Question

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.

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.

Answer 1

To find the silver coin cost of a carnival ticket, observe transactions, create equations, and solve simultaneously. If the proportion of gold coins and tickets are the same in both observations, the cost can't be determined. Adjust the formula if changes are given back in transactions.

Step-by-step explanation:

To determine the cost of one ticket in silver coins based on observations:

1. From the first customer's transaction, we have the equation 10S + 3G = 2T where S is the value of one silver coin, G is the value of one gold coin, and T is the value of one ticket.
2. From the second customer's transaction, we have the equation 20S + 2G = 3T.
3. We need to solve these equations simultaneously to find the value of S, G, and T. Since the gold coin's value and the number of tickets are in different proportions in each equation, we can eliminate G and find a relation between S and T to determine the cost of a ticket.

General formula for the number of silver coins for one ticket:

S = (S1T2 - S2T1) / (T1G2 - T2G1)

However, if T1G2 = T2G1, we won't be able to determine the value of S because the equation becomes undefined.

For transactions involving change:

• First, adjust the original equations by adding S1 and S2 on both sides to account for the change given back. The new equations will now represent only the net cost of the tickets.
• Follow the same process as before to find the cost of a ticket in terms of silver coins.

The modified formula including change is:

S = ((G1T2 + S2T1) - (G2T1 + S1T2)) / (T1G2 - T2G1)