Does Luke 13:24 give us a more vivid picture of how many people will be saved?

While I believe that it is through faith alone we are saved, the only caveat I would like to put forward is the parable of the sheep and the goats in Matthew 25.

31 “When the Son of Man comes in his glory, and all the angels with him, he will sit on his glorious throne. 32 All the nations will be gathered before him, and he will separate the people one from another as a shepherd separates the sheep from the goats. 33 He will put the sheep on his right and the goats on his left. 34 “Then the King will say to those on his right, ‘Come, you who are blessed by my Father; take your inheritance, the kingdom prepared for you since the creation of the world. 35 For I was hungry and you gave me something to eat, I was thirsty and you gave me something to drink, I was a stranger and you invited me in, 36 I needed clothes and you clothed me, I was sick and you looked after me, I was in prison and you came to visit me.’ 37 “Then the righteous will answer him, ‘Lord, when did we see you hungry and feed you, or thirsty and give you something to drink? 38 When did we see you a stranger and invite you in, or needing clothes and clothe you? 39 When did we see you sick or in prison and go to visit you?’ 40 “The King will reply, ‘Truly I tell you, whatever you did for one of the least of these brothers and sisters of mine, you did for me.’ 41 “Then he will say to those on his left, ‘Depart from me, you who are cursed, into the eternal fire prepared for the devil and his angels. 42 For I was hungry and you gave me nothing to eat, I was thirsty and you gave me nothing to drink, 43 I was a stranger and you did not invite me in, I needed clothes and you did not clothe me, I was sick and in prison and you did not look after me.’ 44 “They also will answer, ‘Lord, when did we see you hungry or thirsty or a stranger or needing clothes or sick or in prison, and did not help you?’ 45 “He will reply, ‘Truly I tell you, whatever you did not do for one of the least of these, you did not do for me.’ 46 “Then they will go away to eternal punishment, but the righteous to eternal life.”

There is an implication that faith alone is not enough (without going into James 2:26.)

Luke 13:22-28 (NIV)

Then Jesus went through the towns and villages, teaching as he made his way to Jerusalem. Someone asked him, “Lord, are only a few people going to be saved?” He said to them, “Make every effort to enter through the narrow door, because many, I tell you, will try to enter and will not be able to. Once the owner of the house gets up and closes the door, you will stand outside knocking and pleading, ‘Sir, open the door for us.’ “But he will answer, ‘I don’t know you or where you come from.’ “Then you will say, ‘We ate and drank with you, and you taught in our streets.’ “But he will reply, ‘I don’t know you or where you come from. Away from me, all you evildoers!’ “There will be weeping there, and gnashing of teeth, when you see Abraham, Isaac and Jacob and all the prophets in the kingdom of God, but you yourselves thrown out.

When Jesus was asked how many people will be saved, Jesus did not reply directly that only a few will be saved. Jesus simply replied that many will try to enter but will fail, thereby answering indirectly that only a few people will be saved.

If we read the entire passage, we can understand that Jesus was speaking about the Jews specifically. The Jews at that time were trying very hard to enter the kingdom of God but their efforts were only outward and no effect internally. The people emphasizes more on rules and regulations rather than loving God and loving others.

“Woe to you, teachers of the law and Pharisees, you hypocrites! You clean the outside of the cup and dish, but inside they are full of greed and self-indulgence. Blind Pharisee! First clean the inside of the cup and dish, and then the outside also will be clean. (Matthew 23:25-26, NIV)

The people were trying very hard to remain ceremonially clean and to fulfill their religious duties. They paid offerings and tithes faithfully, they observed the Sabbath day strictly. They were perfect outwardly but sinful inwardly. They hearts were filled with hatred and greed.

The same situation is going on today among many Christians. We try to attain salvation by our own deeds and we try to enter heaven by our own works. In contrast, salvation is a free gift from God and there nothing we can do about it. All we have to do you receive it with faith, let our old self die and put on the new self.

You were taught, with regard to your former way of life, to put off your old self, which is being corrupted by its deceitful desires; to be made new in the attitude of your minds; and to put on the new self, created to be like God in true righteousness and holiness. (Ephesians 4:22-24, NIV)

Unless we are changed internally, put off our old nature and live a new live, we can never enter the kingdom of God. We need to born again spiritually. We need a transformation in our life. It's not about paying tithe or going to church every Sunday. Your piety alone cannot save you. Your hard work alone cannot save you. If you are only trying to be saved, you are like those who try but fail. God doesn't care about how much you try to make him happy but instead, God wants you to strife hard to change yourself.