In an election between two candidates, one received $60\%$ of the total valid votes. $25\%$ of the total votes were invalid. If the total number of votes cast was $15,000$, find the number of valid votes received by the other candidate.

Let's calculate the votes step-by-step:
1. Total votes cast:

$$\text{Total votes}=15,000$$

2. Valid votes:
Since $25\%$ of the votes were invalid, the valid votes constitute $100\%-25\%=75\%$ of the total votes.

$$\text{Total valid votes}=15,000\times \frac{75}{100}=11,250$$

3. Votes received by the second candidate:
The first candidate received $60\%$ of the valid votes, which means the second candidate received $100\%-60\%=40\%$ of the valid votes.

$$\text{Valid votes for the second candidate}=11,250\times \frac{40}{100}=4,500$$

Therefore, the other candidate received $4,500$ valid votes.