If you fit the quantiles to a known distribution, you can calculate any percentile with the distribution's quantile function, which is the inverse of the CDF. However, with only 3 quantiles, any 3-parameter distribution will fit, so you need to choose the distribution beforehand. If possible you should get some raw data or more quantiles. See this link also has some handy R code for fitting quantiles to a distribution using optim()
and the distribution's quantile function.
I've found that income/salary data are best fit by a generalized (aka shifted, aka 3-parameter) log-logistic distribution. The log-logistic also has the advantage of having a closed-form quantile function which is easy to calculate and easy for the optimization library to use. I ended up having to write my own shifted log-logistic quantile function after not finding exactly what I wanted in available R packages:
# Shifted/generalized log-logistic quantile function
# http://en.wikipedia.org/wiki/Shifted_log-logistic_distribution
# The qllog3() function from package FAdist appears to be a different parameterization
# location = mu, scale = sigma, shape = xi
qshllogis <- function(p,location=0,scale=1, shape=0) {
if(shape == 0) {
# Revert to logistic distribution
return( qlogis(p,location,scale) );
}
else {
return(scale * ( (1/p - 1)^(-shape) - 1) / shape + location);
}
}