This year my teacher read us the book The Phantom Tollbooth, and since then I've been having strange dreams about drving in usuaual places and paying tolls unusual characters.Last night I dreamed about highway in the clouds near the mythical Mt. Olympus. At my first tollbooth a mythical goddess asked for one-half my gold pieces plus two more. At the next tollbooth,a Greek hero demanded one-half my remaining gold pieces plus two more. Then at third booth,Zenus himself asked me for one-half my remaining gold pieces plus two more.As I approaches another toolbooth I began to worry that I wouldn't have enough gold pieces for all the tollbooths. I wondered what they did to people without enough gold pieces. I looked im my purse and nocited I only had four pieces left. I was really beginning to worry, when my mom woke me up by calling me to breakfast. How many gold pieces did I start with in this dream?

Question

This year my teacher read us the book The Phantom Tollbooth, and since then I've been having strange dreams about drving in usuaual places and paying tolls unusual characters.Last night I dreamed about highway in the clouds near the mythical Mt. Olympus. At my first tollbooth a mythical goddess asked for one-half my gold pieces plus two more. At the next tollbooth,a Greek hero demanded one-half my remaining gold pieces plus two more. Then at third booth,Zenus himself asked me for one-half my remaining gold pieces plus two more.As I approaches another toolbooth I began to worry that I wouldn't have enough gold pieces for all the tollbooths. I wondered what they did to people without enough gold pieces. I looked im my purse and nocited I only had four pieces left. I was really beginning to worry, when my mom woke me up by calling me to breakfast. How many gold pieces did I start with in this dream?

Thanks very much!

Answer 1

You ended up with 4 gold pieces.

Zenus asked you for 1/2 your gold + 2. So you lost 1/2 of your gold and then 2 more.

First add 2 to 4, yielding 6. Now divide by 1/2, or multiply by 2, which yields 12.

Now, try to work out the other 2 "toll booths" in the same fashion to determine the gold at the start.

Answer 2

To determine the number of gold pieces you started with in the dream, we can work backwards by solving the equations given in the tollbooth scenarios.

Let's go through each tollbooth step by step.

At the first tollbooth, the goddess asked for one-half of your gold pieces plus two more. Let's denote the number of gold pieces you started with as G. The equation for the goddess tollbooth can be written as:

(G / 2) + 2

At the second tollbooth, the Greek hero demanded one-half of your remaining gold pieces plus two more. Since you've already given some gold pieces at the first tollbooth, you now have (G - [(G / 2) + 2]) gold pieces left. The equation for the Greek hero tollbooth can be written as:

((G - [(G / 2) + 2]) / 2) + 2

Finally, at the third tollbooth, Zeus himself asked for one-half of your remaining gold pieces plus two more. Since you've already given some gold pieces at the first two tollbooths, you now have (G - [(G / 2) + 2] - [((G - [(G / 2) + 2]) / 2) + 2]) gold pieces left. The equation for Zeus' tollbooth can be written as:

(((G - [(G / 2) + 2]) - [((G - [(G / 2) + 2]) / 2) + 2]) / 2) + 2

Now you need to find the value of G, the number of gold pieces you started with, that satisfies all three equations.

The equation for the number of gold pieces you had left after the third tollbooth is 4, based on the information given. So we can set up the following equation:

(((G - [(G / 2) + 2]) - [((G - [(G / 2) + 2]) / 2) + 2]) / 2) + 2 = 4

Simplifying this equation will give us the value of G, which represents the number of gold pieces you started with in the dream.

Please note that the above explanation assumes a linear deduction of gold pieces at each tollbooth and does not account for any potential rounding or truncation of fractional gold pieces in the dream scenario.