3.03   Least Common Denominators (LCD)

from Basic Algebra: One Step at a Time © 2002

P. 257-264

Dr. Robert J. Rapalje

Seminole State College of Florida

 

ANSWERS TO ALL EXERCISES ARE INCLUDED AT THE END OF THIS PAGE

Addition and subtraction of fractions requires a completely different procedure from the multiplication and division of the last section. Fractions cannot be added (or subtracted) unless they have a common denominator. Fractions that have a common denominator are added in the same way that you combine like terms. Just like adding 3 apples plus 2 apples (which equals 5 apples), adding 3 sevenths plus 2 sevenths equals 5 sevenths. This is written in math symbols:

.

When you add or subtract fractions with a common denominator, you add or subtract the numerators, and place that answer over the common denominator. And of course remember, you are still not allowed (and never will be allowed!) to divide by zero!

PRINCIPLE:   

 

            

            PRINCIPLE:         Before you can add or subtract fractions,  you must

                                         have a common denominator.

 

  If the fractions do not have a common denominator, then the first order of business must be to find a common denominator. (It must really be lost! People have been looking for it for decades!) A common denominator is a number or an expression such that each denominator divides into the common denominator evenly. For example, to add , the common denominator of 6 may be used, since both 2 and 3 divide evenly into 6. However, notice that any multiple of 6, like 12, 18, 24, 36, etc., are also divisible by 2 and 3, so these could also be considered "common denominators" for this addition problem. Finally, we can say that of all these common denominators, 6 is the least (lowest) common denominator (LCD), and in most cases the least common denominator is the best one to use.

       

        DEFINITIONS:       A common denominator is a number or quantity such that

                                      each denominator divides evenly into it.

 

                                     The least (lowest) common denominator (LCD) is the

                                     smallest of all the common denominators.

 

 

 

Many "least common denominators" are rather obvious. While there are many explanations and methods of finding LCDs, a good approach is to begin with "obvious" examples (intuitive method). From these examples, you will discover a strategy that will allow you to quickly and easily find most LCDs. Then, after further probing, you will discover a method that works to find all LCDs, even those you do not intuitively know. This method will not only guarantee your ability to find LCDs regardless of how complicated it may be--it will ensure your understanding of the entire concept of LCDs--the what, the how, and the why!

EXERCISES. In the following exercises, by trial and error, find the least common denominators (LCD). Remember you are trying to find the smallest possible number that each of the denominators divides into evenly. For your convenience, the answers to this page are provided at the bottom of the page.

 

1. LCD = _____                     2. LCD = _____                       3. LCD = _____

4. LCD = _____                     5. LCD = _____                      6. LCD = _____

7. LCD = _____                   8. LCD = _____                      9. LCD = _____

10. LCD = _____              11. LCD = _____                   12. LCD = _____

13. LCD = _____                                             14. LCD = _____

15. LCD = _____                                      16. LCD = _____

17. LCD = _____                 18. LCD = _____                       19. LCD = _____

20. LCD = _____                 21. LCD = _____                       22. LCD = _____

23. LCD = _____                                            24. LCD = _____

 

ANSWERS:

1.4; 2.6; 3.8; 4.6; 5.12; 6.12; 7.12; 8.12; 9.10; 10.20; 11.60; 12.24; 13.6; 14.12; 15.12; 16.24; 17.6; 18.12; 19.10; 20.15; 21.20; 22.35; 23.30; 24.70.

What is the pattern in #1-16? _________________________________

What is the pattern in #17-24?_________________________________

25. LCD = _____                26. LCD = _____                       27. LCD = _____

28. LCD = _____               29. LCD = _____                   30. LCD = _____

 

#31 - 34 are harder to do by trial and error. This underscores the need for systematic strategies that will follow these exercises. (Before getting frustrated, you may want to sneak a peek ahead!)

 

31. LCD = _____                                                32. LCD = _____

 

33. LCD = ______                                            34. LCD = _____

 

From these exercises you probably noticed that sometimes the LCD is the larger (or largest) of the numbers (as in #1-16). Sometimes the LCD is the product of the numbers (as in #17-24). Sometimes you just have to use trial and error looking at multiples of the denominators (as in #25-34). Sometimes you could spend hours looking (as in #35-40)!

An easy way to find the LCD is to begin with the largest denominator--see if that is the LCD. If it is not, then take multiples of that largest denominator, trying each one in order, until you find one that "works"--that is, until you find one that each of the other denominators divides into evenly.

For example, if the denominators are 6, 12, and 9, begin with the largest number which is 12. Since 12 is not divisible by 9, try multiples of 12: 24, 36, 48, 60, etc. Notice that 24 does not work either, but 36 does work, since it is divisible by 6, 12, and 9. Therefore, the LCD is 36. This is a quick and easy way to find most LCDs. Unfortunately, some problems could take a long time to do this way.

