2. 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
3. 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.
Example A. Find the LCM of 4 and 6.
LCM and LCD
4. 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.
Example A. Find the LCM of 4 and 6.
The multiples of 4 are 4, 8, 12, 16, 20, 24, …
LCM and LCD
5. 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.
Example A. Find the LCM of 4 and 6.
The multiples of 4 are 4, 8, 12, 16, 20, 24, …
The multiples of 6 are 6, 12, 18, 24, 30,…
LCM and LCD
6. 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.
Example A. Find the LCM of 4 and 6.
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
7. 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.
Example A. Find the LCM of 4 and 6.
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
8. 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.
Example A. Find the LCM of 4 and 6.
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
We may improve the above listing-method for finding the LCM.
9. 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.
Example A. Find the LCM of 4 and 6.
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
We may improve the above listing-method for finding the LCM.
Given two or more numbers, we start with the largest number,
10. 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.
Example A. Find the LCM of 4 and 6.
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
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.
11. 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.
Example A. Find the LCM of 4 and 6.
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
Example B. Find the LCM of 8, 9, and 12.
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.
12. 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.
Example A. Find the LCM of 4 and 6.
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
Example B. Find the LCM of 8, 9, and 12.
The largest number is 12 and the multiples of 12 are 12, 24,
36, 48, 60, 72, 84 …
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.
13. 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.
Example A. Find the LCM of 4 and 6.
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
Example B. Find the LCM of 8, 9, and 12.
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.
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.
14. 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.
Example A. Find the LCM of 4 and 6.
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
Example B. Find the LCM of 8, 9, and 12.
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.
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.
15. But when the LCM is large, the listing method is cumbersome.
LCM and LCD
16. 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
17. To construct the LCM:
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
18. To construct the LCM:
a. Factor each number completely
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
19. To construct the LCM:
a. Factor each number completely
b. For each prime factor, take the highest power appearing in
the factorizations.
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
20. To construct the LCM:
a. Factor each number completely
b. For each prime factor, take the highest power appearing in
the factorizations. The LCM is their product.
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
21. To construct the LCM:
a. Factor each number completely
b. For each prime factor, take the highest power appearing in
the factorizations. The LCM is their product.
Example C. Construct the LCM of {8, 15, 18}.
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
22. To construct the LCM:
a. Factor each number completely
b. For each prime factor, take the highest power appearing in
the factorizations. The LCM is their product.
Example C. Construct the LCM of {8, 15, 18}.
Factor each number completely,
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
23. To construct the LCM:
a. Factor each number completely
b. For each prime factor, take the highest power appearing in
the factorizations. The LCM is their product.
Example C. Construct the LCM of {8, 15, 18}.
Factor each number completely,
8 = 23
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
24. To construct the LCM:
a. Factor each number completely
b. For each prime factor, take the highest power appearing in
the factorizations. The LCM is their product.
Example C. Construct the LCM of {8, 15, 18}.
Factor each number completely,
8 = 23
15 = 3 * 5
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
25. To construct the LCM:
a. Factor each number completely
b. For each prime factor, take the highest power appearing in
the factorizations. The LCM is their product.
Example C. Construct the LCM of {8, 15, 18}.
Factor each number completely,
8 = 23
15 = 3 * 5
18 = 2 * 32
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
26. To construct the LCM:
a. Factor each number completely
b. For each prime factor, take the highest power appearing in
the factorizations. The LCM is their product.
Example C. Construct the LCM of {8, 15, 18}.
Factor each number completely,
8 = 23
15 = 3 * 5
18 = 2 * 32
From the factorization select the highest degree of each prime
factor:
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
27. To construct the LCM:
a. Factor each number completely
b. For each prime factor, take the highest power appearing in
the factorizations. The LCM is their product.
Example C. Construct the LCM of {8, 15, 18}.
Factor each number completely,
8 = 23
15 = 3 * 5
18 = 2 * 32
From the factorization select the highest degree of each prime
factor:
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
28. To construct the LCM:
a. Factor each number completely
b. For each prime factor, take the highest power appearing in
the factorizations. The LCM is their product.
Example C. Construct the LCM of {8, 15, 18}.
Factor each number completely,
8 = 23
15 = 3 * 5
18 = 2 * 32
From the factorization select the highest degree of each prime
factor
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
29. To construct the LCM:
a. Factor each number completely
b. For each prime factor, take the highest power appearing in
the factorizations. The LCM is their product.
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
Example C. Construct the LCM of {8, 15, 18}.
Factor each number completely,
8 = 23
15 = 3 * 5
18 = 2 * 32
From the factorization select the highest degree of each prime
factor: 23, 32, 5,
30. To construct the LCM:
a. Factor each number completely
b. For each prime factor, take the highest power appearing in
the factorizations. The LCM is their product.
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
Example C. Construct the LCM of {8, 15, 18}.
Factor each number completely,
8 = 23
15 = 3 * 5
18 = 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.
31. To construct the LCM:
a. Factor each number completely
b. For each prime factor, take the highest power appearing in
the factorizations. The LCM is their product.
