3 lcm and lcd, addition and subtraction of fractions

$Page 1: 3 lcm and lcd, addition and subtraction of fractions$
LCM and LCD

Definition of LCMThe 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


Example A. Find the LCM of 4 and 6.

LCM and LCD



The multiples of 4 are 4, 8, 12, 16, 20, 24, …

LCM and LCD



The multiples of 4 are 4, 8, 12, 16, 20, 24, … The multiples of 6 are 6, 12, 18, 24, 30,…

LCM and LCD



The multiples of 4 are 4, 8, 12, 16, 20, 24, … The multiples of 6 are 6, 12, 18, 24, 30,… The smallest of the common multiples is 12,

LCM and LCD



The multiples of 4 are 4, 8, 12, 16, 20, 24, … The multiples of 6 are 6, 12, 18, 24, 30,… The smallest of the common multiples is 12, so LCM{4, 6 } = 12.

LCM and LCD




LCM and LCD

We may improve the above listing-method for finding the LCM.



The multiples of 4 are 4, 8, 12, 16, 20, 24, … The multiples of 6 are 6, 12, 18, 24, 30,… The smallest of the common multiples is 12, then LCM{4, 6 }= 12.

LCM and LCD

We may improve the above listing-method for finding the LCM. Given two or more numbers, we start with the largest number,




LCM and LCD

We may improve the above listing-method for finding the LCM. Given two or more numbers, we start with the largest number, list its multiples in order until we find the one that can divide all the other numbers.




LCM and LCD

Example B. Find the LCM of 8, 9, and 12.





LCM and LCD


The largest number is 12 and the multiples of 12 are 12, 24, 36, 48, 60, 72, 84 …





LCM and LCD


The largest number is 12 and the multiples of 12 are 12, 24, 36, 48, 60, 72, 84 … The first number that is also a multiple of 8 and 9 is 72.





LCM and LCD


The largest number is 12 and the multiples of 12 are 12, 24, 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.


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

LCM and LCD

But when the LCM is large, the listing method is cumbersome. It's easier to find the LCM by constructing it instead.

LCM and LCD

To construct the LCM:


LCM and LCD

To construct the LCM: a. Factor each number completely


LCM and LCD

To construct the LCM: a. Factor each number completely 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




Factor each number completely, 8 = 23


LCM and LCD





15 = 3 * 5


LCM and LCD





15 = 3 * 518 = 2 * 32


LCM and LCD





15 = 3 * 518 = 2 * 32 From the factorization select the highest degree of each prime factor:


LCM and LCD





15 = 3 * 518 = 2 * 32 From the factorization select the highest degree of each prime factor:


LCM and LCD





15 = 3 * 518 = 2 * 32 From the factorization select the highest degree of each prime factor


LCM and LCD




LCM and LCD



15 = 3 * 518 = 2 * 32 From the factorization select the highest degree of each prime factor: 23, 32, 5,




LCM and LCD



15 = 3 * 518 = 2 * 32 From the factorization select the highest degree of each prime factor: 23, 32, 5, then LCM{8, 15, 18} = 23

*32*5 = 8*9*5 = 360.




LCM and LCD




*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).




*32*5 = 8*9*5 = 360.






*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 anapplication of the LCM.

LCM and LCDExample 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. 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

Mary

In picture:

Joe


LCM and LCD

Mary Chuck

In picture:

Joe


LCM and LCD

We find the LCM of 1/3, 1/4, 1/6 by searching.

Mary Chuck

In picture:

Joe


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


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, … Since 12 is also the multiple of 3 and 4, then 12 is the LCM.

Mary Chuck

In picture:

Joe


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, … Since 12 is also the multiple of 3 and 4, then 12 is the LCM. Hence we should cut it into 12 slices

Mary Chuck

In picture:

Joe



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 1213 * 12 = 4 slices

13

slices

Mary: of the 1214

14 * 12 = 3 slices

Chuck: of the 1216

16 * 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 or34

16

.

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


LCM and LCD

In the above example, we found that is the same .

13

412


LCM and LCD


13

412

The following theorem tells us how to convert the denominator of a fraction to a fraction with a different denominator.


LCM and LCD


13

412


Multiplier Theorem:


LCM and LCD


13

412


Multiplier Theorem:

To convert the fraction into a fraction with denominator d,

the new numerator is * d.

ab

ab


LCM and LCD


13

412


Multiplier Theorem:



ab

ab

Example E. Convert to a fraction with denominator 48.916


LCM and LCD


13

412


Multiplier Theorem:



ab

ab

Example E. Convert to a fraction with denominator 48.

