Can you solve this stolen money riddle

arithmeticeconomicsword problem

So, a basic word problem has gone viral on FaceBook quite a few times now and many people remain convinced of their own incorrect answers.

Suppose a man walks into a store to steal a $\$100$ bill from the cash register. Shortly thereafter, he purchases $\$70$ worth of goods using the same $\$100$ bill he stole. The owner gives him $\$30$ in change. How much money did the owner lose?

The answer is very obviously $\$-100$, according to this problem. You can either add and subtract all the negatives and positives or you can realize the thief only gained two things: $\$70$ in goods and $\$30$ in change. However, $\$-100$ is not truly accurate because the owner must have bought the goods for less than he is selling in accordance of his profit margin goals. The “at cost” value of goods will be less than $\$70$ in order to make profit, therefore we can only assume he lost somewhat less than $\$100$ in total because we are not given all the necessary information to give an exact answer. But I'll be lenient and entertain the question to friends.

Anyway, would anyone like to provide an algebraic expression that represents this word problem very nicely or perhaps even a proof?

Thanks.

Best Answer

There are 5 events:

  1. Owner bought the goods from somewhere with $k$ dollars.
  2. Owner’s 100 dollars was stolen
  3. Owner received 100 dollars from the thief
  4. Owner gave the thief a change of 30 dollars
  5. Owner gave the thief the goods

Let $M(n)$ be the change of amount of cash of owner due to the $n$th event.

Let $P(n)$ be the change of the total value of the owner’s properties due to the $n$th event.

Clearly,

  1. $M(1)=-k,P(1)=+70$
  2. $M(2)=-100, P(2)=0$
  3. $M(3)=+100, P(3)=0$
  4. $M(4)=-30, P(4)=0$
  5. $M(5)=0, P(5)=-70$

The total change in amount of cash ($\Delta M$) is $$\Delta M=\sum^5_{j=1}M(j)=-30-k$$

Similarly, $$\Delta P=0$$

So the owner in total lost $30+k$ dollars.

However, we know $0<k<70$. Thus, $$\color{red}{-30>\Delta M+\Delta P>-100}$$

In other words, the owner’s loss is less than $100$ dollars, but more than $30$ dollars.

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