LCM of Fractions : Educational Articles | Edugain Germany

## LCM of Fractions

The Least Common Multiple (LCM) of two integers x and y, is the smallest positive integer that is a multiple of both x and y. Generally LCM is used for adding fractions where denominators are not same. Most of the kids know how to calculate LCM, but I was surprised to learn that most of the kids are not aware of the physical significance of LCM.

I was interacting with some of the grade 6 students and I asked them about LCM, and all kids in class said that they know LCM very well. So I asked them to find LCM of 1/2 and 1/3 and surprisingly no one could answer. They were trying to apply regular method of finding LCM of integers, and of-course that did not help them in finding LCM of fractions.

Problem was that kids just learn the method of solving questions in textbooks, and do not pay much attention to theory. Had any student used the definition of LCM (and not the regular method of finding LCM) they could have easily solved this question.

Anyway lets forget the regular method of finding LCM and try to solve this using definition of LCM.

So as per the definition of LCM, we have to find a number which can be fully divided by 1/2 and 1/3. If you think about it you will find that answer is 1. Since one is fully divisible by both 1/2 and 1/3.

1/ (1/2) = 2
1/ (1/3) = 3

Now lets take another example. Find LCM of 1/6 and 1/9.
Answer for this one is 1/3, since,
(1/3) / (1/6) = 2
(1/3) / (1/9) = 3

Ok, these were simple cases so we could do just by thinking about it, but we might have to do this for more complex fractions. Now lets formalize a method to find LCM of any two fractions.

Lets try to find LCM of (a/b) and (c/d). If b and d were same it was easy to find LCM since if denominators are same, we just need to find LCM of numerators, hence LCM of (a/b) and (c/b) would be LCM(a,c)/b. So we have to first make denominators of both the fractions same.

So here are the steps to find LCM of a/b and c/d

1. Find the LCM of b and d = LCM(b,d)
2. Multiply numerator and denominator of first fraction by LCM(b,d)/b.
Multiply numerator and denominator of first fraction by LCM(b,d)/d.
After this multiplication, denominator of both fractions are same.
3. Find LCM of new numerators.

Lets see this using example of 2/9 and 8/21.

1. LCM of 9 and 21 = 63,
2. Now multiply first fraction by 63/9 = 7
Multiply second fraction by 63/21 = 3
So now first fractions is (2 x 7)/(9 x 7) = 14/63
and second fraction is (8 x 3)/(21 x 3) = 24/63
3. Now since both denominators are same, LCM of numerators 14 and 24 = 168.
4. Hence LCM of 2/9 and 6/21 is (168/63)
After simplification 168/63 = 8/3

Now lets check if our answer (8/3) is correct or not by dividing this by 2/9 and 8/21.
(8/3) / (2/9) = 12
(8/3) / (8/21) = 7

You can see that this is fully divisible by both fractions.

Now we have understood the concept, lets try to find a formula for quickly solving this.
If you observe the new numerators after multiplication, they are (a/b)*LCM(b,d) and (c/d)*LCM(b,d).
LCM ( (a/b)*LCM(b,d) , (c/d)*LCM(b,d) ) / LCM(b,d)

After simplification this will come down to a simple formula,

LCM( (a/b) , (c/d) ) = LCM(a,c)/HCF(b,d)

### Practice LCM of Fractions

Above worksheet is just a sample with fixed questions. To generate worksheet or practice test with new questions every time use following links.

1. Posted by Ruchita| 2011-06-26 11:27:20

Another masterpiece from ExamHelp.

2. Posted by sastry| 2011-07-12 07:47:44

superb, i am a parent. i never came to know this concept earlier. thank u for bringing clarity to students

3. Posted by Shubhashish| 2011-07-16 11:28:03

Wow

4. Posted by Nyvaeh| 2011-07-19 19:57:10

A million thkans for posting this information.

5. Posted by Shilpa| 2011-08-06 22:21:25

Really well written article.

