lcm and lcd

1. LCM and LCD

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. 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}

62. 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