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
Example C. Construct the LCM of {8, 15, 18}.
Factor each number completely,
8 = 23
15 = 3 * 5
18 = 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.
32. To construct the LCM:
a. Factor each number completely
b. For each prime factor, take the highest power appearing in
the factorizations. The LCM is their product.
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
The LCM of the denominators of a list of fractions is called the
least common denominator (LCD).
Example C. Construct the LCM of {8, 15, 18}.
Factor each number completely,
8 = 23
15 = 3 * 5
18 = 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.
33. To construct the LCM:
a. Factor each number completely
b. For each prime factor, take the highest power appearing in
the factorizations. The LCM is their product.
Example C. Construct the LCM of {8, 15, 18}.
Factor each number completely,
8 = 23
15 = 3 * 5
18 = 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.
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
The LCM of the denominators of a list of fractions is called the
least common denominator (LCD). Following is an
application of the LCM.
34. Example D. From one pizza, Joe wants 1/3, Mary wants 1/4
and Chuck wants 1/6.
LCM and LCD
35. Example D. From one pizza, Joe wants 1/3, Mary wants 1/4
and Chuck wants 1/6.
LCM and LCD
Mary Chuck
In picture:
Joe
36. 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?
LCM and LCD
Mary Chuck
In picture:
Joe
37. 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
38. 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
We find the LCM of 1/3, 1/4, 1/6 by searching.
Mary Chuck
In picture:
Joe
39. 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
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
40. 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
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
41. 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
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
42. 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?
Joe gets 12*
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 and
Mary Chuck
In picture:
Joe
1
3
43. 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?
Joe gets 12* = 4 slices
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 and
Mary Chuck
In picture:
Joe
1
3
44. 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?
Joe gets 12* = 4 slices 1
4
Mary gets 12*
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 and
Mary Chuck
In picture:
Joe
1
3
45. 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?
Joe gets 12* = 4 slices 1
4
Mary gets 12* = 3 slices
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 and
Mary Chuck
In picture:
Joe
1
3
46. 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?
Joe gets 12* = 4 slices 1
4
Mary gets 12* = 3 slices
1
6
Chuck gets 12*
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 and
Mary Chuck
In picture:
Joe
1
3
47. 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?
Joe gets 12* = 4 slices 1
4
Mary gets 12* = 3 slices
1
6
Chuck gets 12* = 2 slices
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 and
Mary Chuck
In picture:
Joe
1
3
48. 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?
Joe gets 12* = 4 slices 1
4
Mary gets 12* = 3 slices
1
6
Chuck gets 12* = 2 slices
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 and
Mary Chuck
In picture:
Joe
1
3
In total, that is 4 + 2 + 3 = 9 slices,
49. 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?
Joe gets 12* = 4 slices 1
4
Mary gets 12* = 3 slices
1
6
Chuck gets 12* = 2 slices
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 and
Mary Chuck
In picture:
Joe
1
3
In total, that is 4 + 2 + 3 = 9 slices, or of the pizza.
9
12
50. 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?
Joe gets 12* = 4 slices 1
4
Mary gets 12* = 3 slices
1
6
Chuck gets 12* = 2 slices
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 and
Mary Chuck
In picture:
Joe
1
3
In total, that is 4 + 2 + 3 = 9 slices, or of the pizza.
9
12 =
3
4
51. 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
52. 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
In the above example, we found that is the same .
1
3
4
12
53. 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
In the above example, we found that is the same .
1
3
4
12
The following theorem tells us how to convert the denominator
of a fraction to a fraction with a different denominator.
54. 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
In the above example, we found that is the same .
1
3
4
12
The following theorem tells us how to convert the denominator
of a fraction to a fraction with a different denominator.
Multiplier Theorem:
55. 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
In the above example, we found that is the same .
1
3
4
12
The following theorem tells us how to convert the denominator
of a fraction to a fraction with a different denominator.
Multiplier Theorem:
To convert the fraction into a fraction with denominator d,
the new numerator is * d.
a
b
a
b
56. 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
In the above example, we found that is the same .
1
3
4
12
The following theorem tells us how to convert the denominator
of a fraction to a fraction with a different denominator.
Multiplier Theorem:
To convert the fraction into a fraction with denominator d,
the new numerator is * d.
a
b
a
b
Example D: Convert to a fraction with denominator 48.
9
16
57. 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
In the above example, we found that is the same .
1
3
4
12
The following theorem tells us how to convert the denominator
of a fraction to a fraction with a different denominator.
Multiplier Theorem:
To convert the fraction into a fraction with denominator d,
the new numerator is * d.
a
b
a
b
Example D: Convert to a fraction with denominator 48.
The new denominator is 48,
9
16
58. 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
In the above example, we found that is the same .
1
3
4
12
The following theorem tells us how to convert the denominator
of a fraction to a fraction with a different denominator.
Multiplier Theorem:
To convert the fraction into a fraction with denominator d,
the new numerator is * d.
a
b
a
b
Example D: Convert to a fraction with denominator 48.
The new denominator is 48, then the new numerator is
48*
9
16
9
16
59. 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
In the above example, we found that is the same .