Perhaps you also correctly noticed that if the denominators have no common factors, then the LCD is the product of the denominators. Most importantly, perhaps you noticed that every LCD is built using the prime factors of the denominators involved. The following examples should add details to this idea, and enable you to develop a systematic way of finding the LCD, based upon the factors of the denominators. Begin by factoring each denominator into its prime factors.

EXAMPLE 1:

Solution:        In the LCD, you need factors of 2 and 5, but the extra 2 is not needed.

                       LCD = 2×5 = 10.

EXAMPLE 2:

Solution:       The denominators consist of prime factors 2, 5, and 7. Therefore,

                      LCD = 2×5×7 = 70.

EXAMPLE 3:

Solution:       In this case the prime factors are 2 and 3.  Since you probably already know the LCD is

                      18,  you will need factors of 2 and of 32 (the highest power of the 3 factors).

EXAMPLE 4:

Solution:     The prime factors are 2 and 5. If you take 23 (the highest power of 2) that will include the

                    other 22.   Therefore, LCD = 23×5 = 40.

EXAMPLE 5:

Solution:      This time the highest power of 2 is 3, and the highest power of 3 is 2. Therefore,

                     LCD = 23×32 = 8×9 = 72.

 

Find the LCDs by factoring each denominator into prime factors.

35. LCD = _____                                 36. LCD = _____

 

 

37. LCD = ______                               38. LCD = _____

 

 

39. LCD = _____                                 40. LCD = _____

 

 

This technique is particularly useful where variables are involved.

EXAMPLE 6.

Solution:     The LCD for 3 and 8 is obviously 24. The highest power of X is X3.

                    LCD = 24X3.

 

EXAMPLE 7.

Solution:     LCD = 3X3.Y4.

 

EXAMPLE 8.

Solution:     The factors are 3, X, and X+3.  Notice that X+3 is a distinct factor.

                    LCD = 3X(X+3).

 

EXAMPLE 9.

Solution:     First factor each denominator.

                   

                    Then the factors are (X+2), (X+1), and (X+3). (Any order!)

                    LCD = (X+1)(X+2)(X+3).

 

EXAMPLE 10.

Solution:     First factor each denominator.

                   

                    The prime factors are (X+2) and (X+3). (X+3) to the highest power is (X+3)2.

                    LCD = (X+2)(X+3)2.

 

EXERCISES. In the following, find the least common denominator (LCD).

41.                              42.                               43.

 

 

 

44.                              45.                     46.

 

 

 

47.                                       48.

 

 

49.                                             50.

 

 

51.                            52.

 

 

53.                                      54.

                                        

     LCD = ___________________                                LCD = ___________________

 

55.                                        56.

                                   

      LCD = ___________________                               LCD = ___________________

 

57.                              58.

    =                                                                                 =

 

     LCD = ______________                                          LCD = ______________

 

59.                                         60.

     =                                                                               =

     LCD = ______________                                         LCD = ______________

 

61.                               62.

      =                                                                              =

     LCD = ______________                                        LCD = ______________

 

ANSWERS 3.03

 p. 259 - 264:

             1. 4; 2. 6; 3. 8; 4. 6; 5. 12; 6. 12; 7. 12; 8. 12; 9. 10;    10. 20; 11. 60; 12. 24; 13. 6; 14. 12; 15. 12; 16. 24; 17. 6;  18. 12; 19. 10; 20. 15; 21. 20; 22. 35; 23. 30; 24. 70; #1-16, LCD is the largest denominator; #17-24, LCD is the product of the denominators; 25. 12; 26. 24; 27. 30; 28. 30; 29. 50; 30. 40; 31. 180; 32. 200; 33. 108; 34. 450; 35. 150; 36. 540;  37. 864; 38. 312; 39. 294; 40. 360; 41. 144; 42. 240; 43. 900;  44. 360; 45. 10X4; 46. 40X3; 47. 50X4Y3;  48. 450X2Y5;  49. 60X7(X+3);  50. 60X3(X+4)2;  51. 60X3(X-5)3;  52. 45X5(X-2)6; 53. X(X+2)(X+3); 54. X(X-5)(X+5); 55. (X-1)(X-6)(X+6);  56. (X‑2)(X+2)(X-3);  57. (X+4)(X+5)(X+2);  58. X(X-4)(X+4);  59. X(X-1)2; 60. (X‑2)2(X+2);  61. (X-4)2(X-1);  62. (X+5)2(X-2).

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Dr. Robert J. Rapalje Altamonte Springs Campus
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