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Expensive Dollars: The Math of Money



In this installment we are examining the various math topics around money. Now money in general is an extremely broad topic that has a vast amount of theory and entire professions dedicated to different aspects of money (accountants, book keepers, loan officers, bank employees, etc.) Our discussion will focus on basics essential for young people to get a foothold and see a few different simple math uses in action. We’ll cover what money is, income and expenses.

 What is Money?


Before we talk about money we should understand what money or currency is. In a general sense, money or currency is made up. But one can touch it and hold it, how can it be made up? Money is made up in the sense that money is merely a social contract that a group has decided on. A value is assigned that maps goods to currency. Once this value is agreed upon, commerce, or an exchange of goods, proceeds from there. You can have commerce without money by simply trading item for item. This is called bartering.


Bartering is simple to visualize. If one person raises chickens and another person raises pigs, they can trade each other. Say the chicken farmer agrees to trade 5 chickens for 1 pig. They make the exchange and enjoy the bounty of their new trade. This works out well because the pig farmer has pigs and wants chickens while the chicken farmer has chickens and wants pigs. This simple system of trading begins to break down under a few different scenarios.


Pigs have a season. They take a certain amount of time and care before they are ready to sell. After a certain period of time they become extremely large and it takes excessive resources to feed them. Very old pigs may lose their savor and not be as good for consumption. So there is a time frame when it makes sense to sell the pig. But what if the pig farmer does not need chicken at that time? What if no chickens are ready for trade? How does the pig farmer eat chicken when it is not the right time to sell the pigs? The farmers certainly could agree to have a delay in their exchange. An arrangement like: I will give you a pig now for chicken later. They could also make preservation efforts like curing a ham for future sell, etc. But all of these things take time. Money allows the farmers find a buyer during the appropriate season and use currency to purchase things later when they have no goods for sale.

Fractional Amounts

Related to the problem of storage is fractional amounts. What if the chicken farmer can only part with three of his chickens right now? Does the pig farmer give him 3/5 of a pig and keep the rest? In this example this division may actually be possible. The chicken farmer could trade for just the most desirable portions of the pig. But what about the other way around? The pig farmer has only one pig to trade and he needs 2 chickens and some and milk. He might still be able to divide up the cuts of meat but what if the dairy and the chicken farm are far apart and transporting the divided product before it spoils is difficult? Now there are certainly ways around this. The pig farmer could hire a helper (but what would he pay the helper in, pigs too?) or agree to meet at a central place, but things get complicated really fast. Currency can solve this problem by allowing the chicken farmer to transform his goods (chickens) into currency over time. Perhaps the farmer can sell three chickens for $3 now and use $2 from a previous chicken exchange to fund his pig purchase. In the same line of thinking the pig farmer can sell his pig for $5 and pick up chickens when he is that area and milk the next day when he travels to the other part of the region.

No Direct Exchange

Consider a third party who makes lantern oil. Both the pig and chicken farmers need the oil to light their homes. The chicken farmer has chicken to offer for oil and the pig farmer has pigs to offer for oil. But the oil maker isn’t very fond of meat and prefers to eat fresh vegetables. Now they certainly could find another farmer who grows vegetables but things can get complicated. Say the vegetable farmer prefers chicken over pork. The pig farmer could certainly trade a pig for some chickens, chickens for vegetables and vegetables for lantern oil. But this whole exchange takes time to coordinate and execute.  On the other hand, if this area has an agreement on a currency, the pig farmer can sell to an interested buyer for money and buy goods from the lantern oil maker directly. The oil maker can then use the money to buy as many fresh salads as he wants directly from the vegetable farmer.

Desirable Properties of Money

If we can all agree that this piece of paper or that piece of metal has some value, then money or currency is formed. Currency solves a problem by being easy to store, easy to break into pieces and easy to exchange among different parties. But, as we will see, this invention can also introduce different problems like stress and  debt.



Now that we have this common form of exchange called money, we need some way to acquire this resource to exchange for things we need and want. Acquiring money is called income. In the most general sense, income is the result of exchanging something you have with someone who wants what you have, for money. Some common things we trade are time, energy, knowledge, and resources.

Time, energy and knowledge are common things people do in jobs to make income. A home owner might pay someone with a lot of strength to dig a ditch. An employer might pay someone to deliver fliers to advertise a business while they stay in the office to handle calls. Another person might get paid to help someone who is injured feel better. There are as many jobs as there are kinds of people.

Our farmers discussed earlier were exchanging resources they had developed over time. A computer chip maker might make a computer part and sell it to interested parties. A person who has a home or property may rent the location (allow others to occupy the space) in exchange for money.As with jobs, there are as many types of resources that can be gathered and exchanged for income as there are kinds of people.

Let’s look at a few different calculations we can do with Income

Wage over Different Time Periods

In the case of a job, some amount of money is exchanged for some result or time. An employer might say a job is $10 an hour and another is $50,000 a year. Translating these numbers between different time frames like $/hour, $/month and $/year can be important for different contexts. Just as we did with our sports discussion, we can use simple math to move from one set of units to another. Some key things to consider:

  • While there are 356 days in the year, a person is unlikely to work each and every one of those days. So we need to consider how many days in a given year will be worked.
  • The number of hours a person can work is determined by budgets, labor laws and the bounds of human energy. So while there may be 24 hours in a day, you may only be eligible for pay during 8 of them.
  • Gross income is different than net income. We will talk about taxes in the expense section, but in short, in most regions there is some amount of money that never hits your pocket. It is taken out in the form of taxes. Net income is how much you actually have at the end of the day.

