A man bought a horse and a carriage for Rs. $80,000$. He sold the horse at a gain of $10\%$ and the carriage at a loss of $5\%$. He gained $1\%$ on his whole transaction. What is the cost price of the horse?

Let the cost price of the horse be $x$. Then, the cost price of the carriage will be $80000-x$.
According to the question, the net gain on the entire transaction is $1\%$.

$$\text{Profit from Horse}-\text{Loss from Carriage}=\text{Total Gain}$$

$$\left(\frac{10}{100}\cdot x\right)-\left(\frac{5}{100}\cdot (80000-x)\right)=\frac{1}{100}\cdot 80000$$

Multiply the entire equation by $100$ to eliminate denominators:

$$10x-5(80000-x)=80000$$

$$10x-400000+5x=80000$$

$$15x=480000$$

$$x=\frac{480000}{15}=32000$$

Hence, the cost price of the horse is $\text{Rs. }32,000$.