Two Newly Defined Functions

$\begin{aligned} (22*k)\#(k*33)&=&(\frac{1}{22}+\frac{1}{k})\#(\frac{1}{k}+\frac{1}{33}) \\ &=& \frac{\frac{1}{22}+\frac{1}{k}+\frac{1}{33}+\frac{1}{k}}{\frac{1}{22}+\frac{1}{k}-\frac{1}{33}-\frac{1}{k}} \\ &=& \frac{\frac{5}{66}+\frac{2}{k}}{\frac{1}{66}} \\ &=& 5+\frac{\frac{2}{k}}{\frac{1}{66}}=27 \\ \end{aligned}$

So,

$\frac 2k \div \frac1{66} = 22 \quad\Rightarrow\quad \frac{132}k = 22 \quad\Rightarrow\quad 22k=132 \quad\Rightarrow k=\boxed{6}$

(22 * k) # (k * 33) = 27

[(1/22 + 1/k) + (1/k + 1/33)]/[(1/22 + 1/k) - (1/k + 1/33)] = 27

(5/66 + 2/k)/(1/66) = 27

5/66 + 2/k = 27/66

22/11k = 22/66

Simplifying the eq.

11k = 66

So therefore,

k = 6

22 * k) # (k * 33) = 27 than [(1/22 + 1/k) + (1/k + 1/33)]/[(1/22 + 1/k) - (1/k + 1/33)] = 27; (5/66 + 2/k)/(1/66) = 27; 132/k = 22; so k = 132 /22; k = 6

22 * k) # (k * 33) = 27 than [(1/22 + 1/k) + (1/k + 1/33)]/[(1/22 + 1/k) - (1/k + 1/33)] = 27; (5/66 + 2/k)/(1/66) = 27; 132/k = 22; so k = 132 /22; k = 6 .

(22 * k)#(k * 33)=27
(1/22 + 1/k)#(1/k + 1/33)=27
[(k+22)/22k]#[(33+k)/33k]=27
[(3k+66+2k+66)/66k] / [(3k+66-2k-66)/66k]=27
[(5k+132)/66k] / (1/66)=27
[(5k+132)/66k]*66=27
(5k+132)/k=27
5k+132=27k
22k=132
k=6