Question 58
The first COVID-19 infections in a care home were identified one week ago. Since then, the number of infected residents has increased fivefold, so it is now five times the number one week ago. This morning, $21$ of the infected residents were taken to hospital, leaving $9.\overline{09}\%$ of the residents remaining in the home infected. This evening, another $9$ infected residents will be taken to hospital, so only $5\%$ of the residents who remain will be infected. How many residents were infected one week ago, and how many residents were in the home in total at that time?