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Once we have calculated the discounted price, we could also fi nd the single equivalent
discount, using the formula:
NP LP(1 d)
$63.75 $100(1 d)
0.6375 1 d
0.3625 d
d 36.25%
As mentioned above, we cannot just add the percentages and conclude that this equates to a
40% discount. The reason is that the first 25% discount applies to the original price, while
the second 15% of a percent of the reduced price. Since they are not percents of the same
thing, they cannot be added.
Nonetheless, discounts are sometimes presented in this incorrect way anyway. If a store
advertises “take an additional 20% off of prices that have already been discounted by 50%”
they may mean successive discounts of 50% and 20% (equivalent to a single discount of
60%), or they may mean discounts totaling 50% 20% 70% off the original prices.^2
Additional discounts are also often offered in response to competition.Example 8.3.9 You work for a company that manufactures photovoltaic (solar)
panels. Your company’s 70-watt panel lists for $295, and you offer an 18% trade
discount. A competing company lists its 70-watt panel for $305 and offers a 25%
trade discount. Ingraham Solar Systems, one of your main customers, tells you that it
is considering switching to your competitor’s panels because of their lower cost. What
additional discount do you need to offer to match your competitor’s price?You are selling your panels for (82%)($295) $241.90. Your competitor is selling for
(75%)($305) $228.75. To further discount your panels to match this price, you would
need to drop your current net price by $241.90 $228.75 $13.15. As a percent of
$241.90, this works out to $13.15/$241.90 5.44%.Alternatively, we can work this out by using the formula treating the current $241.90 as the
list price and calculating the discount to get the price to $228.75.
NP LP(1 d)
$228.75 $241.90(1 d)
0.9456387 1 d
d 5.44%Cash Discounts
A merchant may require customers to pay for their orders in advance, before the merchan-
dise is shipped. It is not unusual, though, for a seller to sell items on credit, especially to
customers with whom the seller has (or hopes to build) a good relationship. When mer-
chandise is sold on credit, the seller sends the buyer an invoice for the goods, but does not
require the invoice be paid prior to shipment.
Even when credit is extended for a sale, though, the seller naturally still wants to be paid
promptly. Late payments may incur interest charges. To encourage prompt payment, many
sellers offer the carrot of a discount in addition to the stick of interest charges. A cash dis-
count is a discount offered for prompt payment. This discount usually would usually apply
only to the price of the goods themselves, not to shipping or other added-on charges.
The period of time during which the cash discount is available to the buyer is called the
discount period. The cash discount offered will often be denoted on an invoice with the percent
discount offered followed by a slash and then the length of the discount period. For example,
“3/10” would indicate a cash discount of 3% is offered for a 10-day period. If a seller offers a
more complicated deal, this can be indicated similarly. “3/10, 1/30” indicates that a 3% discount
if offered for payment within 10 days, and a 1% discount if the bill is paid within 30 days (from
the start of the discount period; in this case, it would mean between the 11th and 30th day).(^2) While the latter is technically incorrect, I have personally never seen fi t to stand on principle and demand
a smaller discount.
8.3 Series and Trade Discounts 355