Understanding the Mathematics of Personal Finance

CHANGES

When a number that you’re interested in (the cost of a pound of coffee or the cost of a new home) changes, it’s often more relevant to look at the percent change than it is to look at the absolute numbers.

For example, if you ’ ve been paying $3.00 a pound for coffee and the price changes by $2.00 up to $5.00 a pound, this is a relatively big change. On the other hand, if you’ve been considering purchasing a new car for $25,000 and the price changes by $2.00 to $25,002, relatively speaking, this is not a big difference.[2]

The standard way of calculating percent change is by subtracting the new value from the old value, and then by dividing this difference by the old value:

New value - Old value
Old value

The 100 multiplier is just to change the fractional quantity into a percentage. Example: The pound of coffee mentioned above. The price was (the old value) $3.00, and the price now is (the new value) $5.00 so that

Remember that the units of the new and old values must be the same. Don’t have the new value in pennies, or francs, or any other units, if the old value is in dollars (and vice versa). Also, because we’re dividing dollars by dollars, or francs by francs, the percent change is called dimensionless t It has no units. This is reas­suring, because if we were to first convert our dollar amounts into, say euros, and then calculate the percent change, we should certainly expect to get the same answer.

If the new value is smaller than the old value, then the percent change will be a negative number.

Example: The price of a new car dropped from $30,000 to $27,000. The percent change is

Looking at changes as percent changes or fractional changes (just don’t multiply by 100) helps us to put things in perspective. We are comparing how much a number changes to how much the number used to be.

Percent change is not symmetrical in that changing a number and then changing it back doesn’t give you the same results. For example, if I have something that used

to cost $100 and now it costs $150, the percent change was 50%. However, if the price then returns from $150 back to $100, we’ve reversed the titles of old and new values. In this latter case, the percent change is -33.3%.

In some situations we don’t have an old value or a new value. Consider the statement “My bank account seems to swing back and forth between $700 and $800.” How do you calculate the percent change?

In this case, where the numbers seem to be taking turns being the old and new numbers, it makes sense to calculate the average of the two numbers,

and then to talk about a percent variation or sometimes a “percent swing” by sub­tracting the smaller number from the larger number and then dividing the result by this average:

In this case, the two percent change numbers about the average are

100 $900 - $800 $800

and

100 $700 - $800 $800

This is often written as ±12.5%, which is read as “plus or minus 12.5%.”

One last little item that makes writing and reading about small changes more convenient is that a small change in a number (not a percent change) is often denoted by the Greek uppercase letter delta (A). If, for example, we have a cost of something that changes from an old value of $125.00 to a new value of $127.00, we would write

ACost = $127.00 - $125.00 = $2.00.

As above, the convention is to subtract the old value from the new value. If the new value is smaller than the old value, the result is negative. Note that ACost has the same units as the two numbers used to calculate it (in this example, dollars), and consequently, the two numbers must have the same units (both be dollars, or both be cents, or francs, or euros, and so on).

Sometimes, it’s necessary to work these problems backward. Suppose I tell you that a store is having a “30% off sale” on all items. What would the sale price be on an $80 purse? What we’re looking at here is

100 New price - $80 = -30%.

I’m sure I could awe you with my prowess at algebraic manipulation, but it’s really not called for. This can easily be solved in either of two ways:

1. If we’re reducing the price by 30%, then we’re keeping 70% of the original price: 0.7($80) = $56.

2. Thirty percent of $80 is $24. Reducing an $80 price by $24 leaves $80 - $24 = $56.

These simple calculations work correctly because we’re looking for the new price. If we have to go the other way, the problem is a bit trickier. For example, “A purse that was reduced by 30% is now selling for $56. What was the original price?” Without trying to awe you (or more likely to bore you) with the derivation, the formula you need here is

New price
% Change +1'
100 +

Putting in the appropriate numbers,