Chapter 5 | Mathematics of Merchandising
5.1 |
141
Introduction
Merchandising
Merchandising is a process that involves the buying and selling of goods (merchandise) from
the manufacturer, to the final sale of those goods to the consumers. The process can include
discounting, promoting, advertising, etc. Typically, manufactured goods are sold to consumers
through a chain of distributors, wholesalers, and retailers. This is called a merchandising chain
or distribution chain.
is an essential part of
commerce and involves
the buying and selling of
goods.
Distribution Chain
■■ The
illustration above is a typical distribution chain for most organizations. Some
organizations may skip one or two members in this chain. For example, in some organizations,
retailers may obtain goods directly from the manufacturer while in others, distributors may
act as wholesalers and supply directly to retailers.
■■ Each
manufacturer could have numerous distributors, wholesalers,
retailers, or consumers located locally or in different countries depending
on the type and nature of the organization, product, and consumer.
■■ Each member in this chain makes profits, except the consumer, as the
Having fewer members
in the distribution chain
will usually result in
a lower selling price
of the product to the
consumers.
consumer is only a buyer and not a seller.
■■ The selling price of a product at each stage of the distribution chain depends on many factors
such as demand, competition, the cost of the product, operating costs, or the profit necessary
to stay in business.
5.2 |
Trade Discounts
The list price is the price
quoted by a supplier for
a product.
A trade discount is a
reduction on the list price
of a product.
Manufacturers usually determine the price at which a product can be sold to the consumer. This
is called the manufacturer's suggested retail price (MSRP), catalogue price, or list price.
When a manufacturer sells the product to the distributor, the manufacturer deducts a certain
amount from their list price, usually a percent of the list price. This is known as a trade discount.
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Chapter 5 | Mathematics of Merchandising
The net price is the
price of a product after
deducting trade discounts
from the list price.
The distributor in turn gives a trade discount on their list price to the wholesaler and the wholesaler
gives a trade discount to the retailer. The price each of them pays for the product after the trade
discount is called their net price (N).
As the product goes through the distribution chain, the trade discount becomes smaller and smaller.
Trade discounts are generally offered to increase sales, to reward valuable customers, to encourage
large quantity purchases, to promote a new or seasonal product, etc.
Single Trade Discount
When one discount is offered by the seller to the buyer, it is called a single trade discount. The amount
of trade discount is calculated by multiplying the single trade discount rate (d) and the list price (L).
Amount of trade discount = Trade discount rate (d) # List price (L)
This can be written as the following formula:
Formula 5.2(a)
Trade discount rate,
Amount of Trade Discount
Amount of trade discount = d # L
The net price (N) is calculated by subtracting the amount of trade discount (d # L) from the list
price (L).
Net price (N) = List Price (L) - Amount of trade discount (d # L)
N = L - d # L
Taking the common factor 'L', results in the following formula for Net Price,
Formula 5.2(b)
Net Price
N = L(1 - d)
Note: (1 - d) is the complement of the discount rate (d). It is known as the Net Cost Factor (NCF).
The NCF is the percent of the list price that has to be paid by the buyer.
NCF = 100% - discount% = (1 - d)
Therefore, the Net Price can also be calculated by multiplying the list price by the complement of
the discount rate (or NCF).
Example 5.2(a)
N = L(1 - d)
N = L(NCF)
Calculating the Amount of Trade Discount and the Net Price
The list price of an item is $640. The trade discount is 20%. Calculate the amount of trade discount
and the net price of the item.
Solution
L = $640.00,
d = 20% = 0.20
Amount of trade discount = d # L
= 640.00 # 0.20 = $128.00
Net Price, N = List price (L) - Amount of trade discount (d # L)
= 640.00 - 128.00 = $512.00
Chapter 5 | Mathematics of Merchandising
Solution
continued
Or
Net Price, N = L(1 - d) = 640.00(1 - 0.20)
= 640.00 # 0.80 = $512.00
Amount of trade discount = List price (L) - Net price (N)
= 640.00 - 512.00 = $128.00
Therefore, the amount of trade discount is $128.00 and the net price of the item is $512.00.
