The half-life of a radioactive substance is 15 hours. If at some instance, the sample has a mass of 512 g…

Analyze the question: The half-life is 15 hours. We need the time for 7/8 of the sample to decay, which means 1/8 of the sample remains.

Concept: After each half-life, the remaining mass is halved:

N = N₀ × (1/2)ⁿ, where n = t / T½

Solution:

• Fraction remaining = 1 − 7/8 = 1/8
• (1/2)ⁿ = 1/8 = (1/2)³, so n = 3
• t = n × T½ = 3 × 15 = 45 hours

Check with the mass (512 g):

• After 15 h: 256 g
• After 30 h: 128 g
• After 45 h: 64 g

The decayed mass is 512 − 64 = 448 g, and 448/512 = 7/8. ✔

Answer: C. 45 hours

Tip: When a question says a fraction "has decayed," always subtract it from 1 to get the fraction remaining. Then write that remainder as a power of ½ (1/2, 1/4, 1/8, 1/16, ...). The power is the number of half-lives. The mass given (512 g) is not needed here, since only the fraction matters.