A certain city in the king’s empire had failed to pay its taxes to the royal treasury. Angry at the long overdue liability, the king came with a mighty army to collect the debt. When the king was ten miles from the city, the mayor and the city administrators came out to meet him. Since their treasury was empty, they asked if he would be willing to cancel the debt. Feeling sorry for them, the king reduced the amount due by one-third.
When the king came closer, the merchants and storekeepers of the city came to greet him and asked him to release them of the remaining debt. Feeling sorry for them, the king canceled another one-third of the liability. When the king arrived at the city gate, the entire population came out to welcome him. Seeing this massive demonstration of loyalty, the king forgave the entire debt!
The Tur (Orach Chaim 581) says: On Erev Rosh Hashanah, the devout and the pious Jews fast. In their merit, Hashem forgives ‘one-third’ of the Jewish people’s transgressions. During the Aseres Yemei Teshuvah, the Ten Days of Repentance (which are the days between Rosh Hashanah and Yom Kippur) even some average Jews fast, and in their merit, Hashem forgives another ‘third’ of Jewish people’s transgression. Comes Yom Kippur, where all the Jewish people fast – where Hashem forgives all of our transgressions. (Rosh Hashanah and Yom Kippur secrets)