# chapter 05

```Chapter 5
Mathematics
of
Merchandising
Copyright &copy; 2022 McGraw-Hill Education Limited.
Microsoft&reg; PowerPoint&reg; Presentation by
Hussam Jawad and Howard Umrah, Durham College.
Previously updated by Julie Howse, St. Lawrence College
Learning Objectives
LO1 Calculate the net price of an item after single
LO2 Calculate a single discount rate that is
equivalent to a series of discounts.
LO3 Understand the ordinary dating notation for
the terms of payment of an invoice.
LO4 Calculate the amount of the cash discount for
which a payment qualifies.
LO5 Solve merchandise pricing problems involving
markup and markdown.
5-2
Introduction
• Products for purchase by the consumer
undergo the marketing chain. As products
move from the manufacturer to the
consumer, they are marked up, marked
down and under go trade discounts.
• This chapter explores calculations needed
for product distribution and merchandising.
5-3
• Goods move from a
manufacturer to the
ultimate consumer
through a distribution
chain or merchandising
chain.
• We need a way to
determine the prices of
goods along the chain.
5-4
• Typically, the distributor
will have a set of list
prices for the goods
(chosen to approximate
the retail price).
• The distributor will offer
discount off of the list
prices.
5-5
• The net price is the list price minus the trade discount.
Net price = List price – Amount of trade discount
Net price = List price – (Rate of trade discount x List price)
Net price = List price(1– Rate of trade discount)
Let N = Net price
L = List price
d = Rate of trade discount
This shortens our equation to:
5-6
Net price = List price – Amount of trade discount
Net price = List price – (Rate of trade discount x List price)
Net price = List price(1– Rate of trade discount)
Let N = Net price
L = List price
d = Rate of trade discount
5-7
Example: 5.1A
A wholesaler lists an item at \$117 less 20%. What is
the amount of the discount and the net price to a
retailer?
Given : 𝐿=\$ 117, 𝑑=0.20
Amount of discount=𝑑𝐿=( 0.20 ) ( \$ 117 )=\$ 𝟐𝟑 . 𝟒𝟎
5-8
Example: 5.1B
After a trade discount of 30%, a garage is able to
purchase windshield wiper blades for a net price of
\$19.46. What is the list price of the blades?
Given : 𝑑=0.30 , 𝑁=\$ 19.46
Substituting these values in our formula , we have
5-9
Multiple Discounts
• Sometimes, a seller/vendor may offer
multiple discounts.
• i.e. the usual trade discount plus a volume
purchase discount and a promotional discount
• In this case, the discounts are compounded
• In a case with three discounts, we could
calculate the net price as follows:
5-10
Equivalent Single Discount Rate
• An equivalent discount rate is the single
discount rate that gives the same net price
as the combined effect of multiple
discounts.
• This rate will always be less than the simple
sum of the multiple discount rates.
5-11
Example: Equivalent Discount Rate
A wholesaler offers a trade discount of 20% and a
promotional discount of 10%. How much would a retailer
pay for \$100 worth of goods? What is the equivalent
discount rate?
Given : 𝐿=\$ 100 , 𝑑 1=0.20 , 𝑑 2=0.10
\$ 100 − \$ 72
Equivalent discount=
&times; 100 %=𝟐𝟖 %
\$ 100
5-12
Example: Equivalent Discount Rate
Spring Sale
50% OFF
+ 30% OFF
+ an additional 20% OFF (with our email)
What is the single
equivalent discount
rate?
5-13
Skill Check
i.
ii.
iii.
Calculate the sale price of an item with
cost of \$36.90 after a 15% trade discount.
A supplier bought merchandise from the
manufacturer totaling \$6050. The supplier
was given a series of discounts, 8%, 5% and
3% respectively. What was the final cost of
the merchandise?
Determine the equivalent rate of discount
of an \$60 item that underwent discounts
of 20% and 15%.
5-14
Cash Discounts &amp; Terms of Payment
• Transactions within the merchandising chain
• Following a transaction, the vendor will send
the buyer an invoice which includes:
• Items purchased
• Unit prices
• Shipping cost
• Taxes
• Amount due
5-15
Invoice Example
Items purchased
Unit
price
Discount
rates
Shipping cost, Taxes,
Amount due
5-16
Terms of Payment
An invoice normally states the terms of payment,
including:
• Credit period – the length of time until the
invoice is due.
• Cash discount rate – discount applied for early
payment.
• Cash discount period – the time period when the
cash discount is available.
• The date on which both the credit period and
the discount period begin.
5-17
Dating Notation
• With ordinary dating, both the credit period
and the discount period are measured from
the invoice date (day 0).
• For example, if an invoice was dated July 3rd
with a 30-day credit period, then July 4th
would be considered day 1 and August 2nd
would be day 30.
• Payments made on August 3rd or later would
be charged the overdue-account penalty.
5-18
Terms of Payment
• Read as “two ten, net
thirty”.
• A 2% discount will be
applied to payments
2/10, n/30
Cash discount Discount period Credit period
(2%)
(10 days)
(30 days)
5-19
Common Practices With
Dating
• If the last day of the discount period or the
credit period falls on a non-business day, the
period is extended to the next business day.