6. Posted by vps| 2011-08-17 19:30:43

Very interesting points you have observed , thankyou for putting up.

7. Posted by Nancy Joseph| 2011-08-31 02:04:53

Another Method:

Rule: First express the given fractions in their lowest form.

H.C.F= H.C.F of Numerators / L.C.M of denominators

L.C.M= L.C.M of Numerators / H.C.F of denominator

Hope dis helps..... :)

8. Posted by VISHAL KUMAR THAKUR| 2011-08-31 21:36:28

????? ??? ?? ???

9. Posted by VISHAL KUMAR THAKUR| 2011-08-31 21:41:49

???? ?? EXAM ?? ????? ?????? ???? ??? ?? ?LCM ??????? ?? ??? ???? IDEA ?? ?

10. Posted by HS| 2011-09-02 19:56:37

Good points

11. Posted by solo| 2011-09-16 20:39:34

Nice post.

12. Posted by JosephTorson| 2011-12-07 01:28:13

Cool blogpost to my mind. Thank u for posting this data.

13. Posted by Jerry Skonczewski| 2012-02-24 21:21:51

Thank the author very much for this astonishing content. Great work!

14. Posted by Umang| 2012-03-22 22:32:00

I dont know how she/he supposed 1/(1/2) = 2 and 1/(1/3) = 3 ... answer are 1/(1/2) = 1/2 and 1/(1/3) = 1/3.... bcos we are trying to distribute 1 in 1/2 equal parts and in other case we are trying to distribute 1 in 1/3 equal parts. Hope I understood it properly. anyways daring to put of my thought in public.

15. Posted by Arunima| 2012-03-26 12:11:34

In the last line, should it not be LCM(b,d) instead of HCF(b,d)?

16. Posted by saad| 2012-05-19 23:54:25

tell me h.c.f and l.c.m abbreviation.

17. Posted by ambien| 2012-07-16 18:09:17

Simply Amazing...

18. Posted by Deependra| 2012-08-15 09:24:22

Today's article is very popular for the children, because it is very nice.I think your article will be famous , then your website will be famous.Thankyou for this article. I will be never scared of maths.

19. Posted by Deependra| 2012-08-15 09:28:30

I LOVE THIS TYPE OF ARTICLE.

20. Posted by alisse| 2013-07-15 11:05:09

im sorry..u understood wrong
we r dividing 1 into 1/2 equal parts..so we get 2 parts each, of 1/2
(1/2)*2=1 (or) 1=1/2+1/2
similarly for 1 in 1/3 equal parts ie. 3 parts each of 1/3
so 1/(1/2) = 2 and 1/(1/3) = 3 is correct.

21. Posted by alisse| 2013-07-15 11:08:23

lcm=lowest common multiple
hcf=highest common factor

22. Posted by chris| 2013-10-01 17:18:51

2 3
- (x-1) - - (x+2)=1
3 4

cant understand how it works

2(x-1) 3(x+2) 1
------- - ------- = - (l.c.m.=12)
3 4 1

how to work and get the no 8 and 9

8(x-1) - 9 (x+2) =12
- -
- -

this is an example took it from a book

23. Posted by Tiesta| 2013-10-29 13:11:23

If LCM is a positive integer by definition, then how come LCM of 1/6 and 1/9 is 1/3 which is a fraction?

24. Posted by AMAR| 2014-01-30 08:20:27

Thanks

25. Posted by abcd| 2014-02-01 22:59:53

thx

26. Posted by J.K.Gupta| 2014-02-15 19:02:25

great

27. Posted by Aman Sharma| 2014-03-18 12:36:35

Thanks For The Information

28. Posted by Nathan| 2014-04-14 22:28:03

Can you generalize it to more than 2 fractions, so I could find the LCM of any size set of fractions?

29. Posted by munib| 2014-05-28 15:34:42

very good explaination

30. Posted by AASHIMA| 2014-06-11 16:47:29

its betrrer!!! thnx

31. Posted by Ashish| 2014-06-12 15:10:40

Thank You v.much!!!

