LCM of Fractions

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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.
  4. The answer is LCM(numerators)/LCM(b,d)

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).
So answer is
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)

 

Comments: 33

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  • Ruchita

    Another masterpiece from ExamHelp.

     
     
     
    • किताब देख ना यार

       
  • sastry

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

     
     
     
    • A million thkans for posting this information.

       
  • Shubhashish

    Wow

     
     
     
  • Shilpa

    Really well written article.

    I am a teacher and would recommend others to read this article.

     
     
     
  • vps

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

     
     
     
  • Nancy Joseph

    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….. :)

     
     
     
    • its betrrer!!! thnx

       
  • VISHAL KUMAR THA..

    बैँक के EXAM मे कितना प्रश्न इससे आता है ।LCM निकालने का कोई आसान IDEA दे ।

     
     
     
  • HS

    Good points

     
     
     
  • solo

    Nice post.

     
     
     
  • JosephTorson

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

     
     
     
  • Jerry Skonczewski

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

     
     
     
  • Umang

    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.

     
     
     
    • 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.

       
  • Arunima

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

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

     
     
     
    • lcm=lowest common multiple
      hcf=highest common factor

       
  • ambien

    Simply Amazing…

     
     
     
  • Deependra

    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.

     
     
     
  • Deependra

    I LOVE THIS TYPE OF ARTICLE.

     
     
     
  • chris

    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

     
     
     
  • Tiesta

    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?

     
     
     
  • AMAR

    Thanks

     
     
     
  • abcd

    thx

     
     
     
  • J.K.Gupta

    great

     
     
     
  • Aman Sharma

    Thanks For The Information

     
     
     
  • Nathan

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

     
     
     
  • munib

    very good explaination

     
     
     
  • Ashish

    Thank You v.much!!!

     
     
     
  • shantanu

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

     
     
     
  • Pranjul Bhatt

    what if fractions contain irrational number?

     
     
     
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