Let’s analyze each comparison one by one and explain the reasoning:

a) $\sqrt{5}$ and 2.5

1. Calculate $\sqrt{5} \approx 2.236$.
2. Compare: $2.236 < 2.5$.
3. Conclusion: $\sqrt{5} < 2.5$.

Reasoning: Since $\sqrt{5} \approx 2.236$, which is less than 2.5, $\sqrt{5} < 2.5$.

b) $\sqrt{10}$ and $3 \frac{2}{3}$

1. Calculate $\sqrt{10} \approx 3.162$.
2. Convert $3 \frac{2}{3}$ to a decimal: $3 \frac{2}{3} = 3.666$.
3. Compare: $3.162 < 3.666$.
4. Conclusion: $\sqrt{10} < 3 \frac{2}{3}$.

Reasoning: Since $\sqrt{10} \approx 3.162$, which is less than $3.666$, we have $\sqrt{10} < 3 \frac{2}{3}$.

c) 113% and $\sqrt{1.2996}$

1. Convert 113% to a decimal: 113% = 1.13.
2. Calculate $\sqrt{1.2996} \approx 1.14$.
3. Compare: $1.13 < 1.14$.
4. Conclusion: $113\% < \sqrt{1.2996}$.

Reasoning: Since $1.13 < 1.14$, $113\% < \sqrt{1.2996}$.

d) 0.03% and $\frac{3}{1000}$

1. Convert 0.03% to a decimal: $0.03\% = 0.0003$.
2. Convert $\frac{3}{1000}$ to a decimal: $\frac{3}{1000} = 0.003$.
3. Compare: $0.0003 < 0.003$.
4. Conclusion: $0.03\% < \frac{3}{1000}$.

Reasoning: Since 0.0003 is less than 0.003, $0.03\% < \frac{3}{1000}$.

e) 0.0009 and 3%

1. Convert 3% to a decimal: $3\% = 0.03$.
2. Compare: $0.0009 < 0.03$.
3. Conclusion: $0.0009 < 3\%$.

Reasoning: Since 0.0009 is less than 0.03, $0.0009 < 3\%$.

Would you like further explanation on any of these comparisons?

Here are some related questions to consider:

1. How can we approximate square roots without a calculator?
2. What are other methods for comparing percentages and decimals?
3. How do we convert fractions to decimals efficiently?
4. Why is it important to understand percentage conversions in mathematics?
5. When is it more efficient to compare numbers directly as fractions or decimals?

Tip: Always convert percentages and fractions to decimals for easier comparison.