8, 9} b. {6, 9, 10} c. {4, 9, 10}
d. {6, 8, 10}
4. a.{4, 8, 15} b. {8, 9, 12} c. {6, 9, 15}
5. a.{6, 8, 15} b. {8, 9, 15} c. {6, 9, 16}
6. a.{8, 12, 15} b. { 9, 12, 15} c. { 9, 12, 16}
7. a.{8, 12, 18} b. {8, 12, 20} c. { 12, 15, 16}
8. a.{8, 12, 15, 18} b. {8, 12, 16, 20}
9. a.{8, 15, 18, 20} b. {9, 16, 20, 24}

B. Convert the fractions to fractions with the given
denominators.
10. Convert to denominator 12.
2
3 ,
3
4 ,
5
6 ,
7
4
1
6 ,
3
4 ,
5
6 ,
3
8
7
12 ,
5
4 ,
8
9 ,
11
6
9
10 ,
7
12 ,
13
5 ,
11
15
LCM and LCD

3 lcm and lcd, addition and subtraction of fractions

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