gmat1220 wrote:
VeritasPrepKarishma wrote:
Also remember, if a and b are 2 numbers whose HCF and LCM is given,
a*b = HCF*LCM
(Try to figure out why this must be true.)
karishma
I never appreciated the formulas for LCM and HCF of the fractions. I just don't seem to get these formulas, can you throw some light, how?
LCM of a fraction is - LCM of numerator/HCF of denominator. 21/2 is the LCM of 3/4 and 7/6
HCF of a fraction is - HCF of numerator/LCM of denominator. 1/12 is the HCF of 3/4 and 7/6
This looked daunting even during my school days until I started using LCM to compare fractions wherein it did make sense. But not the LCM of the fractions.
LCM/HCF is an important concept as basis for other concepts but LCM/HCF of fractions isn't that important. Over the years, I remember using this concept very rarely. Nevertheless, of course no harm in having a clear understanding.
Algebraic approach
Consider 2 fractions a/b and c/d in their lowest form, their LCM (L1/L2) and HCF (H1/H2) (also in their lowest forms)
LCM should be divisible by both numbers so
L1/L2 is divisible by a/b. This implies L1*b/L2*a is an integer. Since a/b and L1/L2 are in their lowest form, L1 must be divisible by a and b must be divisible by L2.
Similarly, L1 must be divisible by c and d must be divisible by L2.
L1, the numerator of LCM, must be divisible by both a and c and hence should be the LCM of a and c, the numerators. (L1 cannot be just any multiple of a and c; it must be the lowest common multiple so that L1/L2 is the
Lowest Multiple of the two fractions)
b and d both must be divisible by L2 and hence L2 must be their HCF. (Not just any common factor but the
highest common factor so that L1/L2 is the lowest multiple possible)
Using similar reasoning, you can figure out why we find HCF of fractions the way we do.
Now let me give you some feelers. They are more important than the algebraic explanation above. They build intuition.
Let me remind you first that LCM is the lowest common multiple. It is that smallest number which is divisible by both the given numbers.
Say, I have two fractions: 1/4 and 1/2. What is their LCM? It's 1/2 because 1/2 is the smallest fraction which is divisible by both 1/2 and 1/4. (If this is tricky to see, think about their equivalents in decimal form 1/2 = 0.50 and 1/4 = 0.25. You can see that 0.50 is the smallest common multiple they have)
But 1/2 = 2/4. LCM of 2/4 and 1/4 will obviously be 2/4....
What is HCF? It is that greatest number which is common between the two fractions. Again, let's take 1/2 and 1/4. What is greatest common fraction between them? 1/4 (Note that 1/2 and 1/4 are both divisible by other fractions too e.g. 1/8, 1/24 etc but 1/4 is the greatest such common fraction)
On the same lines, what will be the LCM of 2/3 and 1/8. We know that 2/3 = 16/24 and 1/8 = 3/24. what do you think their LCM will be?
16*3/24 = 48/24 = 2
Also, think what will be the HCF of 2/3 and 1/8. We know that 2/3 = 16/24 and 1/8 = 3/24. What is common between the two fractions? 1/24
LCM is a fraction greater than (or equal to) both the fractions. When you take the LCM of the numerator and HCF of the denominator, you are making a fraction greater than (or equal to) either one of the numbers.
HCF is a fraction smaller than (or equal to) both the fractions. When you take the HCF of the numerator and LCM of the denominator, you are making a fraction smaller than (or equal to) either one of the numbers.
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Karishma
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