1
3
4
12
The following theorem tells us how to convert the denominator
of a fraction to a fraction with a different denominator.
Multiplier Theorem:
To convert the fraction into a fraction with denominator d,
the new numerator is * d.
a
b
a
b
Example D: Convert to a fraction with denominator 48.
The new denominator is 48, then the new numerator is
48*
9
16
9
16
3
60. 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
In the above example, we found that is the same .
1
3
4
12
The following theorem tells us how to convert the denominator
of a fraction to a fraction with a different denominator.
Multiplier Theorem:
To convert the fraction into a fraction with denominator d,
the new numerator is * d.
a
b
a
b
Example D: Convert to a fraction with denominator 48.
The new denominator is 48, then the new numerator is
48* = 27 so =
9
16
9
16
3
9
16
27
48 .
61. Suppose a pizza is cut into 4 equal slices
Addition and Subtraction of Fractions
62. Suppose a pizza is cut into 4 equal slices
Addition and Subtraction of Fractions
63. Suppose a pizza is cut into 4 equal slices and Joe takes
one slice or ¼ of the pizza,
Addition and Subtraction of Fractions
1
4
64. 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,
Addition and Subtraction of Fractions
1
4
2
4
65. 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
+
Addition and Subtraction of Fractions
1
4
2
4
1
4
2
4
66. 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
+
Addition and Subtraction of Fractions
1
4
2
4
= 3
4
of the entire pizza.
1
4
2
4
67. 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
+
Addition and Subtraction of Fractions
1
4
2
4
= 3
4
of the entire pizza. In picture:
+ =
1
4
2
4
3
4
68. 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
+
Addition and Subtraction of Fractions
Addition and Subtraction of Fractions With the Same
Denominator
1
4
2
4
= 3
4
of the entire pizza. In picture:
+ =
1
4
2
4
3
4
69. 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
+
Addition and Subtraction of Fractions
Addition and Subtraction of Fractions With the Same
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
of the entire pizza. In picture:
+ =
1
4
2
4
3
4
70. 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
+
Addition and Subtraction of Fractions
Addition and Subtraction of Fractions With the Same
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
of the entire pizza. In picture:
±
a
d
b
d
= a ± b
d
+ =
1
4
2
4
3
4
71. 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
+
Addition and Subtraction of Fractions
Addition and Subtraction of Fractions With the Same
Denominator
To add or subtract fractions of the same denominator, keep
the same denominator, add or subtract the numerators
,then simplify the result.
1
4
2
4
= 3
4
of the entire pizza. In picture:
±
a
d
b
d
= a ± b
d
+ =
1
4
2
4
3
4
81. Example A:
a. 7
12
+ =
7 + 11
12
18
12
= 3
2
18/6
12/6
=
+ = 8 + 4 – 2
15
= 2
3
=b.
Addition and Subtraction of Fractions
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.
82. Example A:
a. 7
12
+ =
7 + 11
12
18
12
= 3
2
18/6
12/6
=
+ = 8 + 4 – 2
15
= 2
3
=b.
Addition and Subtraction of Fractions
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. For example
1
2
83. Example A:
a. 7
12
+ =
7 + 11
12
18
12
= 3
2
18/6
12/6
=
+ = 8 + 4 – 2
15
= 2
3
=b.
Addition and Subtraction of Fractions
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. For example
+
1
2
1
3
84. Example A:
a. 7
12
+ =
7 + 11
12
18
12
= 3
2
18/6
12/6
=
+ = 8 + 4 – 2
15
= 2
3
=b.
Addition and Subtraction of Fractions
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. For example
+
1
2
1
3
=
?
?
85. Example A:
a. 7
12
+ =
7 + 11
12
18
12
= 3
2
18/6
12/6
=
+ = 8 + 4 – 2
15
= 2
3
=b.
Addition and Subtraction of Fractions
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. For example
To add them, first find the LCD of ½ and 1/3, which is 6.
+
1
2
1
3
=
?
?
86. Example A:
a. 7
12
+ =
7 + 11
12
18
12
= 3
2
18/6
12/6
=
+ = 8 + 4 – 2
15
= 2
3
=b.
Addition and Subtraction of Fractions
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. For example
To add them, first find the LCD of ½ and 1/3, which is 6.
We then cut each pizza into 6 slices.
+
1
2
1
3
=
?
?
87. Example A:
a. 7
12
+ =
7 + 11
12
18
12
= 3
2
18/6
12/6
=
+ = 8 + 4 – 2
15
= 2
3
=b.
Addition and Subtraction of Fractions
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. For example
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.
+
1
2
1
3
=
?
?
88. Example A:
a. 7
12
+ =
7 + 11
12
18
12
= 3
2
18/6
12/6
=
+ = 8 + 4 – 2
15
= 2
3
=b.
Addition and Subtraction of Fractions
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. For example
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,
1
2
=
3
6
1
3
=
2
6
+
1
2
1
3
=
?
?
89. Example A:
a. 7
12
+ =
7 + 11
12
18
12
= 3
2
18/6
12/6
=
+ = 8 + 4 – 2
15
= 2
3
=b.