• If no cash discount is offered, only the “net”
figure for the credit period is given (for
example, n/15 or n/30).
• If a net figure for the credit period is not stated,
it is understood that the credit period ends 20
days after the end of the discount period. For
example, “2/10” by itself implies “2/10, n/30.”
5-20
Calculating Cash
Discounts
• We can use the formula we have already
learned to help with cash discounts.
• Substitute the invoice amount for L and the
cash discount rate for d.
• N will be the full payment that will settle the
invoice within the discount period.
5-21
Example: Early Payment Discount
An invoice for \$1,000 with terms 2/10, n/30 is dated June
3rd. What payment will settle the invoice on June 7th?
Given : 𝐿=\$ 1,000 , 𝑑=0.02
5-22
Skill Check
i.
An invoice for \$6,000 with terms 3/10, n/30 is
dated August 5th. What payment will settle the
invoice on August 10th?
ii.
An invoice for \$2,000 with terms 2/10, n/30 is
dated May 3rd. What payment will settle the
invoice on May 31st?
iii.
An invoice for \$3,000 with terms 2/10, 1/20,
n/30 is dated May 3rd. What payment will settle
the invoice on May 21st?
5-23
Markup
• The markup or gross profit is the amount
added to the cost of an item to arrive at its
selling price.
𝑆 = 𝐶+ 𝑀
• The markup must be large enough to cover a
portion of overall operating expenses and
contribute toward the overall operating
profit.
𝑀=𝐸 + 𝑃
5-24
Markup
Let us define the following symbols:
S
C
M
E
P
= Selling price (per unit)
= Cost (per unit)
= Markup (per unit)
= Overhead or operating expenses (per unit)
= Operating profit (per unit)
Given that
Then, we end up with:
5-25
Markup Diagram
• The two ways of expressing selling price
(either as C+M or C+E+P) are shown below.
𝑆 = 𝐶+ 𝑀
𝑆 = 𝐶+ 𝐸 + 𝑃
5-26
Markup
• Merchandisers prefer to think of markup in
terms of percentages, using either:
or
(Also known as gross profit margin)
4-27
Example: Markup
Coastal Marine is importing a new line of inflatable boats at a unit cost of \$1860.
Coastal estimates that operating expenses per unit will be 30% of cost.
a) What should the markup and selling price be if Coastal’s desired operating
profit per unit is 25% of cost?
b) What are Coastal’s rate of markup on cost and rate of markup on selling price
for the boats?
5-28
Example: Markup
Product “Z” has a unit cost of \$245. If the retailer wants 30% rate of markup on
selling price, what is dollar amount of markup and what is selling price?
therefore M=0.30 S
5-29
Markdown
• A markdown is a reduction in the selling
price of an item.
• Markdowns may be used for a variety of
reasons including:
• to reduce excess inventory
• to increase sales volume during sales
4-30
Markdown
Substituting D for the amount of markdown, S
for the regular selling price and S(reduced) for
the reduced selling price gives:
The rate of markdown is the markdown
expressed as a percentage of the regular
selling price:
5-31
Markdown
We can also calculate the reduced
selling price (or sale price) using
our original discounting formula
restated as:
5-32
Example 5.4A: Markdown
Toby’s Cycle Shop advertises a 20% markdown on an Alpine mountain bike
regularly priced at \$445. Cycle City’s regular selling price for the same model of
bike is \$429.
a) What is the reduced price at Toby’s?
b) What rate of markdown would Cycle City have to offer to match Toby’s reduced
price?
4-33
Example 5.4C: Markdown
K&amp;M Clearance Centre sells premium coffee makers for 60% off the regular selling
price, which represents a discount of \$95.40. What are the regular and reduced
selling prices?
We are given : 𝑅𝑎𝑡𝑒𝑜𝑓 𝑚𝑎𝑟𝑘𝑑𝑜𝑤𝑛=60 % 𝑎𝑛𝑑 𝐷=\$ 95.40
5-34
Skill Check
Noble furniture store is importing a new line of
sofas at a unit cost of \$750. Noble estimates that
operating expenses per unit will be 25% of cost.
a) What should the markup and selling price be
if Noble’s desired operating profit per unit is
10% of cost?
b) What are Noble’s rate of markup on cost and
rate of markup on selling price for the sofas?
5-35
Skill Check
Noble’s furniture store advertises a 20%
sectionals regularly priced at \$1550. Lennon’s
regular selling price for the same sectional is
\$1300.
a) What is the reduced price at Noble’s?
b) What rate of markdown would Lennon
have to offer to match Noble’s reduced
price?
5-36
Skill Check
Noble’s furniture store advertises a 20%
sectionals regularly priced at \$1750. Lennon’s
regular selling price for the same sectional is
\$1300.
a) What is the reduced price at Noble’s?
b) What rate of markdown would Noble have
to offer to match Lennon’s reduced price?
5-37
Chapter 5
End of Chapter
Copyright &copy; 2022 McGraw-Hill Education Limited.
Microsoft&reg; PowerPoint&reg; Presentation by
Hussam Jawad and Howard Umrah, Durham College.
Previously updated by Julie Howse, St. Lawrence College
```