$A _{ t }= A _{0} \times e ^{- k _{ A } t }$

$A _{ t }= A _{0} \times e ^{\left(\frac{-\ln 2}{100} \times t \right)}$

$B _{ t }= B _{0} \times e ^{\left(\frac{-\ln 2}{50} \times t \right)}$

$A _{0}= B _{0}$

$At =4 B _{ t }$

$e ^{-\frac{\ln 2}{100} \times t }=4 \times e ^{-\frac{\ln 2}{50} \times t }$

$e ^{\frac{\ln 2}{100} \times t}=4$

$e ^{\frac{\ln 2}{100} \times t}=4$

$\frac{\ln 2}{100} \times t=\ln 4=2 \ln 2$

$t=200 \,sec$