The mean proportional between $0.02$ and $0.32$ is

### Concept & Formula The mean proportional $x$ between two quantities $a$ and $b$ is given by the geometric mean formula: $$ x = \sqrt{a \times b} $$ When dealing with decimals, convert them to fractions for easier and safer calculation. ### Step-by-Step Solution 1. Identify the values: $a = 0.02$ and $b = 0.32$. 2. Express the decimals as fractions: $a = \frac{2}{100}$ and $b = \frac{32}{100}$ 3. Substitute into the mean proportional formula: $$ x = \sqrt{\frac{2}{100} \times \frac{32}{100}} $$ 4. Multiply the numerators and denominators together: $$ x = \sqrt{\frac{64}{10000}} $$ 5. Extract the square roots of the numerator and denominator independently: $x = \frac{\sqrt{64}}{\sqrt{10000}} = \frac{8}{100}$ 6. Convert back to decimal format: $x = 0.08$ ### Exam Strategy & Shortcut Multiply the significant figures: $2 \times 32 = 64$. The square root of 64 is 8. Since there are 4 decimal places in total between the two original numbers (2 in 0.02 and 2 in 0.32), their square root will have exactly half that amount, which is 2 decimal places. Thus, 8 becomes 0.08. ### Common Pitfall Miscounting the decimal places in the final step. Many students mistakenly answer 0.8 (which would only be correct if the original product had exactly two decimal places, not four). ### Final Answer Therefore, the correct answer is **$0.08$**.