Add the unit fractions one at a time, reducing after each addition:
procedure toRat(e; var m,n)
    require 0 <= e.i <= N and every stored denominator is positive
    m := 0
    n := 1
    j := 0
    while j < e.i do
        q := e.x[j]
        m := m*q+n
        n := n*q
        Reduce(m,n)
        j := j+1
The loop invariant is that is the reduced fraction equal to the sum of the first terms, with . The update is exactly , and the greatest common divisor reduction preserves it. The loop variant decreases to zero. The empty sum gives .
An arbitrary stored Egyptian sum can be at least one, so this uses the broader nonnegative-numerator contract of the implementation in part (a), not an unjustified assumption that every intermediate fraction is proper.

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