LCM IN RATIOS

Saumyojit

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There are three ratios 1/4:1/3:1/5 now to convert them into simplified ratio format i see every term needs to be multiplied with lcm 60. I know multiplying with 60 will get rid of the fractional part . I know that (1/4)*60 ; (1/3)*60 ; (1/5)*60 like this it is happening but i can't satisfy my brain how it is being applied or the logic behind this process other than to get rid of fractional part!

ANALOGY: When i see the above step i feel like 60 rupees was divided among 3 persons in the according ratio . Now i am finding the actual money in rs of each share.

ANALOGY 2:Like we do lcm of addition
suppose 1/4 + 1/3 +1/5 first we convert all the num and denom prorportionately to new values with denom being 60 1/4=15/60;1/3=20/60;1/5=12/60;then add them ; the sum divided by only one 60. We do the step 1/4=15/60 as the fraction have to kept same (equivalent).i find a logic in this process.

There is another method of lcm in addition suggested by my friend which i was not aware of: every term needs to be multiplied with lcm 60. (1/4)*60+(1/3)*60+(1/5)*60 similar to the ratio operation ; then 15+20+12 by 60.

@Dr.Peterson @JeffM
 

JeffM

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I am not sure that there is a logic to this process. Or if there is, you have already discovered it.

Many people find fractions unintuitive. They find it much easier to think in terms of whole numbers. Consequently, it is conventional to present ratios as whole numbers. So I think it is more a matter of human psychology than logical necessity.
 

Saumyojit

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I am not sure that there is a logic to this process. Or if there is, you have already discovered it
(1/4)*60 ; (1/3)*60 ; (1/5)*60 how this is done . i cant understnad the process. lcm * every term of ratio . Do i need to rote learn the process?
 

JeffM

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This may be repetitious. If so, I apologize.

As far as I am concerned, whole number ratios are simply one method for presenting data. I cannot think of anything (other than the hint given to atomic theory by whole number ratios) that requires presentation in ratio form. However, for many people, a whole number ratio is easy to understand. In that sense, the logic is that the best presentation is the one that is both accurate and most comprehensible for the prospective audience, and in some cases that means whole number ratios.

\(\displaystyle 1/4 : 1/3\) is not in whole numbers.

If we multiply those fractions by 4, we get

\(\displaystyle 1 : 4/3.\). Not both whole numbers.

If we multiply those numbers by 3, we get

\(\displaystyle 3/4 : 1.\). Not both whole numbers.

But if we multiply both fractions by ANY common multiple, we know that both products will be whole numbers. To keep the numbers small and more intuitively comprehensible, we use the least common multiple

\(\displaystyle 12 * \dfrac{1}{4} = 3 \text { and } 12 * \dfrac{1}{3} = 4.\)

\(\displaystyle 3 : 4\) indicates the same relative relationship as \(\displaystyle 1/4 : 1/3.\)

But in this problem, we have two ratios. After we multiply by 12, we have this:

\(\displaystyle 3 : 4 : 12/5.\) Not all whole numbers.

So we multiply by 5 and get

\(\displaystyle 15 : 20 : 12.\)

Or we can do it in one step by multiplying by 60 because 12 * 5 = 60.

Did you notice that \(\displaystyle 15 + 20 + 12 = 47 < 60.\)

Consequently, I do not like this presentation for data like these. I'd prefer 25%, 33%, and 12%, which obviously do not add up to 100%. Of course you have to do what the problem asks you to do.
 

Jomo

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1/4 cup of item A :1/3 cup of item B :1/5 cup of item C

Image in a big bowl you put 1/4 cup of item A, 1/3 cup of item B and1/5 cup of item C. So you lived up to the given ratio. Now do this 59 more times. You still lived up to the ratio. In the end you put into the bowl 15 cups of item A, 20 cups of item B and 12 cups of item C. This gives you a 15:20:12 ratio which must be the same as 1/4:1/3 :1/5.
 

Saumyojit

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\(\displaystyle 3 : 4\) indicates the same relative relationship as \(\displaystyle 1/4 : 1/3.\)
Did you notice that \(\displaystyle 15 + 20 + 12 = 47 < 60.\)
How 3:4 is in relative relationship to 1/4:1/3 ..coz 3:4 comes from multiplying 60 with 1/4:1/3.