Addition and Subtraction of Fractions
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. For example
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,
1
2
=
3
6
1
3
=
2
6
+
1
2
1
3
=
?
?
90. Example A:
a. 7
12
+ =
7 + 11
12
18
12
= 3
2
18/6
12/6
=
+ = 8 + 4 – 2
15
= 2
3
=b.
Addition and Subtraction of Fractions
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. For example
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,
+
1
2
=
3
6
1
3
=
2
6
3
6
1
3
=
?
?
91. Example A:
a. 7
12
+ =
7 + 11
12
18
12
= 3
2
18/6
12/6
=
+ = 8 + 4 – 2
15
= 2
3
=b.
Addition and Subtraction of Fractions
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. For example
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,
+
1
2
=
3
6
1
3
=
2
6
3
6
1
3
=
?
?
92. Example A:
a. 7
12
+ =
7 + 11
12
18
12
= 3
2
18/6
12/6
=
+ = 8 + 4 – 2
15
= 2
3
=b.
Addition and Subtraction of Fractions
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. For example
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,
+
1
2
=
3
6
1
3
=
2
6
3
6
2
6
=
?
?
93. Example A:
a. 7
12
+ =
7 + 11
12
18
12
= 3
2
18/6
12/6
=
+ = 8 + 4 – 2
15
= 2
3
=b.
Addition and Subtraction of Fractions
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. For example
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,
+
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
94. We need to convert fractions of different denominators to
a common denominator in order to add or subtract them.
Addition and Subtraction of Fractions
95. We need to convert fractions of different denominators to
a common denominator in order to add or subtract them.
The easiest common denominator to use is the LCD, the least
common denominator.
Addition and Subtraction of Fractions
96. We need to convert fractions of different denominators to
a common denominator in order to add or subtract them.
The easiest common denominator to use is the LCD, the least
common denominator. We list the steps below.
Addition and Subtraction of Fractions
97. We need to convert fractions of different denominators to
a common denominator in order to add or subtract them.
The easiest common denominator to use is the LCD, the least
common denominator. We list the steps below.
Addition and Subtraction of Fractions
Addition and Subtraction of Fractions With the Different
Denominator
98. We need to convert fractions of different denominators to
a common denominator in order to add or subtract them.
The easiest common denominator to use is the LCD, the least
common denominator. We list the steps below.
Addition and Subtraction of Fractions
Addition and Subtraction of Fractions With the Different
Denominator
1. Find their LCD
99. We need to convert fractions of different denominators to
a common denominator in order to add or subtract them.
The easiest common denominator to use is the LCD, the least
common denominator. We list the steps below.
Addition and Subtraction of Fractions
Addition and Subtraction of Fractions With the Different
Denominator
1. Find their LCD
2. Convert all the different-denominator-fractions to the have
the LCD as the denominator.
100. We need to convert fractions of different denominators to
a common denominator in order to add or subtract them.
The easiest common denominator to use is the LCD, the least
common denominator. We list the steps below.
Addition and Subtraction of Fractions
Addition and Subtraction of Fractions With the Different
Denominator
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 the
result.
101. We need to convert fractions of different denominators to
a common denominator in order to add or subtract them.
The easiest common denominator to use is the LCD, the least
common denominator. We list the steps below.
Addition and Subtraction of Fractions
Addition and Subtraction of Fractions With the Different
Denominator
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 the
result.
Example B:
5
6
3
8
+a.
102. We need to convert fractions of different denominators to
a common denominator in order to add or subtract them.
The easiest common denominator to use is the LCD, the least
common denominator. We list the steps below.
Addition and Subtraction of Fractions
Addition and Subtraction of Fractions With the Different
Denominator
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 the
result.
Example B:
5
6
3
8
+a.
Step 1: To find the LCD, list the multiples of 8
103. We need to convert fractions of different denominators to
a common denominator in order to add or subtract them.
The easiest common denominator to use is the LCD, the least
common denominator. We list the steps below.
Addition and Subtraction of Fractions
Addition and Subtraction of Fractions With the Different
Denominator
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 the
result.
Example B:
5
6
3
8
+a.
Step 1: To find the LCD, list the multiples of 8 which are
8, 16, 24, ..
104. We need to convert fractions of different denominators to
a common denominator in order to add or subtract them.
The easiest common denominator to use is the LCD, the least
common denominator. We list the steps below.
Addition and Subtraction of Fractions
Addition and Subtraction of Fractions With the Different
Denominator
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 the
result.
Example B:
5
6
3
8
+a.
Step 1: To find the LCD, list the multiples of 8 which are
8, 16, 24, .. we see that the LCD is 24.
105. Addition and Subtraction of Fractions
Step 2: Convert each fraction to have 24 as the denominator.
106. Addition and Subtraction of Fractions
Step 2: Convert each fraction to have 24 as the denominator.
For , the new numerator is 24 * = 20,
5
6
5
6
107. Addition and Subtraction of Fractions
Step 2: Convert each fraction to have 24 as the denominator.
