- On 2011-06-26
- By Edugain

## 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**

- Find the LCM of
**b**and**d**=**LCM(b,d)** - 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. - Find LCM of new numerators.
- The answer is
**LCM(numerators)/LCM(b,d)**

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

- LCM of 9 and 21 = 63,
- 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 - Now since both denominators are same, LCM of numerators 14 and 24 = 168.
- 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)

Another masterpiece from ExamHelp.

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

Wow

A million thkans for posting this information.

Really well written article.

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

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

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..... :)

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

Good points

Nice post.

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

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

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.

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.

Simply Amazing...

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.

I LOVE THIS TYPE OF ARTICLE.

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.

lcm=lowest common multiple

hcf=highest common factor

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

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?

Thanks

thx

great

Thanks For The Information

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

very good explaination

its betrrer!!! thnx

Thank You v.much!!!

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

what if fractions contain irrational number?

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

another easy way is

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

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.

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

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

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.

HElp this 2/3 and 1/6

coolllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll!

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)

Your message ..Nice concept

thanx for post

cool

hi admin i need answer for 37 1/2 and 45 LCM

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)