Two radioactive isotopes, A and B, have half-lives of 20 minutes and 40 minutes, respectively. If both samples start with an identical number of nuclei, determine the ratio of the number of decayed nuclei of A to those of B after a period of 80 minutes.

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EEJ MAIN Physics QUESTION #7072

Question 1

Two radioactive isotopes, $A$ and $B$, have half-lives of $20$ minutes and $40$ minutes, respectively. If both samples start with an identical number of nuclei, determine the ratio of the number of decayed nuclei of $A$ to those of $B$ after a period of $80$ minutes.

$4:1$
$1:4$
$5:4$✔️
$1:16$

Correct Answer Explanation

After $80$ minutes, $A$ undergoes $4$ half-lives ($80/20$), leaving $N_A = N_0/2^4 = N_0/16$. The number of decayed nuclei for $A$ is $N_0 - N_0/16 = 15N_0/16$. $B$ undergoes $2$ half-lives ($80/40$), leaving $N_B = N_0/2^2 = N_0/4$. The number of decayed nuclei for $B$ is $N_0 - N_0/4 = 3N_0/4 = 12N_0/16$. The ratio of decayed nuclei is $15:12$, which simplifies to $5:4$.

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