Can you solve this stolen money riddle?

Question

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.

Answer 1

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.

Answer 2

The owner was stolen $\$100$ and he lost $\$100$, full stop. Whether this money was taken from his cash register, from his future benefit or from under his mattress is irrelevant.

Answer 3

We'll ignore the storekeepers actual wholesale cost of the purloined goods. For all we know the it's a not for profit and there IS no markup. Whatever. Let's say the store starts with $500 cash (C) and $500 goods (G).Lady steals $100 cash- the store's down $100. Lady comes back, gives the $100 back- the store's even. She walks with $70 in goods and $30 cash. The store's out $100. So, C + G = total value of the shop. C - 100 then C + 100 then (G- 70 + C-30) = -100

Answer 4

Agree with the post from OP, and to incorporate the profit from selling the goods the best answer - in my opinion - should be: 100 usd minus the profit of selling the goods.

To the people saying that profit should not be included, I think it is incorrect to neglect this solely because no info was given. Profit is crucial in answering the question: "how much money the owner lose?".

To give an illustrative example to my point. Consider the following quiz: "A dealership sells a porsche and a ferrari. The profit for selling a porsche is 5k usd. How much profit did the dealership make with selling the porsche and ferrari?"

Would you answer 5k usd and assume no profit was taken in selling the ferrari just because no profit for the Ferrari was given in the question? Or would you think that an answer like "5k + the profit of the ferrari" is more appropiate?

Answer 5

OK, first people are trying to answer this riddle like it's a bottom-line question. It's a top-line question - don't complicate a simple riddle. Second, people seem to be looking at this linearly. The truth is very little math (none really) to solve the basic question of the riddle. The purchase transaction is irrelevant to the store owner's loss. I'll explain: Let's say the store's till starts at $200. The thief steals $100 leaving the till at $100. Later, he goes to purchase $70 worth of goods with the stolen money. The store takes the $100 and gives him $30 in change. The till is now at $170. The goods weere paid for. It doesn't matter where the money came from, it's still real money and the purchase is a wash. If the thief did not steal the $100 in the first place, the till would be at $270 after the transaction. Still short the original theft. The register view would look like this: -100 theft (end of first event, this actually shows up at the end of the day when they count the drawer and sales)

(-$70) goods purchase rung into the register. The register is now looking for payment. (+$100) payment for good (-$30) change back (end of second event/transaction) = 0 loss to the store. This is a complete money for goods exchange.

The store is still missing the $100 from the register and when sales receipts and inventory is balanced, they will show the goods were properly sold and not shrinkage.

And if you really insist that this should be a bottom line question, despite there being no COGS data, there's no loss of goods in this scenario and the profit from the sale does not offset the loss.

All you really need to know to answer this riddle is how much the thief stole from the register drawer. The purchase is irrelevant to the store's loss. The only thing the purchase changes is what the thief ultimately walks out the door with.