For , the new numerator is 24 * = 20, hence
5
6
5
6
5
6
=
20
24
108. Addition and Subtraction of Fractions
Step 2: Convert each fraction to have 24 as the denominator.
For , the new numerator is 24 * = 20, hence
5
6
5
6
5
6
=
20
24
For , the new numerator is 24 * = 9,
3
8
3
8
109. Addition and Subtraction of Fractions
Step 2: Convert each fraction to have 24 as the denominator.
For , the new numerator is 24 * = 20, hence
5
6
5
6
5
6
=
20
24
For , the new numerator is 24 * = 9, hence
3
8
3
8
3
8
=
9
24
110. Addition and Subtraction of Fractions
Step 2: Convert each fraction to have 24 as the denominator.
For , the new numerator is 24 * = 20, hence
5
6
5
6
5
6
=
20
24
For , the new numerator is 24 * = 9, hence
3
8
3
8
3
8
=
9
24
Step 3: Add the converted fractions.
111. Addition and Subtraction of Fractions
Step 2: Convert each fraction to have 24 as the denominator.
For , the new numerator is 24 * = 20, hence
5
6
5
6
5
6
=
20
24
For , the new numerator is 24 * = 9, hence
3
8
3
8
3
8
=
9
24
Step 3: Add the converted fractions.
5
6
3
8
+ =
20
24
+
9
24
112. Addition and Subtraction of Fractions
Step 2: Convert each fraction to have 24 as the denominator.
For , the new numerator is 24 * = 20, hence
5
6
5
6
5
6
=
20
24
For , the new numerator is 24 * = 9, hence
3
8
3
8
3
8
=
9
24
Step 3: Add the converted fractions.
5
6
3
8
+ =
20
24
+
9
24
=
29
24
113. Addition and Subtraction of Fractions
Step 2: Convert each fraction to have 24 as the denominator.
For , the new numerator is 24 * = 20, hence
5
6
5
6
5
6
=
20
24
For , the new numerator is 24 * = 9, hence
3
8
3
8
3
8
=
9
24
Step 3: Add the converted fractions.
5
6
3
8
+ =
20
24
+
9
24
=
29
24
b.
7
12
5
8
+ –
16
9
114. Addition and Subtraction of Fractions
Step 2: Convert each fraction to have 24 as the denominator.
For , the new numerator is 24 * = 20, hence
5
6
5
6
5
6
=
20
24
For , the new numerator is 24 * = 9, hence
3
8
3
8
3
8
=
9
24
Step 3: Add the converted fractions.
5
6
3
8
+ =
20
24
+
9
24
=
29
24
b.
7
12
5
8
+ The LCD is 48.–
16
9
115. Addition and Subtraction of Fractions
Step 2: Convert each fraction to have 24 as the denominator.
For , the new numerator is 24 * = 20, hence
5
6
5
6
5
6
=
20
24
For , the new numerator is 24 * = 9, hence
3
8
3
8
3
8
=
9
24
Step 3: Add the converted fractions.
5
6
3
8
+ =
20
24
+
9
24
=
29
24
b.
7
12
5
8
+ The LCD is 48.–
16
9
Convert:
116. Addition and Subtraction of Fractions
Step 2: Convert each fraction to have 24 as the denominator.
For , the new numerator is 24 * = 20, hence
5
6
5
6
5
6
=
20
24
For , the new numerator is 24 * = 9, hence
3
8
3
8
3
8
=
9
24
Step 3: Add the converted fractions.
5
6
3
8
+ =
20
24
+
9
24
=
29
24
b.
7
12
5
8
+ The LCD is 48.
= 28
–
16
9
Convert:
7
12
48 *
117. Addition and Subtraction of Fractions
Step 2: Convert each fraction to have 24 as the denominator.
For , the new numerator is 24 * = 20, hence
5
6
5
6
5
6
=
20
24
For , the new numerator is 24 * = 9, hence
3
8
3
8
3
8
=
9
24
Step 3: Add the converted fractions.
5
6
3
8
+ =
20
24
+
9
24
=
29
24
b.
7
12
5
8
+ The LCD is 48.
= 28
–
16
9
Convert:
7
12
48 * so
7
12
= 28
48
118. Addition and Subtraction of Fractions
Step 2: Convert each fraction to have 24 as the denominator.
For , the new numerator is 24 * = 20, hence
5
6
5
6
5
6
=
20
24
For , the new numerator is 24 * = 9, hence
3
8
3
8
3
8
=
9
24
Step 3: Add the converted fractions.
5
6
3
8
+ =
20
24
+
9
24
=
29
24
b.
7
12
5
8
+ The LCD is 48.
= 28
–
16
9
Convert:
7
12
48 * so
7
12
= 28
48
= 30
5
8
48 *
119. Addition and Subtraction of Fractions
Step 2: Convert each fraction to have 24 as the denominator.
For , the new numerator is 24 * = 20, hence
5
6
5
6
5
6
=
20
24
For , the new numerator is 24 * = 9, hence
3
8
3
8
3
8
=
9
24
Step 3: Add the converted fractions.
5
6
3
8
+ =
20
24
+
9
24
=
29
24
b.
