Arithmetic/Accounting balancing $ 100 spent-retained.

Question

A person has 100 $.

The person spends 40 $ and retains 60 $.

The person spends 30 $ and retains 30 $.

The person spends 18 $ and retains 12 $.

The person spends 12 $ and retains 0 $.

Total spent = $100.

Total retained = $102.

Explain the inequality in the balancing, why total spent is not equal to total retained. Where does the extra $2 come from ?

Answer 1

Basically, each individual dollar may only be spent once, but may be retained multiple times. We retain $60 the first time, then retain $30 out of that $60 a second time, then retain $12 out of that $30 a third time, etc. Some of those dollars have been retained several times. It is merely a coincidence that the sum of money retained in each step adds up to $102 which happens to be close to $100.

For example, what if the spending was instead as follows:

-spend $40, retain $60

-spend $30, retain $30

-spend $20, retain $10

-spend $10, retain $0

Now, both spend and retain are equal at $100 a piece. To make it more obvious, let's look at another example:

-spend $1, retain $99

-spend $1, retain $98,

-spend $1, retain $97

-spend $97, retain $0

Notice how spend is still equal to $100, but now retain is $99 + $98 + $97 = $294. They are not even close to the same. This is, again, because the each individual dollar may only be spent once but may be retained multiple times.