What did u mean by"15 + 20 + 12 = 47 < 60" that means my perception/analogy of sharing 60 dollars into according ratio among 3 persons is wrong as the each share money does not add up to 60. right? oh yes i should have noticed it by summing up the each share that it is not eqaul to 60 ..!
@JeffM
 

JeffM

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It means that, for your analogy to work, there must be more than three shares. As you presented the problem, you were simply given proportions; there was no indication of a relationship to any total. Your analogy can work if there are more than three shares.

Person A: 15 dollars
Person B: 20 dollars
Person C: 12 dollars
Others: 13 dollars.

\(\displaystyle 15 : 20 \equiv 15/60 : 20\60 \equiv 1/4 : 1/3\)

Ratio between A's and B's respective shares.

\(\displaystyle 20 : 12 \equiv 20/60 : 12\60 \equiv 1/3 : 1/5\)

Ratio between B's and C's respective shares.

The total is irrelevant to the relationship between the parts

In your first post, you said "nothing other than getting rid of the fractions." That is exactly correct.

I also said in my second post that I did not like this presentation for the problem as described; I do not find it particularly enlightening.
 

Saumyojit

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In your first post, you said "nothing other than getting rid of the fractions." That is exactly correct.
@JeffM
So the bottomline to convert from fractional ratio to whole ratio i have to multiply by lcm every term as multiplying with a exact mutliple will yield me a whole no quotient and divde by lcm for viceversa . other than this There is no logic here its just the way it is .
For the money eg part i just felt an analogy that time ; yes 13 dollars have to be there ...so i think u cleared everything right?
 

Jomo

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How 3:4 is in relative relationship to 1/4:1/3 ..coz 3:4 comes from multiplying 60 with 1/4:1/3.

What did u mean by"15 + 20 + 12 = 47 < 60" that means my perception/analogy of sharing 60 dollars into according ratio among 3 persons is wrong as the each share money does not add up to 60. right? oh yes i should have noticed it by summing up the each share that it is not eqaul to 60 ..!
@JeffM
Just because you multiply two numbers by 60 does not mean that the numbers will add up to 60. This will only happen when the two numbers initially adds up to 1.

Here is why it makes no sense that the numbers must add up to 60:
Take 1/3:1/4 and multiply by 60. You get 20:30.
Leaving the 1st number the same consider 1/3:3/4. Multiply by 60 yields 20:45
Now how can we multiply 1/3 by 60 and multiply two different numbers by 60 (like 1/4 and 3/4) yet the sum be the same? Note that the two sums starts off with the same number but the sums are different: 20+30 does not equal 20+45 and neither equal 60!
 
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JeffM

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@JeffM
So the bottomline to convert from fractional ratio to whole ratio i have to multiply by lcm every term as multiplying with a exact mutliple will yield me a whole no quotient and divde by lcm for viceversa . other than this There is no logic here its just the way it is .
For the money eg part i just felt an analogy that time ; yes 13 dollars have to be there ...so i think u cleared everything right?
I hope so. It is usual to express ratios in terms of whole numbers to make them understandable. That may take a bit of arithmetic, but no profound mathematical idea is involved.
 

Saumyojit

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Take 1/3:1/4 and multiply by 60. You get 20:30.
it will be 20:15.
this will only happen when the two numbers initially adds up to 1.
the two nos means two nos in the original fractional ratio ? : like if it was 1/2:1/2 -->1/2 +1/2=1 (30 + 30=adds up to 60).
Now how can we multiply 1/3 by 60 and multiply two different numbers by 60 (like 1/4 and 3/4) yet the sum be the same? Note that the two sums starts off with the same number but the sums are different: 20+30 does not equal 20+45 and neither equal 60!
What intution are u trying to give me with this lines.I get that 1/3 is same in both the ratios and the other two terms are diffrnt but can't get what are u trying to say
@Dr.Peterson may be doc can say something intutive which may help with my original doubt why lcm * evry term
 
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