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Constructing-a-Least-Common-Denominator

Constructing a Least Common Denominator (LCD)
When Adding or Subtracting Fractions with Unlike Denominators

When adding or subtracting fractions we need both fractions to have the same denominator. When this happens
we say that the fractions have Like Denominators . In order to keep our computations to a minimum we will be
using the Least Common Denominator (LCD) as our like (or common) denominators.

There are three cases to consider:
Case 1: One denominator is a multiple of the other denominator.
Case 2: The two denominators are relatively prime ( equivalently , GCF = 1)
Case 3: The two denominators have a GCF ≠ 1.


Case 1: One denominator is a multiple of the other denominator.
Construction of the LCD : Use the larger of the two denominators as your LCD.

Examples:

12 is a multiple of 4. 24 is a multiple of 2. 60 is a multiple of 5. 24 is a multiple of 3. 75 is a multiple of 15.
Use LCD = 12. Use LCD = 24. Use LCD = 60. Use LCD = 24. Use LCD = 75.

Case 2: The two denominators are relatively prime (equivalently, GCF = 1)
Construction of the LCD: Multiply the two denominators together to get your LCD.

Examples:

GCF(15, 4) = 1,
and 15 × 4 = 60.
GCF(3, 7) = 1,
and 3 × 7 = 21.
GCF(7, 6) = 1,
and 7 × 6 = 42.
GCF(9, 4) = 1,
and 9 × 4 = 36.
GCF(21, 2) = 1,
and 21 × 2 = 42.
Use LCD = 60. Use LCD = 21. Use LCD = 42. Use LCD = 36. Use LCD = 42.

Case 3: The two denominators have a GCF other than 1.
Construction of the LCD: Use the prime factorization method (explained in class) to construct the LCD.

Examples:

10 = 2 • 5 and
25 = 52 , so the
LCD = 2•52
9 = 32 and
15 = 3• 5, so the
LCD = 32 •5
18 = 2 •32 and
12 = 22 •3 , so the
LCD = 22•32
14 = 2•7 and
21 = 3 •7, so the
LCD = 2 •3• 7
22 = 2• 11 and
33 = 3• 11, so the
LCD = 2 •3• 11
Use LCD = 50. Use LCD = 45. Use LCD = 36. Use LCD = 42. Use LCD = 66.

Practice Identifying the
Least Common Denominator (LCD)


The Least Common Denominator (LCD) is the Least Common Multiple ( LCM ) of the denominators.

In cases 1 and 2, the LCD, if computed, would be the simpler forms described in the handout. Case 3 is the
only case that actually requires an official computation for the LCD.

From before:
Case 1:
One denominator is a multiple of the other denominator.
1. Use the larger of the two denominators as your LCD.
Case 2:
The two denominators are relatively prime (equivalently, GCF = 1)
1. Multiply the two denominators together to get your LCD.
Case 3:
The two denominators have a GCF ≠ 1.
1. Use the Prime Factorization Method to construct the LCD.


Indicate the cases to which the denominators belong. Then, indicate the LCD.
•Do your scratch work for Case 3 on another sheet of paper.





Solutions :

1.) Case 3; LCD = 42 2.) Case 2; LCD = 77 3.) Case 3; LCD = 150 4.) Case 3; LCD = 24
5.) Case 1; LCD = 70 6.) Case 2; LCD = 60 7.) Case 3; LCD = 400 8.) Case 2; LCD = 120

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