A trader sold a bicycle at a profit of $10\%$. Had he bought the bicycle at $10\%$ less price and sold it at a price $\text{Rs. }12$ more, he would have gained $25\%$. What is the cost price of the bicycle?

Let the initial cost price $({\text{CP}}_1)$ be $100x$.
Since it is sold at a profit of $10\%$, the initial selling price $({\text{SP}}_1)$ is:

$${\text{SP}}_1=110x$$

Condition 2: Had he bought it at $10\%$ less price:

$$\text{New Cost Price }({\text{CP}}_2)=100x-10\%\text{ of }100x=90x$$

And sold it for $\text{Rs. }12$ more:

$$\text{New Selling Price }({\text{SP}}_2)=110x+12$$

The new profit gained is $25\%$:

$${\text{SP}}_2={\text{CP}}_2\times \left(1+\frac{25}{100}\right)$$

$$110x+12=90x\times 1.25$$

$$110x+12=112.5x$$

$$112.5x-110x=12$$

$$2.5x=12$$

$$x=\frac{12}{2.5}=4.8$$

The original cost price of the bicycle is:

$${\text{CP}}_1=100x=100\times 4.8=\text{Rs. }480$$