make_rat <- function(n, d) {
    g <- gcd(n, d)
    pair(n / g, d / g)
} # constructor
numer    <- function(x) { head(x) } # selector
denom    <- function(x) { tail(x) } # selector

add_rat <- function(x, y) {
  make_rat(numer(x) * denom(y) + numer(y) * denom(x),
           denom(x) * denom(y))
}
sub_rat <- function(x, y) {
  make_rat(numer(x) * denom(y) - numer(y) * denom(x),
           denom(x) * denom(y))
}
mul_rat <- function(x, y) {
  make_rat(numer(x) * numer(y),
           denom(x) * denom(y))
}
div_rat <- function(x, y) {
  make_rat(numer(x) * denom(y),
           denom(x) * numer(y))
}
equal_rat <- function(x, y) {
  numer(x) * denom(y) == numer(y) * denom(x);
}