Moving from an hourly rate to a gross yearly rate looks like the following:



Similarly we can move from from a gross yearly rate to an hourly rate as follows:



Profit Margin

When goods are sold the seller tries to make a profit. That is to say they wish to bring in more money than they spent to make the item. When the seller spends more money than they bring in that is called a loss. If they neither make money nor lose money it is called a wash. Profits are important because they insulate against future loses. A business or individual who is selling things will often want to calculate how much income they are producing compared to how much they are spending. A profit margin is a ratio of profits to total sales

Let’s consider a girl named Suzie. Suzie 7 years old but she has studied all her life on how to make delicious baked goods. She wants to open a small stand to sell her cookie marvel, Suzie’s Scrumptious Sugar Cookies. Suzie is also a smart business women. She sat down and calculated all of her costs, everything from flour, sugar, to how much gas it takes to run the oven and energy to cool the house after a long day of baking. She estimates each cookie to cost $0.75 to make. She decides to sell each cookie for $1. Her first day goes great and she sells all 100 of the cookies she baked. Let’s calculate her profit margin.

  • Suzie spent $75 to make the cookies
  • Suzie took in $100 in sales
  • Suzie’s net result was $25 in profit

25/100 gives us a value of .25 or 25%. Suzie has a very healthy profit margin of 25%.  A few weeks of good margins like this and Suzie will have the capitol to add new cookies or expand her operations to other neighborhoods.

Importance of Choices

Whatever way one decides to generate income, there are a series of choices that lead up to the options that are available.  Some choices include

  • Self Improvement Efforts
  • Time management
  • Resource Management
  • Relationships

Each of these choices may lead to more or less income opportunities. For example, the choice to study a difficult subject while others party may open up many profitable career paths not available to others. Making a good investment might lead to additional income opportunities while a bad or poorly thought out investment might remove income opportunities.

These choices and many more will add together. The following graphic helps to illustrate decisions we can make in what we do with our time:




Expenses are simply money that you have to payout. It might be for a good you need but don’t have, it might be for something you can’t do or it might be part of a social agreement you made. We will limit this discussion to two expenses that might be confusing or new.


A tax is an expense required of a person, usually by some larger entity like a city, state or federal government. Taxes are expressed as a percentage of an amount or a fixed value. It might be a percentage of a person’s income, percentage of a sales price or a bridge toll. Taxes are not always called taxes. Sometimes they are called fees, tariffs, duties, tolls, etc. but they mostly behave in a similar way by requiring an expense by some party. Taxes are generally collected with the stated purpose that the individual benefits from some benefit available to all. For example, a person might pay a gas tax when they fill up their car and that tax might be used to build and repair roads. Taxes make a lot of adults pretty heated and there are a lot of active discussions on this topic each day all across the world. People discuss who should be taxed, what should be done with the money, and what amount to tax. This post is not a political post so we we will just talk about the math.

Let’s revisit our two workers from the income section. One made $20,000 a year and the other made $50,000 a year. The United States currently has a variable tax rate that is determined by a complex set of calculations. We’ll leave those complexities to the accountants and just use their gross income without any modifiers to determine a tax from a tax table. We can use these numbers to determine an effective tax rate


If you are still following along, it is important to note these are only federal taxes and there may be many more types of taxes taken on a pay check before it makes it to the earner. Another thing we can do with this is we can revisit the calculations for hourly wage to figure out how much people really make when taxes are factored in:




When discussing interest, it is important to ponder: What do banks sell? An initial response might be that a bank does not sell anything. They just hold your money right? That would not be an accurate observation. A bank collects depositor’s funds and sells them to other people in the form of loans. To say differently, the bank sells money today for more money tomorrow. The increase the bank collects is called interest. Interest can come in the form of a fixed percentage or a value that changes over time.

Simple interest just applies the percentage to the loan. So if you borrowed $1,000 today and agreed to pay 5% simple interest 1 month from now, you would have to pay back $1,050 once the month passes.

Many loans however, do not have simple interest. Instead the interest compounds over time. What this means is that a given interest calculation includes previous interest values. Consider this thought experiment. You start with $1. The you add 10% to that and call it the new value. The you repeat again and again. The first 10 entries would look like:


Now let’s look at that over 100 cycles, this time with just the graph:


Imagine for a second that the one dollar in the example was a pack of gum that was put on a credit card. Imagine that credit card had a modest interest rate (for credit cards of 10%) and that the credit card was was forgotten for a little over 8 years. Through compounding interest thousands of dollars would be due. Now this scenario is a bit contrived because a credit card would have a limit and the company would expect payment well before the balance reached this amount, but it does show what compounding interest can do over time.

Speaking of minimum payments, credit card companies will often lure unsuspecting card users with a “minimum payment”. What they do not disclose is that the minimum payment will often be just barely above the interest fee. So each month that goes by the balance is only reduced by a small fraction. This can cause some cards to take up to 30 years to payoff!

Credit lines, such as credit cards are not intrinsically evil. They can be a great help when used appropriately. For example, they can help businesses with cash flow problems who have money but do not have it at the times their debtors need to be paid. Credit can also help with large purchases like cars, houses or education that are important but would otherwise be unobtainable. But the compounding interest of credit cards means that it is very easy to get trapped by the weight of debt. Care must be taken where the terms can be met without spiraling costs.

Loans and complex debt scenarios are often tackled best with the assistance of computers. A computer spread sheet or other software can be configured to evaluate thousands of tedious calculations and determine different outcomes. In the resources section a few different examples are provided for exploration.



We can now close our discussion on Money. We learned a little about why currency systems emerged over trading systems. We also learned a little bit about what income is ad a few simple calculations related to hourly wages and profit margins. Finally, we learned about two types of expenses called taxes and interest. In the references below there are more detailed exercises to practice math in these areas.





Categories: Math
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