Six numbers a, b, c, d, e, f are such that ab = 1, bc = $$\frac{1}{2}$$​, cd = 6, de = 2 and ef = $$\frac{1}{2}$$​. What is the value of (ad : be : cf)?

Correct option is C

Given:

We are given the following equations:
ab = 1
bc = $$\frac{1}{2}$$​​
cd = 6
de = 2
ef = $$\frac{1}{2}$$​​
We need to find the value of (ad : be : cf).

Formula Used:

We used the following steps:
Express each variable in terms of 'a'.
Substitute and solve for each variable.
Calculate the values of ad, be, and cf, then simplify the ratio.

Solution:

Solve for each variable in terms of 'a'.
From equation (1), ab = 1, we get b = $$\frac{1}{a}$$​​
From equation (2), bc = $$\frac{1}{2}$$​, substituting b = $$\frac{1}{a}$$​ gives c = $$\frac{a}{2}$$​.

From equation (3), cd = 6, substituting c = $$\frac{a}{2}$$​ gives d = $$\frac{12}{a}$$ ​

From equation (4), de = 2, substituting d = $$\frac{12}{a }$$​gives e =$$\frac{a}{6}$$​.

From equation (5), ef = $$\frac{ 1}{2}$$​, substituting e = $$\frac{a}{6}$$​ gives $$f = \frac{3}{a}.$$​​
Calculate ad, be, and cf.
​$$ad = a \times (\frac{12}{a}) = 12 \\\ \\be = (\frac{1}{a}) \times (\frac{a}{6}) = \frac{1}{6} \\\ \\cf = (\frac{a}{2}) \times (\frac{3}{a}) = \frac{3}{2}$$​​
Find the ratio ad : be : cf.
ad : be : cf = $$12 : \frac{1}{6} : \frac{3}{2}$$​​
To simplify this, multiply each term by 6 (LCM of 6 and 2):
72 : 1 : 9
Thus, the ratio ad : be : cf is 72 : 1 : 9.