7
12
5
8
+ The LCD is 48.
= 28
–
16
9
Convert:
7
12
48 * so
7
12
= 28
48
= 30
5
8
48 * so
5
8
= 30
48
120. Addition and Subtraction of Fractions
Step 2: Convert each fraction to have 24 as the denominator.
For , the new numerator is 24 * = 20, hence
5
6
5
6
5
6
=
20
24
For , the new numerator is 24 * = 9, hence
3
8
3
8
3
8
=
9
24
Step 3: Add the converted fractions.
5
6
3
8
+ =
20
24
+
9
24
=
29
24
b.
7
12
5
8
+ The LCD is 48.
= 28
–
16
9
Convert:
7
12
48 * so
7
12
= 28
48
= 30
5
8
48 * so
5
8
= 30
48
= 27
9
16
48 *
121. Addition and Subtraction of Fractions
Step 2: Convert each fraction to have 24 as the denominator.
For , the new numerator is 24 * = 20, hence
5
6
5
6
5
6
=
20
24
For , the new numerator is 24 * = 9, hence
3
8
3
8
3
8
=
9
24
Step 3: Add the converted fractions.
5
6
3
8
+ =
20
24
+
9
24
=
29
24
b.
7
12
5
8
+ The LCD is 48.
= 28
–
16
9
Convert:
7
12
48 * so
7
12
= 28
48
= 30
5
8
48 * so
5
8
= 30
48
= 27
9
16
48 * so
9
16
=
27
48
122. Addition and Subtraction of Fractions
Step 2: Convert each fraction to have 24 as the denominator.
For , the new numerator is 24 * = 20, hence
5
6
5
6
5
6
=
20
24
For , the new numerator is 24 * = 9, hence
3
8
3
8
3
8
=
9
24
Step 3: Add the converted fractions.
5
6
3
8
+ =
20
24
+
9
24
=
29
24
b.
7
12
5
8
+ The LCD is 48.
= 28=
–
16
9
Convert:
7
12
48 * so
7
12
= 28
48
= 30
5
8
48 * so
5
8
= 30
48
= 27
9
16
48 * so
9
16
=
27
48
28
48
+
30
48
–
27
48
123. Addition and Subtraction of Fractions
Step 2: Convert each fraction to have 24 as the denominator.
For , the new numerator is 24 * = 20, hence
5
6
5
6
5
6
=
20
24
For , the new numerator is 24 * = 9, hence
3
8
3
8
3
8
=
9
24
Step 3: Add the converted fractions.
5
6
3
8
+ =
20
24
+
9
24
=
29
24
b.
7
12
5
8
+ The LCD is 48.
= 28=
–
16
9
48
28 + 30 – 27
Convert:
7
12
48 * so
7
12
= 28
48
= 30
5
8
48 * so
5
8
= 30
48
= 27
9
16
48 * so
9
16
=
27
48
28
48
+
30
48
–
27
48
=
124. Addition and Subtraction of Fractions
Step 2: Convert each fraction to have 24 as the denominator.
For , the new numerator is 24 * = 20, hence
5
6
5
6
5
6
=
20
24
For , the new numerator is 24 * = 9, hence
3
8
3
8
3
8
=
9
24
Step 3: Add the converted fractions.
5
6
3
8
+ =
20
24
+
9
24
=
29
24
b.
7
12
5
8
+ The LCD is 48.
= 28=
–
16
9
48
=
28 + 30 – 27
Convert:
7
12
48 * so
7
12
= 28
48
= 30
5
8
48 * so
5
8
= 30
48
= 27
9
16
48 * so
9
16
=
27
48
28
48
+
30
48
–
27
48
=
31
48
125. Addition and Subtraction of Fractions
Step 2: Convert each fraction to have 24 as the denominator.
For , the new numerator is 24 * = 20, hence
5
6
5
6
5
6
=
20
24
For , the new numerator is 24 * = 9, hence
3
8
3
8
3
8
=
9
24
Step 3: Add the converted fractions.
5
6
3
8
+ =
20
24
+
9
24
=
29
24
b.
7
12
5
8
+ The LCD is 48.
= 28=
–
16
9
48
=
28 + 30 – 27
Convert:
7
12
48 * so
7
12
= 28
48
= 30
5
8
48 * so
5
8
= 30
48
= 27
9
16
48 * so
9
16
=
27
48
28
48
+
30
48
–
27
48
=
31
48
126. We introduce the following Multiplier-Method to add or subtract
fractions to reduce the amount of repetitive copying.
Addition and Subtraction of Fractions
127. We introduce the following Multiplier-Method to add or subtract
fractions to reduce the amount of repetitive copying.
This method is based on the fact that if we multiply the quantity
x by a, then divide by a, we get back x.
Addition and Subtraction of Fractions
128. We introduce the following Multiplier-Method to add or subtract
fractions to reduce the amount of repetitive copying.
This method is based on the fact that if we multiply the quantity
x by a, then divide by a, we get back x.
For example, 2 * 5 / 5
Addition and Subtraction of Fractions
129. We introduce the following Multiplier-Method to add or subtract
fractions to reduce the amount of repetitive copying.
