The Mathematics of Money

Copyright © 2008, The McGraw-Hill Companies, Inc.

Example 9.1.4 On a trip to his favorite local discount store, Jack bought a DVD

player costing $79.95, a toaster for $29.95, and a case of cranberry juice for $12.75.

The sales tax rate on all purchases is 6^1 ⁄ 4 %. Find the total cost of his purchases

including tax.

If we calculate the total price of each item individually we get:

DVD player: T  P(1 
 r) $79.95(1.0625)  $84.95

Toaster: T  P(1 
 r)  $29.95(1.0625)  $31.82

Cranberry juice: T  P(1 
 r) $12.75(1.0625)  $13.55

Adding these up we get a total of $84.95 
 $31.82 
 13.55  $130.32

If we total the items fi rst and then calculate the price with tax, we get:

Total before tax  $79.95 
 $29.95 
 $12.75  $122.65

With tax: T  P(1 
 r)  $122.65(1.0625)  $130.32

Occasionally these two different methods may give very slightly different results because

of rounding. If we did not need to round, the two methods would always give exactly the

same answer, but the need to round can cause minor discrepancies. With the first method,

we rounded three times, with the second we rounded only once, and that sometimes may

create a difference of a penny or two. This never amounts to enough money to be much of

a business concern (although state or local regulations may require a business to calculate

the tax charged to a customer in one way or another). At most it may be a matter of a few

cents difference. The answers given in this book are calculated by calculating sales tax on

totals, not on individual items separately, since this is the method most commonly used in

actual practice.

Of course, it does matter if there are different tax rates on some items than on others. In

that case, the items must be grouped together based on their tax rate.

Example 9.1.5 Rework Example 9.1.4 if the sales tax rate on food is 2^1 ⁄ 2 %.

While we can combine the prices of items that will be charged the same rate, items with

differing rates must be kept separate.

The DVD player and toaster cost a total of $79.95 
 $29.95  $109.90.

Nonfood items: T  P(1 
 r)  $109.90(1.0625)  $116.77

Food item: T  P(1 
 r)  $12.75(1.025)  $13.07

Total with tax: $116.77 
 $13.07  $129.84.

Students who have covered Chapter 8.1 should note the similarities between sales tax

and markup calculations. They are essentially the same thing, mathematically speaking.

You may also notice that the markup formula given in Section 8.1 is essentially the same

as the sales tax formula given in this chapter, the two differing only in the letters used.

Finding a Price before Tax

Sometimes, we may need to work things in the opposite direction. We may know the price

including tax and need to figure out the price before tax. For example, a business owner

may know her business’s total sales receipts for the day, but want to know how much of

those receipts actually represent income for the business as opposed to taxes collected for

the government. The following example will illustrate another such situation, as well as the

mathematics needed to answer the question.

Example 9.1.6 Ardana is shopping for a new car. She fi gures she can afford a $325

monthly payment, and on the basis of this payment and the rates she can get on a car

loan from her credit union, she has determined she can borrow $21,900. With no down

payment or trade-in, she started shopping, fi guring she could afford to pay $21,900

for her new car.

Unfortunately, she forgot that in the county where she lives the sales tax rate is 8.75%.

She now realizes that this total must include the sales tax, and so the price she can

9.1 Sales Taxes 379