Foreign Carnival Systems Algebra Problem

You are visiting a foreign country, and you want to buy a ticket to the carnival. This country has two types of coins: gold and silver. You have plenty of silver coins with you, but no gold coins. You cannot read any of the writing and do not speak the language in this country, so you don’t know how much the coins are worth, and you don’t know what the carnival tickets cost. You also do not trust that the carnival operator will charge you a fair price when you buy your ticket.
Then you have a great idea! You realize that if you wait around for some other people to buy tickets first, you will be able to observe their transactions. That is, you can see how many gold and silver coins they pay, and how many tickets they receive.

a. You observe the following transactions:

- The first customer paid 10 silver coins, 3 gold coins, and got 2 tickets
- The second customer paid 20 silver coins, 2 gold coins, and got 3 tickets

Based on these observations, how many silver coins does a ticket cost?

b. Try to generalize this. Assume that you observe these transactions:

- The first customer you observe pays S1 silver coins, G1 gold coins, and receives T1 tickets
- The second customer you observe pays S2 silver coins, G2 gold coins, and receives T2 tickets

Write a formula that calculates the number of silver coins you should pay for 1 ticket. You can use the variables S1, G1, T1, S2, G2, and T2, and it should look as simple as possible.

c. After seeing these two transactions, under what circumstances will you not be able to determine how many silver coins to pay for a ticket?

d. Say instead you observed the following:

- The first customer pays G1 gold coins, and receives T1 tickets and S1 silver coins as change
- The second customer pays G2 gold coins, and receives T2 tickets and S2 silver coins as change

Describe a quick way to modify your solution to question (b.) in order to write a new formula for the number of silver coins a ticket costs. Write the formula.

The answer is accepted.
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