This method is based on the fact that if we multiply the quantity
x by a, then divide by a, we get back x.
For example, 2 * 5 / 5 = 10/5 = 2,
Addition and Subtraction of Fractions
130. We introduce the following Multiplier-Method to add or subtract
fractions to reduce the amount of repetitive copying.
This method is based on the fact that if we multiply the quantity
x by a, then divide by a, we get back x.
For example, 2 * 5 / 5 = 10/5 = 2, 3 * 8 / 8 =
Addition and Subtraction of Fractions
131. We introduce the following Multiplier-Method to add or subtract
fractions to reduce the amount of repetitive copying.
This method is based on the fact that if we multiply the quantity
x by a, then divide by a, we get back x.
For example, 2 * 5 / 5 = 10/5 = 2, 3 * 8 / 8 = 24/8 = 3.
Addition and Subtraction of Fractions
132. We introduce the following Multiplier-Method to add or subtract
fractions to reduce the amount of repetitive copying.
This method is based on the fact that if we multiply the quantity
x by a, then divide by a, we get back x.
For example, 2 * 5 / 5 = 10/5 = 2, 3 * 8 / 8 = 24/8 = 3.
Addition and Subtraction of Fractions
Multiplier Method (for adding and subtracting fractions)
133. We introduce the following Multiplier-Method to add or subtract
fractions to reduce the amount of repetitive copying.
This method is based on the fact that if we multiply the quantity
x by a, then divide by a, we get back x.
For example, 2 * 5 / 5 = 10/5 = 2, 3 * 8 / 8 = 24/8 = 3.
Addition and Subtraction of Fractions
Multiplier Method (for adding and subtracting fractions)
To add or subtract fractions, multiply the problem by the LCD
(expand it distributive using law), then divide by the LCD.
134. Example C. a.
We introduce the following Multiplier-Method to add or subtract
fractions to reduce the amount of repetitive copying.
This method is based on the fact that if we multiply the quantity
x by a, then divide by a, we get back x.
For example, 2 * 5 / 5 = 10/5 = 2, 3 * 8 / 8 = 24/8 = 3.
Addition and Subtraction of Fractions
Multiplier Method (for adding and subtracting fractions)
To add or subtract fractions, multiply the problem by the LCD
(expand it distributive using law), then divide by the LCD.
5
6
3
8
+
135. Example C. a.
The LCD is 24.
We introduce the following Multiplier-Method to add or subtract
fractions to reduce the amount of repetitive copying.
This method is based on the fact that if we multiply the quantity
x by a, then divide by a, we get back x.
For example, 2 * 5 / 5 = 10/5 = 2, 3 * 8 / 8 = 24/8 = 3.
Addition and Subtraction of Fractions
Multiplier Method (for adding and subtracting fractions)
To add or subtract fractions, multiply the problem by the LCD
(expand it distributive using law), then divide by the LCD.
5
6
3
8
+
136. Example C. a.
The LCD is 24. Multiply the problem by 24, then divide by 24.
We introduce the following Multiplier-Method to add or subtract
fractions to reduce the amount of repetitive copying.
This method is based on the fact that if we multiply the quantity
x by a, then divide by a, we get back x.
For example, 2 * 5 / 5 = 10/5 = 2, 3 * 8 / 8 = 24/8 = 3.
Addition and Subtraction of Fractions
Multiplier Method (for adding and subtracting fractions)
To add or subtract fractions, multiply the problem by the LCD
(expand it distributive using law), then divide by the LCD.
5
6
3
8
+
137. Example C. a.
The LCD is 24. Multiply the problem by 24, then divide by 24.
5
6
3
8+( ) * 24 / 24
We introduce the following Multiplier-Method to add or subtract
fractions to reduce the amount of repetitive copying.
This method is based on the fact that if we multiply the quantity
x by a, then divide by a, we get back x.
For example, 2 * 5 / 5 = 10/5 = 2, 3 * 8 / 8 = 24/8 = 3.
Addition and Subtraction of Fractions
Multiplier Method (for adding and subtracting fractions)
To add or subtract fractions, multiply the problem by the LCD
(expand it distributive using law), then divide by the LCD.
5
6
3
8
+
138. Example C. a.
The LCD is 24. Multiply the problem by 24, then divide by 24.
5
6
3
8+( ) * 24 / 24
4
We introduce the following Multiplier-Method to add or subtract
fractions to reduce the amount of repetitive copying.
This method is based on the fact that if we multiply the quantity
x by a, then divide by a, we get back x.
For example, 2 * 5 / 5 = 10/5 = 2, 3 * 8 / 8 = 24/8 = 3.
Addition and Subtraction of Fractions
Multiplier Method (for adding and subtracting fractions)
To add or subtract fractions, multiply the problem by the LCD
(expand it distributive using law), then divide by the LCD.
5
6
3
8
+
139. Example C. a.
The LCD is 24. Multiply the problem by 24, then divide by 24.
5
6
3
8+( ) * 24 / 24
4 3
We introduce the following Multiplier-Method to add or subtract
fractions to reduce the amount of repetitive copying.
