library(rootSolve)

d <- read.table(text="indx rate n d
            1   0.12    158 14
            2   0.095   135 9
            3  0.057    123 4
            4  0.033    115 5
            5    0.019  90  4", header=T)

d$real <- with(d, d/n)

opt <- d[ ,c("rate","real", "n")]

# this is close to the correct solution!

scaler <- apply(opt, 1, function(z) uniroot.all(
function(alpha) z[2] - (1 / (1 + alpha * ( (1 - z[1]) / z[1] )) ), interval = c(0,10)))

# check for solution (not fully correct!)

round(crossprod(scaler * opt$real, opt$n)/sum(opt$n), 3) == round(crossprod(round(opt$rate, 3), opt$n)/sum(opt$n), 3)


# using optim() - completely wrong results

infun <- function(data, alpha){ l <- with(data,
( rate - (1 / ( 1 + alpha[1] * ( (1 - real)/real ))) )); return( -sum( l ) ) }

opt_out <- optim(c(0,0), infun, data=opt, method = "BFGS", hessian = TRUE)

with(opt, (1 / ( 1 + opt_out$par[1] * ( (1 - real)/real ))))