B2. In a nursing home for the elderly, the first cases of COVID-19 were identified a week ago. Since then, the number of infected has quintupled, meaning there are now five times as many as there were a week ago. 21 of these infected were taken to the hospital this morning, leaving $9 . \overline{09} \%$ of the remaining residents in the home infected. This evening, another 9 infected will be taken to the hospital, leaving only $5 \%$ of the remaining residents in the home infected. How many were initially infected a week ago, and how many residents were there in the home at that time?
B2. Let there initially be $x$ infected and a total of $y$ caregivers in the house. Of all the infected, $5x$ of them, 21 were taken away, leaving $5x-21$ infected. The number of caregivers left is $y-21$. The proportion of infected is $\frac{5x-21}{y-21}=9.\overline{09}\%$. Converting the repeating decimal $9.\overline{09}$ to a fraction, we get $\frac{100}{11}$ and $9.\overline{09}\%=\frac{1}{11}$. Therefore, the proportion of infected can be written as $\frac{5x-21}{y-21}=\frac{1}{11}$, which can be rearranged to $y=55x-210$. If 9 more infected are taken away, a total of 30 will have been taken away, and the proportion of infected will be $\frac{5x-30}{y-30}=5\%$. Considering that $5\%=\frac{1}{20}$, we rearrange the proportion into the equation $y=100x-570$. By equating both equations, we get $x=8$ and $y=230$, which means there were 230 caregivers in total, of which 8 were infected.
Recording the relationship between the number of infected a week ago and now $x$ and $5x$ ..... 1 point
Recording the number of infected or remaining caregivers $5x-21$ or $y-21$ ..... 1 point
Converting $9.\overline{09}\%$ to $\frac{1}{11}$ ..... 1 point
Writing the correct equation $\frac{5x-21}{y-21}=9.\overline{09}\%$ ..... 1 point
Rearranging the equation to linear form $y=55x-210$ ..... 1 point
Writing the second equation $\frac{5x-30}{y-30}=5\%$ ..... 1 point
Rearranging to linear equation $y=100x-570$ ..... 1 point
Writing the answer. There were 230 caregivers in total, of which 8 were infected. ..... 1 point