1. **State the problem:** We need to find two positive numbers where the difference between them is 125, and one number is 6 times as large as the other. 2. **Define variables:** Let the smaller number be $x$. 3. **Express the larger number:** Since one number is 6 times the other, the larger number is $6x$. 4. **Write the equation for the difference:** The difference between the larger and smaller number is 125, so: $$6x - x = 125$$ 5. **Simplify the equation:** $$5x = 125$$ 6. **Solve for $x$:** $$x = \frac{125}{5}$$ $$x = 25$$ 7. **Find the larger number:** $$6x = 6 \times 25 = 150$$ 8. **Answer:** - The smaller number is $25$. - The larger number is $150$.