SOLUTION: One day a king left his castle with a bag of silver coins to wander his kingdom. To the first peasant he met, he gave one-half his coins plus two more. A little later, he met ano

Question 1151492: One day a king left his castle with a bag of silver coins to wander his kingdom. To the first peasant he met, he gave one-half his coins plus two more. A little later, he met another peasant to whom he gave half his coins plus two more. Walking on, he met a third peasant and again gave half his coins plus two more. Finally, the king went home with two coins left in his bag. How many coins did he have to begin with?
Answer by greenestamps(13200) (Show Source): You can put this solution on YOUR website!

Working this problem "forward", by letting x be the number of coins the king starts with, and trying to form expressions involving x for the number he has aafter meeting each peasant, gets very messy. It is better to work the problem "backwards", letting x be the number of coins he ends with and working back in time to find the number he started with.

You can do that with logical reasoning and simple arithmetic. It might start something like this....

He finished with 2 coins, prior to which he gave the third peasant half his coins plus 2 more. So before he gave the peasant the 2 more, he had 2+2=4; and then before he gave that peasant half his coins, he had 4*2=8 coins.

Then continue working back through the meetings with the other two peasants to find the number the king started with.

But the problem lends itself to a more formal mathematical solution which uses a useful mathematical skill.

Each time the king meets a peasant, he gives the peasant half of the coins plus 2 more. Think of the relationship between the number of coins the king has before each meeting with a peasant and the number he has after the meeting.

If he starts with x coins and gives away half of them plus 2 more, then the number he has left is

Think of this as a function telling how many coins he has after a meeting with a peasant in terms of the number he has before the meeting:

The inverse of a function "gets you back where you started". So the inverse of this function will tell you how many coins the king had before meeting each peasant if you know how many he had after the meeting.

Finding the inverse of a function is easy for simple function like this one.

This function takes the input value, divides it by 2, and then subtracts 2 more.

The inverse function, to get you back where you started, has to do the opposite operations in the opposite order -- add 2 and then multiply by 2:

x --> x+2 --> 2(x+2)

Note that is exactly what I showed above to get from the final 2 coins to the number the king had before meeting the last peasant: 2 --> 2+2=4 --> 2*4=8.

So, now going away from the formal mathematics, working backwards through each meeting with a peasant, the way the number of coins the king has left is determined by the rule "add 2 then multiply by 2". So...

before the third peasant: 2 plus 2 = 4; times 2 is 8
before the second peasant: 8 plus 2 = 10; times 2 is 20
before the first peasant: 20 plus 2 = 22; times 2 = 44

ANSWER: The king started with 44 coins.

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