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

To find the number of valid votes obtained by the second candidate, we can proceed step-by-step:
Step 1: Calculate the total number of valid votes.
The total number of votes polled is $15,000$. Since $25\%$ of the votes were invalid, the remaining $100\%-25\%=75\%$ of the votes were valid.

$$\text{Total Valid Votes}=75\%\text{ of }15,000$$

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

Step 2: Determine the percentage of valid votes won by the other candidate.
One candidate received $60\%$ of the total valid votes. Therefore, the other candidate must have received the remaining percentage of the valid votes:

$$\text{Percentage of valid votes for the other candidate}=100\%-60\%=40\%$$

Step 3: Calculate the number of valid votes for the other candidate.

$$\text{Valid votes got by the other candidate}=40\%\text{ of }11,250$$

$$\text{Valid votes got by the other candidate}=\frac{40}{100}\times 11,250=4,500$$

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