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Gadolinium-153, used to detect osteoporosis, has a half-life of 242 days. Which value is closest to the percentage of Gd-153 remaining in a patient after 2 years?

1. 33%
2. 26%
3. 12.0%
4. 6.25%

Correct answer: 12.0%

Solution

2 years = 2*365 = 730 days. Number of half-lives n = 730/242 ≈ 3.02. Fraction remaining = (1/2)³.02 ≈ (0.5)³ * (0.5)⁰.02 ≈ 0.125 * 0.986 ≈ 0.123, i.e., about 12.3%. The closest option is 12.0%.

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