The new denominator is 48,

916


LCM and LCD


13

412


Multiplier Theorem:



ab

ab


The new denominator is 48, then the new numerator is

48*

916

916


LCM and LCD


13

412


Multiplier Theorem:



ab

ab



48*

916

916

3


LCM and LCD


13

412


Multiplier Theorem:



ab

ab



48* = 27.

916

916

3


LCM and LCD


13

412


Multiplier Theorem:



ab

ab



48* = 27.

916

916

3 916Hence =

2748 .

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,


14

24

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, altogether they take

+


14

24 = 3

4of the entire pizza.

14

24


+


14

24 = 3

4of the entire pizza. In picture:

+ =14

24

34


+


Addition and Subtraction of Fractions With the Same Denominator

14

24 = 3


+ =14

24

34


+



To add or subtract fractions of the same denominator, keep the same denominator, add or subtract the numerators

14

24 = 3


+ =14

24

34


+




14

24 = 3


±ad

bd = a ± b

d

+ =14

24

34


+




,then simplify the result.

14

24 = 3


±ad

bd = a ± b

d

+ =14

24

34

Example F.

a. 712 +


1112

Example F.

a. 712 + =7 + 11

12


1112 =

Example F.

a. 712 + =7 + 11

121812


1112 =

Example F.

a. 712 + =7 + 11

121812 = 18/6

12/6 =


1112 =

Example F.

a. 712 + =7 + 11

121812 = 3

218/612/6 =


1112 =

Example F.

a. 712 + =7 + 11

121812 = 3

218/612/6 =

+ =b.


815

415 – 2

15

1112 =

Example F.

a. 712 + =7 + 11

121812 = 3

218/612/6 =

+ = 8 + 4 – 2 15

=b.


815

415 – 2

15

1112 =

Example F.

a. 712 + =7 + 11

121812 = 3

218/612/6 =

+ = 8 + 4 – 2 15

=b.


815

415 – 2

151015

1112 =

Example F.

a. 712 + =7 + 11

121812 = 3

218/612/6 =

+ = 8 + 4 – 2 15

= 23=b.


815

415 – 2

151015

1112 =

Example F.

a. 712 + =7 + 11

121812 = 3

218/612/6 =

+ = 8 + 4 – 2 15

= 23=b.


815

415 – 2

151015

1112 =

Fractions with different denominators can’t be added directly since the “size” of the fractions don’t match.

Example F.

a. 712 + =7 + 11

121812 = 3

218/612/6 =

+ = 8 + 4 – 2 15

= 23=b.


815

415 – 2

151015

1112 =

Fractions with different denominators can’t be added directly since the “size” of the fractions don’t match. For example

12

Example F.

a. 712 + =7 + 11

121812 = 3

218/612/6 =

+ = 8 + 4 – 2 15

= 23=b.


815

415 – 2

151015

1112 =


+12

13

Example F.

a. 712 + =7 + 11

121812 = 3

218/612/6 =

+ = 8 + 4 – 2 15

= 23=b.


815

415 – 2

151015

1112 =


+12

13

=??


Example F.

a. 712 + =7 + 11

121812 = 3

218/612/6 =

+ = 8 + 4 – 2 15

= 23=b.


815

415 – 2

151015

1112 =


To add them, first find the LCD of ½ and 1/3, which is 6.

+12

13

=??

Example F.

a. 712 + =7 + 11

121812 = 3

218/612/6 =

+ = 8 + 4 – 2 15

= 23=b.


815

415 – 2

151015

1112 =


To add them, first find the LCD of ½ and 1/3, which is 6. We then cut each pizza into 6 slices.

+12

13

=??

Example F.

a. 712 + =7 + 11

121812 = 3

218/612/6 =

+ = 8 + 4 – 2 15

= 23=b.


815

415 – 2

151015

1112 =


To add them, first find the LCD of ½ and 1/3, which is 6. We then cut each pizza into 6 slices. Both fractions may be converted to have the denominator 6.

+12

13

=??

Example F.

a. 712 + =7 + 11

121812 = 3

218/612/6 =

+ = 8 + 4 – 2 15

= 23=b.


815

415 – 2

151015

1112 =


To add them, first find the LCD of ½ and 1/3, which is 6. We then cut each pizza into 6 slices. Both fractions may be converted to have the denominator 6. Specifically,

12 = 3

613 = 2

6

+12

13

=??

Example F.

a. 712 + =7 + 11

121812 = 3

218/612/6 =

+ = 8 + 4 – 2 15

= 23=b.


815

415 – 2

151015

1112 =



12 = 3

613 = 2

6

+12

13

=??

Example F.

a. 712 + =7 + 11

121812 = 3

218/612/6 =

+ = 8 + 4 – 2 15

= 23=b.


815

415 – 2

151015

1112 =



+

12 = 3

613 = 2

6

36

13

=??

