# Answer to the Mysterious Unbalanced Sheets on Loans

The answer is pretty simple. The total amount paid back should be equal to the amount lent out. In both cases (as told by the hunchbacked shopkeeper), \$50 paid back equals \$50 lent out. Consequently, the final balance is just \$0. The shopkeeper, as well as us, shall be satisfied. Both sheets are INDEED BALANCED!

The sum of the money owed each month, have nothing to do with the amount lent out. If we have to assign a meaning to it, then we can divide the sum by four (the number of instalments), and explain it as the average debt held by the two men respectively, to the shopkeeper during a certain period (from the time first instalment is paid, until after all debt is paid off). On the first case, the average debt is \$50 /4 = \$12.5 during the period; and on the second case, the average debt is just slightly higher.

There is another way that we can gain insight. If we add a few “empty” operations, such as “pay nothing” in the first couple of instalments, then followed by the 4 instalments as in the story, (Try this!) Obviously, the sum will rise – but the debt is still paid back.

See the point? No worry for the shopkeeper as long as all money owed has been paid back. During the period the borrower(s) may be worried a bit since he has to remember obligation to pay, and worry about the interest too if he held the debt too long (but no interest in this story – so no worry for all).