\(m=m_0 \times (\frac{1}{2})^\frac{t}{T}\)
m - the mass that remains unchanged, m₀ - the initial mass, t - time, T - the half-life
\(T=269 \ years \\
m_0=60 \ g \\
m=60 \ g - 52.5 \ g=7.5 \ g \\ \\
7.5 = 60 \times (\frac{1}{2})^\frac{t}{269} \\
\frac{7.5}{60} = (\frac{1}{2})^\frac{t}{269} \\
\frac{1}{8}=(\frac{1}{2})^\frac{t}{269} \\
(\frac{1}{2})^3 = (\frac{1}{2})^\frac{t}{269} \\
3=\frac{t}{269} \\
3 \times 269=t \\
t=807\)
It will take 807 years for 52.5 g of a 60 g sample to decay to its daughter isotope.
