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Count coin exchanges when no gold is left

Count each kind of exchange separately

When several operations change an inventory, let a variable count each operation. The final amount of one item and the total number of operations give equations for those variables.

Problem

An exchange booth offers two trades: give 2 gold coins to receive 3 silver coins and 1 copper coin, or give 5 silver coins to receive 3 gold coins and 1 copper coin. A person starts with only silver coins. After several trades, they have no gold coins, 50 copper coins, and fewer silver coins than at the start. How many silver coins were lost?

Answer

The number of silver coins decreased by

10

The two exchange types occur 30 and 20 times, respectively - not equally often.

Step-by-step solution

1. Count the trades

Let $u$ be the number of gold-to-silver trades and $v$ the number of silver-to-gold trades. Every trade produces one copper coin, so $u + v = 50$ .

2. Balance the gold

The person starts and ends with no gold. The first trade uses 2 gold coins and the second creates 3, so $2 u = 3 v$ . Substituting $u = 3 v /2$ into $u + v = 50$ gives $5 v /2 = 50$ . Hence $v = 20$ and $u = 30$ .

3. Balance the silver

The 20 silver-to-gold trades use $5 (20) = 100$ silver coins. The 30 gold-to-silver trades return $3 (30) = 90$ . The net loss is $100 - 90 = 10$ silver coins.

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