$\text{A}$ and $\text{B}$ each can do a work in $75\text{ days}$. They began the work together but $\text{A}$ left after some time and $\text{B}$ completed the remaining work in $23\text{ days}$. After how many days of the start of the work did $\text{A}$ leave?

Let the number of days $\text{A}$ worked before leaving be $x$.
* Work rate of $\text{A}=\frac{1}{75}$ per day
* Work rate of $\text{B}=\frac{1}{75}$ per day

For the first $x$ days, both $\text{A}$ and $\text{B}$ worked together:

$$\text{Combined rate}=\frac{1}{75}+\frac{1}{75}=\frac{2}{75}$$

$$\text{Work done in }x\text{ days}=x\times \frac{2}{75}=\frac{2x}{75}$$

For the last $23\text{ days}$, $\text{B}$ worked completely alone:

$$\text{Work done by B alone}=23\times \frac{1}{75}=\frac{23}{75}$$

The sum of work components equals $1$ total completed job:

$$\frac{2x}{75}+\frac{23}{75}=1$$

$$2x+23=75$$

$$2x=52$$

$$x=26\text{ days}$$

Thus, $\text{A}$ left the work after $26\text{ days}$.