A student’s scores on the first four Marquette Interscholastic Mathematics Competitions were 5, 4, 3, and 3. What score does she need on the fifth contest to finish with an average (mean) score of 4 on the five contests?

(Source: Adapted from Mathematics Teaching in the Middle School, Mar-Apr 1997)

Suppose the first four scores contained two pairs of equal scores, g and h. After the fifth contest, the team has an average of the difference between g and h. If the lowest possible score is 0, then what is the greatest ratio between g and h?