How long was Jesus's ministry?

According to New Advent's Catholic Encyclopedia, Yeshua's ministry lasted about one year. See the writings of the early Church fathers Clement of Alexandria, Tertullian, Origen and Lactantius. It was Eusebius, in the 4th century, who suggested a duration of 3 1/2 years, trying to align it with his interpretation of the prophet Daniel's vision in Chapter 9:20-27.

See "Chronology of the Life of Jesus Christ", under the heading "Relative chronology", sub-headed "The public life of Jesus: its duration".

Luke 3:1 dates the beginning of John the Baptist's mission to 28 or 29 CE. The gospels also say that the last supper was a Passover seder, and that Jesus died before sunset on a Friday. This makes the best candidates for the date of the crucifixion 30 and 33 CE (Humphreys and Waddington, "Dating the crucifixion," Nature, 1983. doi:10.1038/306743a0). For these reasons, most scholars these days seem to agree that the length of Jesus's ministry was either very short (about one year) or else about three years.

There is no secure way to determine which of these possibilities is right.

Although John seems to identify three Passovers during this period (2:13, 6:4, and 13:1), this is a shaky basis on which to rule out the one-year-ministry possibility, since the gospels are literary works and often arranged by topics, not chronology. John in particular is concerned with theology over history, or even narrative. The sequence of events in John is wildly different from what the synoptics give (with the assault on the temple, John 2:13, occurring before the rest of the ministry). Geza Vermes actually argues on similar grounds that Jesus's ministry was less than a year, since Jesus was obligated to make three pilgrimages every year to the temple, and Mark only describes one, on one Passover. (Vermes, The authentic gospel of Jesus, 2003, Penguin paperback edition, p. 371.) John 6:4 is also not present in all manuscripts, which suggests that the added features in his account of the loaves and fishes (going up into the mountains, a Passover feast) are embellishments added long after the fact, in order to theologize the miracle and heighten the aura of spirituality.

Wheat ripens in May, suggesting it was May when the disciples were plucking ears of grain soon after the beginning of Jesus' public mission (Mark 2:23). On this evidence, Jesus began his public mission around the time of the Passover, and of course his crucifixion took place at the time of the Passover. We can know the period of Jesus' public ministry if we know the number of annual Passover celebrations that occurred during this period.

There is general agreement that John's Gospel mentions the annual Passover celebration on at least three separate occasions. The first of these (John 2:13) is consistent with the event mentioned in Mark 2:23. Then, John 6:4 speaks of a second Passover on which Jesus did not travel to Jerusalem, as it occurred during the feeding of the five thousand. A third Passover (John 11:55) marks Jesus' final entry to Jerusalem. A possible fourth Passover is alluded to in John 4:54, where Jesus returns from Judea to Galilee. This would mean that his ministry took between two and three years.

Although John places the feeding of the five thousand at the time of the Passover, the synoptic gospels make no mention of this. Moreover, in Mark's Gospel, the story of feeding the five thousand, narrated in Mark 6:32-44, seems to occur only shortly before Jesus begins his final journey to Jerusalem, and there can not be two Passovers in the same year. The brevity of the synoptic accounts, especially in Mark's Gospel, and the absence of any journey to Jerusalem until the final, fateful trip suggests that Jesus' mission lasted about one year, but the parallel between John 6:4 and Mark 6:32-44 makes it at least possible that the public mission could have been for just two years.

Wikipedia says:

The three Synoptic gospels refer to just one passover during his ministry, while the Gospel of John refers to three passovers, suggesting a period of about three years. However, the Synoptic gospels do not require a ministry that lasted only one year.

Paula Fredriksen, in 'The Historical Jesus, the Scene in the Temple, and the Gospel of John', published in John, Jesus, and History, Volume 2 page 250, says that in Matthew, Mark and Luke, the implied period is about one year and that in John the journeys to Jerusalem require a period of over two years.

1. In the 15th year of Tiberius (Aug. 28AD to Aug. 29AD), John the Baptist began his public ministry (summer of 28AD).
2. At that time Jesus was "about thirty years old", which statement would be precisely true if Jesus were born in 3BC. (Luke 3:23) a. The gospel of John records three Passovers during Jesus' ministry. The first when Jesus was 30½ years old (John 2:13), a second when He was 31½ years old (John 6:4) and His last when He was arrested and killed. (John 13:1) So, we see that His ministry lasted about 2½ years and He died at the age of 32½ years.

I understand that there is some disagreement over how old Jesus was, and how long His earthly ministry lasted, but as your question states, most scholars believe He started at about 30 years old, and it lasted 3-3^1/2 years based on tying events mentioned with other established historical dates.

From http://www.ucg.org/jesus-christ/bible-and-archaeology-jesus-christs-early-ministry/

Archaeologists generally date the start of Christ's ministry to the year A.D. 27. "The beginning of Jesus' public ministry," writes archaeology professor John McRay, "is dated by synchronisms [chronological arrangements of events and people] in the Gospel of Luke (3:1-2). A date of A.D. 27 seems likely . . . The dates mentioned by Luke are rather well established . . ." ( Archaeology and the New Testament, 1997, p. 160).

However, I wouldn't be dogmatic on the age or number of years. Scripture isn't explicit, and there is room for error here, leading to several other interpretations, so it might be best to answer "we don't know, but this is why we think..."

Scholars who have attempted a synchronized harmony of the Gospels seem to indicate the Jesus went up to Jerusalem on three different occasions. From this, the tradition has grown up that his ministry was three years.

Beyond this interpolation, however, there is nothing to suggest how long his earthly ministry actually lasted. As such, it is probably best to say his ministry lasted at least three years.

Then again, after reading Mark keep saying "and he went immediately", you have gotten the impression it was just a few days :). But, overall, it does seem that the various times Jesus shows up in Jerusalem lend weight to at least 3 years.