32. Posted by shantanu| 2014-06-18 23:48:09

LCM of fraction= LCM of numerator/HCF of denominator...it is easy way...

33. Posted by Pranjul Bhatt| 2014-08-06 08:17:03

what if fractions contain irrational number?

34. Posted by Joshi Ravishankar| 2014-10-05 11:31:23

I think you should rethink as irrational numbers cannot be shown as fractions at all. If you are thinking about LCM(3, ?2) than it can be easily done with finding smallest rational multiple of the irrational number.
e.g. the smallest rational multiple of ?2 is 2. (? 2=?2*?2)
than LCM(2,3) = 2*3=6.
so, LCM(3, ?2)=6

35. Posted by sandey| 2014-11-07 17:10:42

another easy way is

lcm of 2 fractions=lcm of numerator / hcf of denominator

36. Posted by Shashank| 2015-02-23 19:18:06

Hello Sir,

I really appreciate for providing such a clear concept but at last step I'm bit confused that After simplification this will come down to a simple formula,

LCM( (a/b) , (c/d) ) = LCM(a,c)/HCF(b,d)

I want to know how this LCM(a,c)/H.C.F.(b,d) came?

As we can see after simplifying this equation
LCM ( (a/b)*LCM(b,d) , (c/d)*LCM(b,d) ) / LCM(b,d)

L.C.M.(b,d) will common n we will get the result

L.C.M.((a/b),(c/d))
then how do we get L.C.M.(a,c)/H.C.F.(b,d).

Thanking in anticipation.

37. Posted by Aiswarya Sreeni| 2015-04-15 15:01:13

Thanks I got the answers of all questions. ..its very simple. ..

38. Posted by shelly| 2015-08-09 08:29:03

I don't get what's that trying to show i asked for LCM worksheet question but not for this type

39. Posted by Sachin Pandey| 2015-08-10 11:07:56

Hey author ,
Check the example you have given there,
There is an error in point 4. It must be 8/21 in place of 6/21.

40. Posted by Matthew| 2015-09-23 07:09:02

HElp this 2/3 and 1/6

41. Posted by jaered| 2015-10-05 16:27:41

coolllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll!

42. Posted by Utpal Kant Singh| 2016-02-07 03:41:20

Can some body explain how can we simplify LCM ( (a/b)*LCM(b,d) , (c/d)*LCM(b,d) ) / LCM(b,d)

to get :

LCM( (a/b) , (c/d) ) = LCM(a,c)/HCF(b,d)

43. Posted by Dr manoj kumjr| 2016-02-10 13:31:33

44. Posted by Dr manoj kumjr| 2016-02-10 13:31:51

45. Posted by shashank| 2016-02-15 14:12:36

thanx for post

46. Posted by ManothamaReddy| 2016-04-26 08:41:23

cool

48. Posted by jai rawal| 2016-07-17 18:01:24

Can some body explain how can we simplify LCM ( (a/b)*LCM(b,d) , (c/d)*LCM(b,d) ) / LCM(b,d)

to get :

LCM( (a/b) , (c/d) ) = LCM(a,c)/HCF(b,d)

49. Posted by Abhi| 2016-07-30 08:59:50

what is the LCM of -2/3 and 2/3

50. Posted by sunil kumar | 2016-08-18 12:18:30

L.C.M.of one and pie

51. Posted by sunil kumar | 2016-08-18 12:18:36

L.C.M.of one and pie

52. Posted by Meshwi | 2016-09-28 12:27:03

53. Posted by Abhi| 2016-10-21 13:58:14

Great sir tganks a lot

54. Posted by Amaan ali| 2016-11-23 14:05:04

But u haven't given the proper derivation of simple formula fir finding LCM ,,, LCM=LCM of numerator /hcf of denominator

55. Posted by Tanmal Dalui| 2016-12-26 05:05:30

great article.I was trying to understand the the basic concept of LCM of fractional number.This article help to understand the concept.Thank you sir