There are Senior officers and Secretaries in an office. If $15$ Secretaries went on leave, then every Secretary has two Senior officers. After that $45$ Senior officers went on leave and now every Senior officer has $5$ Secretaries. Then find out how many Secretaries were there initially.

Let the initial number of Senior officers be $O$ and the initial number of Secretaries be $S$.
* Condition 1: $15$ Secretaries go on leave, so remaining Secretaries = $S-15$.
Now, every Secretary has two Senior officers, which means the number of Senior officers is twice the number of remaining Secretaries:

$$O=2(S-15)⟹O=2S-30\text{— (Equation 1)}$$

* Condition 2: After this, $45$ Senior officers go on leave, so remaining Senior officers = $O-45$.
Now, every Senior officer has $5$ Secretaries, which means the number of Secretaries is $5$ times the number of remaining Senior officers:

$$S-15=5(O-45)\text{— (Equation 2)}$$

Now, substitute Equation 1 into Equation 2:

$$S-15=5((2S-30)-45)$$

$$S-15=5(2S-75)$$

$$S-15=10S-375$$

$$375-15=10S-S$$

$$360=9S$$

$$S=\frac{360}{9}=40$$

Therefore, there were $40$ Secretaries initially.