This method is based on the fact that if we multiply the quantity
x by a, then divide by a, we get back x.
For example, 2 * 5 / 5 = 10/5 = 2, 3 * 8 / 8 = 24/8 = 3.
Addition and Subtraction of Fractions
Multiplier Method (for adding and subtracting fractions)
To add or subtract fractions, multiply the problem by the LCD
(expand it distributive using law), then divide by the LCD.
5
6
3
8
+
140. Example C. a.
The LCD is 24. Multiply the problem by 24, then divide by 24.
5
6
3
8+( ) * 24 / 24 = (4*5 + 3*3) / 24
4 3
We introduce the following Multiplier-Method to add or subtract
fractions to reduce the amount of repetitive copying.
This method is based on the fact that if we multiply the quantity
x by a, then divide by a, we get back x.
For example, 2 * 5 / 5 = 10/5 = 2, 3 * 8 / 8 = 24/8 = 3.
Addition and Subtraction of Fractions
Multiplier Method (for adding and subtracting fractions)
To add or subtract fractions, multiply the problem by the LCD
(expand it distributive using law), then divide by the LCD.
5
6
3
8
+
141. Example C. a.
The LCD is 24. Multiply the problem by 24, then divide by 24.
5
6
3
8+( ) * 24 / 24 = (4*5 + 3*3) / 24 = 29/24 =
4 3 29
24
We introduce the following Multiplier-Method to add or subtract
fractions to reduce the amount of repetitive copying.
This method is based on the fact that if we multiply the quantity
x by a, then divide by a, we get back x.
For example, 2 * 5 / 5 = 10/5 = 2, 3 * 8 / 8 = 24/8 = 3.
Addition and Subtraction of Fractions
Multiplier Method (for adding and subtracting fractions)
To add or subtract fractions, multiply the problem by the LCD
(expand it distributive using law), then divide by the LCD.
5
6
3
8
+
143. The LCD is 48. Multiply the problem by 48, expand the
multiplication, then divide the result by 48.
Addition and Subtraction of Fractions
b.
7
12
5
8
+ –
16
9
144. ( ) * 48 / 48
7
12
5
8+ – 16
9
The LCD is 48. Multiply the problem by 48, expand the
multiplication, then divide the result by 48.
Addition and Subtraction of Fractions
b.
7
12
5
8
+ –
16
9
145. ( ) * 48 / 48
7
12
5
8+ – 16
94
The LCD is 48. Multiply the problem by 48, expand the
multiplication, then divide the result by 48.
Addition and Subtraction of Fractions
b.
7
12
5
8
+ –
16
9
146. ( ) * 48 / 48
67
12
5
8+ – 16
94
The LCD is 48. Multiply the problem by 48, expand the
multiplication, then divide the result by 48.
Addition and Subtraction of Fractions
b.
7
12
5
8
+ –
16
9
147. ( ) * 48 / 48
67
12
5
8+ – 16
94 3
The LCD is 48. Multiply the problem by 48, expand the
multiplication, then divide the result by 48.
Addition and Subtraction of Fractions
b.
7
12
5
8
+ –
16
9
148. ( ) * 48 / 48
= (4*7 + 6*5 – 3*9) / 48
67
12
5
8+ – 16
94 3
The LCD is 48. Multiply the problem by 48, expand the
multiplication, then divide the result by 48.
Addition and Subtraction of Fractions
b.
7
12
5
8
+ –
16
9
149. ( ) * 48 / 48
= (4*7 + 6*5 – 3*9) / 48
= (28 + 30 – 27) / 48
67
12
5
8+ – 16
94 3
The LCD is 48. Multiply the problem by 48, expand the
multiplication, then divide the result by 48.
Addition and Subtraction of Fractions
b.
7
12
5
8
+ –
16
9
150. ( ) * 48 / 48
= (4*7 + 6*5 – 3*9) / 48
= (28 + 30 – 27) / 48
=
67
12
5
8+ – 16
94 3
The LCD is 48. Multiply the problem by 48, expand the
multiplication, then divide the result by 48.
Addition and Subtraction of Fractions
b.
7
12
5
8
+ –
16
9
48
31
151. ( ) * 48 / 48
= (4*7 + 6*5 – 3*9) / 48
= (28 + 30 – 27) / 48
=
67
12
5
8+ – 16
94 3
The LCD is 48. Multiply the problem by 48, expand the
multiplication, then divide the result by 48.
Addition and Subtraction of Fractions
b.
7
12
5
8
+ –
16
9
48
31
152. ( ) * 48 / 48
= (4*7 + 6*5 – 3*9) / 48
= (28 + 30 – 27) / 48
=
67
12
5
8+ – 16
94 3
The LCD is 48. Multiply the problem by 48, expand the
multiplication, then divide the result by 48.
Addition and Subtraction of Fractions
b.
7
12
5
8
+ –
16
9
48
31
153. 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}
154. B. Convert the fractions to fractions with the given
denominators.
10. Convert to denominator 12.
11. Convert to denominator 24.
12. Convert to denominator 36.
13. Convert to denominator 60.
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