Aptitude - Problems on H.C.F and L.C.M - Discussion

If we take LCM collectively, the HCF is 2 and 3, if one by one it's 3. How can we recognise that the HCF should be taken separately?

Answer Should be 3/2800.

Because,
H.C.F of 9, 12, 18, 21 is 3.
L.C.M of 10, 25, 35, 40 is 2800.

So, the answer is 3/2800.

Simply

Here HCF of 9 =3 * 3, 12 = 3 * 2 * 2, 18 = 3 * 3 * 2, and 21 = 3 * 7.
HCF = 3.

LCM of 10 =2 * 5, 25 = 5* 5, 35 = 3 * 7, 40 = 2* 2 * 2 * 5.
LCM = 1400.

Required HCF = HCF/LCM.
= 3/1400.

Actually, the answer is showing 3/2800. That's why getting confused. I also got the result 3/1400.

Thats long process, tell me short cut method.

Why we have taken LCM of denominator?

Can you please explain shortcut?

What is the need for so much of calculations?

Since we know that HCF of any fraction is computed with the following formula = HCF of Numerators/LCM of denominators.

Now see the Options:

Numerators has to be less than the given numbers (9, 12, 18, 21) since, it is HCF. This rules out Options B and D. We are left with options A and C.

Now denominator is the LCM which shall be more than the given number i.e. 10, 25, 35, 40. Therefore Options C is the correct answer.

The concept behind hcf/lcm is that suppose we take 9/10 now there should be a fraction a/b that exactly divides it (9/10) / (a/b) = (9*b) / (10*a) now you take a as hcf of all numbers (numerators only) that will exactly divide 9.

Now for b if you take hcf of denominators it will exactly divide 10 but now there will be a number in denominator of our expression giving us fraction but if we rather take lcm then 10 will go in integral number to divide b and we will get integer.

9/10, 12/25, 18/35, 21/40 is the given problem.
H.C.F of numerators,

9 = 3*3.
12 = 3*2*2.
18 = 2*3*3.
21 = 7*3.

Common least factors of these numbers is 3(h.c.f).

L.C.M of denominators,
10 = 2*5.
25 = 5*5.
35 = 7*5.
40 = 5*2*2*2.

Highest factors of all these numbers is 1400(2*2*2*5*5*7).

So final answer is 3/1400.