You have not provided any sample data to work with, so I invented some. I used functions from the dplyr package to join the data and make a new column. If you need more specific help, please provide samples of your data in a reproducible example as explained in the link below.
library(dplyr)
#>
#> Attaching package: 'dplyr'
#> The following objects are masked from 'package:stats':
#>
#> filter, lag
#> The following objects are masked from 'package:base':
#>
#> intersect, setdiff, setequal, union
#invent data
Admit <- data.frame(cancer_type = c("A", "B", "C"), Admissions = c(123, 264, 97))
Deaths <- data.frame(cancer_type = c("B", "C", "A"), Died = c(57, 23, 60))
#Join the data
AllDat <- inner_join(Admit, Deaths, by = "cancer_type")
AllDat
#> cancer_type Admissions Died
#> 1 A 123 60
#> 2 B 264 57
#> 3 C 97 23
#Calculate the rate
AllDat <- AllDat %>% mutate(Rate = Died/Admissions)
AllDat
#> cancer_type Admissions Died Rate
#> 1 A 123 60 0.4878049
#> 2 B 264 57 0.2159091
#> 3 C 97 23 0.2371134
Created on 2021-06-21 by the reprex package (v0.3.0)