Example F.

a. 712 + =7 + 11

121812 = 3

218/612/6 =

+ = 8 + 4 – 2 15

= 23=b.


815

415 – 2

151015

1112 =



+

12 = 3

613 = 2

6

36

13

=??

Example F.

a. 712 + =7 + 11

121812 = 3

218/612/6 =

+ = 8 + 4 – 2 15

= 23=b.


815

415 – 2

151015

1112 =



+

12 = 3

613 = 2

6

36

26

=??

Example F.

a. 712 + =7 + 11

121812 = 3

218/612/6 =

+ = 8 + 4 – 2 15

= 23=b.


815

415 – 2

151015

1112 =



+36

26

=

12 = 3

613 = 2

6 Hence, 12 + 1

3 = 36 + 2

6 = 56

56

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.

12 + 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

12 + 1

3a.

=

Addition and Subtraction of FractionsWe list the steps of the above traditional method here.












(Multiplier Method) To add or subtract fractions

Example G. Add or subtract using the multiplier–method. 12 + 1

3a.





(Multiplier Method) To add or subtract fractions 1. multiply the problem by the number 1 as “LCD/LCD”,

12 + 1

3


The LCD is 6,so multiply the problem by 6/6 (=1),

12 + 1

3a.







The LCD is 6,so multiply the problem by 6/6 (=1),

12 + 1

3a.






12 + 1

3


The LCD is 6,so multiply the problem by 6/6 (=1),( )6/6

12 + 1

3a.

=






12 + 1

3



( )6 /6

12 + 1

3a.






12 + 1

3



( )6 /6

12 + 1

3a.

=

3 2

5/6


b. 512

16

+ – 169

( ) * 48 / 485

129+ – 6

1


b. 512

16

+ – 169

16

The LCD is 48 so multiply the problem by 48/48,

( ) * 48 / 4835

129

16+ – 614 8


b. 512

16

+ – 169

The LCD is 48 so multiply the problem by 48/48, expand the multiplication, then place the result over 48.

( ) * 48 / 48

= (4* 5 + 3*9 – 8) / 48

= (20 + 27 – 8) / 48

3512

916+ – 6

14 8



b. 512

16

+ – 169

( ) * 48 / 4835

129

16+ – 614 8


b. 512

16

+ – 169

4839= 16

13=


= (4* 5 + 3*9 – 8) / 48

= (20 + 27 – 8) / 48

( ) * 48 / 4835

129

16+ – 614 8


b. 512

16

+ – 169

4839

This methods extend to the ± operations of the rational (fractional) formulas. We will use this method extensively.

= 1613=


= (4* 5 + 3*9 – 8) / 48

= (20 + 27 – 8) / 48

( ) * 48 / 4835

129

16+ – 614 8


b. 512

16

+ – 169

4839

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.

= 1613=


= (4* 5 + 3*9 – 8) / 48

= (20 + 27 – 8) / 48

Cross Multiplication


Cross Multiplication A useful eyeballing-procedure with two fractions is the cross-multiplication.


ab

cd

ab

cd



take their denominators and multiply them diagonally across.

To cross-multiply two given fractions as shown,

We obtain are two products.

ab

cd


a*d b*c





ab

cd


a*d b*c

! Make sure that the denominators cross up with the numerators stay put.





ab

cd


a*d b*c


ab

cdadbc




Do not cross downward as shown here.


ab

cd


a*d b*c


ab

cdadbc




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.

ab

cd

±




ab

cd

± = a*d ±b*c




ab

cd

± = a*d ±b*cbd




ab

cd

Afterwards we reduce if necessary for the simplified answer.

± = a*d ±b*cbd




ab

cd


Example D. Calculate

± = a*d ±b*cbd

35

56 – a.

512

59 – b.




ab

cd



± = a*d ±b*cbd

35

56 – = 5*5 – 6*3

6*5a.

512

59 – =b.




ab

cd



± = a*d ±b*cbd

35

56 – = 5*5 – 6*3

6*57

30=a.

512

59 – =b.




ab

cd



± = a*d ±b*cbd

35

56 – = 5*5 – 6*3

6*57

30=a.

512

59 – =5*12 – 9*5

9*1215108=b.




ab

cd



± = a*d ±b*cbd

35

56 – = 5*5 – 6*3

6*57

30=a.

512

59 – =5*12 – 9*5

9*1215108=b. 5

36=


LCM and LCDExercise 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.




23 ,

34 ,

56 ,

74

16 ,

34 ,

56 ,

38

712 ,

54 ,

89 ,

116

910 ,

712 ,

135 ,

1115

LCM and LCD

Education

3 lcm and lcd, addition and subtraction of fractions