12.5% = 1/8 = (1/2)³, so 3 half-lives have passed.
["Understanding 12.5%: How 12.5% Equals 1/8 and Represents 3 Half-Lives", "In science, mathematics, and everyday life, percentages and fractions are powerful tools for expressing proportions. One particularly helpful equivalence to understand is how the percentage 12.5% corresponds to the fraction (\frac{1}{8}) — and why this reflects precisely 3 half-lives. Let’s explore this relationship in detail.", "### 12.5%: A Common Fraction", "12.5% is the decimal equivalent of 12.5 out of 100. To convert this to a fraction:", "[\n12.5% = \frac{12.5}{100} = \frac{125}{1000} = \frac{1}{8}\n]", "So, 12.5% = (\frac{1}{8}). This means 12.5% represents one part out of eight equal parts.", "### Breaking It Down Using Half-Lives", "Understanding the “three half-lives” concept helps visualize exponential decay, especially in nuclear physics, medicine, and environmental science.", "- A half-life is the time required for a quantity (like a radioactive substance) to reduce to half its initial value.\n- After 1 half-life: ( \frac{1}{2} ) remains\n- After 2 half-lives: ( \frac{1}{4} ) remains\n- After 3 half-lives: ( \frac{1}{8} ) remains", "We see that:", "[\n\left(\frac{1}{2}\right)^3 = \frac{1}{2} \ imes \frac{1}{2} \ imes \frac{1}{2} = \frac{1}{8}\n]", "Therefore, three half-lives equal 12.5% (or (\frac{1}{8})) remaining.", "### Why This Matters", "Recognizing this equivalence enables clearer communication and calculations in scenarios involving radioactive decay, compound dilution, or even savings growth where reductions occur over time. Whether you’re interpreting lab data or explaining scientific principles: knowing that 12.5% = (\frac{1}{8}) and corresponds to 3 half-lives offers a solid foundation.", "### Summary", "- 12.5% = 0.125 = (\frac{1}{8})\n- This fraction results from reducing the whole into eight equal halves\n- After 3 half-lives: ( \left(\frac{1}{2}\right)^3 = \frac{1}{8} ) remains\n- Hence, 12.5% = (\frac{1}{8}) = (1/2)³, meaning exactly 3 half-lives have passed", "Understanding this connection simplifies complex decay processes and reinforces the intuitive value of fractions and percentages in real-world science.", "---", "Keywords: 12.5% = 1/8, (1/2)³, half-lives, exponential decay, radioactive decay, fraction explanations, exponential reduction."]
