If the ratio of two numbers is 3 : 16 and the product of their LCM and their HCF is 432, then the sum of the reciprocals of the LCM and the HCF is:​

Correct option is C

Given:

The ratio of two numbers is 3 : 16

The product of their LCM (Least Common Multiple) and HCF (Highest Common Factor) is 432

Formula Used:

LCM × HCF = Number 1 × Number 2

Solution:

Let the two numbers be 3x and 16x

The product of the two numbers is:

(3x) × (16x) = 432

​$$48x^2 = 432$$​​

​$$x^2 = \frac{432}{48} = 9$$​​

x = 3

Now,  two numbers are: 3x = 9, 16x = 48

The HCF of 9 and 48 is 3

The LCM of 9 and 48 = $$\frac{9 \times 48}{\text{HCF}}$$​= $$\frac{9 \times 48}{3}$$​ = 144 

Now, 

​$$\frac{1}{\text{LCM}} + \frac{1}{\text{HCF}} = \frac{1}{144} + \frac{1}{3}$$​​ = $$\frac{48}{144}$$​​

Thus, the sum is = $$\frac{1}{144} + \frac{48}{144} = \frac{49}{144}$$​​