1. Problem 1: Find the least 6-digit number which is a perfect square. - The smallest 6-digit number is 100000. - We need to find the smallest integer $n$ such that $n^2 \geq 100000$. - Calculate $n = \lceil \sqrt{100000} \rceil$. 2. Calculate $\sqrt{100000}$: $$\sqrt{100000} = \sqrt{10^5} = 10^{2.5} = 316.2277...$$ 3. The smallest integer $n$ is 317. 4. Calculate $317^2$: $$317^2 = 317 \times 317 = 100489$$ 5. Check the options given: 100489, 100588, 100688, 100788. - 100489 is a perfect square and the smallest 6-digit perfect square. 6. Problem 2: If 12.5% of $p$ equals 50% of $q$, find the ratio $p:q$. - Write the equation: $$0.125p = 0.5q$$ - Divide both sides by $q$: $$0.125 \frac{p}{q} = 0.5$$ - Solve for $\frac{p}{q}$: $$\frac{p}{q} = \frac{0.5}{0.125} = 4$$ - Therefore, the ratio is: $$p : q = 4 : 1$$ Final answers: - Least 6-digit perfect square is $100489$. - Ratio $p : q = 4 : 1$.