The distance between New Orleans and Houston is $353$ miles.

At $12:20\mathrm{PM}$, a bus leaves Houston for New Orleans at a speed of $60\mathrm{mph}$.

Forty-five minutes later, a motorcycle leaves New Orleans for Houston at a speed of $72\mathrm{mph}$.

At what time will the bus and the motorcycle pass each other if neither stops nor changes speed?

Determine the time at which the vehicles meet:

The total distance is $353$ miles.

Assume that the time taken by the vehicles to meet after the motor cycle leaves is $t$ hours.

Before the start of the motorcycle, the bus has already moved for $45\min$.

So, the distance traveled by bus in $45$ minutes$=60\times \frac{45}{60}\mathrm{miles}$

In these $t$ hours, there will be some distance covered by the bus and some distance covered by the motorcycle.

So, the total of all three distances will be $353$ miles.

As we know, $\mathrm{Distance}=\mathrm{speed}\times \mathrm{time}$

Express the distance in terms of the time $t$ and solve:

$\begin{array}{rcl}353& =& 60\times \frac{45}{60}+60\times t+72\times t\\ & \Rightarrow & 132t=353-45\\ & \Rightarrow & 132t=308\\ & \Rightarrow & t=\frac{7}{3}\end{array}$

$\frac{7}{3}$ hours is equivalent to $2\mathrm{hr}20\min$.

Calculate the time that is $45$ minutes and $2\mathrm{hr}20\min$ past $12:20\mathrm{PM}$:

$\begin{array}{rcl}\mathrm{Total}\mathrm{Time}& =& 12:20\mathrm{PM}+45\min +2\mathrm{hr}20\min \\ & \Rightarrow & \mathrm{Total}\mathrm{Time}=03:25\mathrm{PM}\end{array}$

Hence, the vehicles meet at $03:25\mathrm{PM}$