Getting the value of $n$

Question

It's strange. I can't get the value of $n$. Could someone give me the step by step way of getting the value of $n$. The answer key says $20$.

$$ 1+\frac{i}{n}=\frac{1+\frac{i}{4}}{1+\frac{i}{5}} $$

Answer 1

Hint

So, you have $$1+\frac{i}{n}=\frac{1+\frac{i}{4}}{1+\frac{i}{5}}$$ Simplify the rhs so $$1+\frac{i}{n}=\frac{5 (i+4)}{4 (i+5)}$$ Substract $1$ from each side; reduce to same denominator to get $\frac{i}{n}$; take the inverse to get $\frac{n}{i}$; multiply both sides by $i$ to get $n$ as a function of $i$.

I am sure that you can take from here.

Answer 2

I have got

$$1+\frac{i}{n}=\frac{20+5i}{20+4i},$$

$$\frac{i}{n}=\frac{i}{2+4i},$$

and thus $n=4(5+i)$.