Example 5.2(b)
Calculating the List Price and the Amount of Trade Discount
After a 40% trade discount, the net price of an item is $420. Calculate the list price of the item
and the amount of trade discount.
Solution
d = 40% = 0.40,
N = $420.00,
L=?
Using Formula 5.2(b),
N = L(1 - d)
Rearranging,
L=
Using Formula 5.2(a),
N
= 420.00 = 420.00 = $700.00
(0.60)
(1 - d)
(1 - 0.40)
Amount of trade discount = d # L
= 0.40 # 700.00 = $280.00
Or
Amount of trade discount = L - N
= 700.00 - 420.00 = $280.00
Therefore, the list price of the item is $700.00 and the amount of trade discount is $280.00.
Example 5.2(c)
Calculating the List Price and the Net Price
An item is purchased after a trade discount of 25% and the amount of trade discount received is
$150. Calculate the list price and net price of the item.
Solution
d = 25% = 0.25, d # L = $150.00,
Using Formula 5.2(a),
L = ?,
N=?
Amount of trade discount = d # L
Rearranging,
L = Amount of trade discount (d # L)
d
Substituting the values,
L =
Using Formula 5.2(b),
N= L(1 - d) = 600.00(1 - 0.25)
150.00 150.00
=
= $600.00
d
0.25
= 600.00(0.75) = $450.00
Or
N= List price - Amount of trade discount
= 600.00 - 150.00 = $450.00
Therefore, the list price is $600.00 and the net price is $450.00.
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Example 5.2(d)
Calculating the Trade Discount Rate
The list price is $500 and the net price is $400 for an item. What is the trade discount rate?
Solution
L = $500.00,
N = $400.00,
d=?
Amount of trade discount = L - N
Amount of trade discount = 500.00 - 400.00 = $100.00
d # L = $100.00
Substituting the values,
Or
d = 100.00 = 100.00 = 0.20 = 20.00%
L
500.00
Using Formula 5.2(b),
N = L(1 - d)
Substituting the values,
400.00 = 500.00(1 - d)
Expanding,
400.00 = 500.00 - 500.00(d)
Rearranging,
500.00(d) = 500.00 - 400.00
d = 100.00 = 0.20 = 20.00%
500.00
Therefore, the trade discount rate is 20.00%.
Series of Trade Discounts
Series of trade
discount rates are
never added to get a
single trade discount
rate.
Trade discounts of
10% followed by 5%,
and followed by 2%,
are not equal to 17%.
As explained earlier, trade discounts are provided for various reasons. At times, a buyer may qualify
to receive more than one trade discount rate on the list price of a product. In which case, the buyer
may be offered a series of trade discount rates (also known as multiple discount rates or chain
discount rates). These series of trade discounts are always applied in sequence, one after the other;
i.e., the second discount is applied on the amount after the first discount and the third discount is
applied on the amount after the first and second discounts.
For example, three trade discounts: a 10% seasonal discount, a 5% large purchase
discount, and a 2% new product discount were offered on a product listed at $1000.
If a buyer is eligible for all three trade discounts, then the net price and amount of
discount is calculated as follows:
The series of trade
discount rates are
usually written as
10%, 5%, and 2%.
Net price after the 1st discount = 1000(1 - 0.10) = 1000 # 0.90 = $900.00
Similarly,
Net price after the 2nd discount = 900(1 - 0.05) = 900 # 0.95 = $855.00
and,
Net price after the 3rd discount = 855(1 - 0.02) = 855 # 0.98 = $837.90
Therefore, Net price after the series of three trade discounts = $837.90
and,
Amount of trade discount = 1000.00 - 837.90 = $162.10
Example 5.2(e)
Calculating the Net Price of an Item After Two Trade Discounts
A manufacturer gives a trade discount of 10% for volume purchases and an additional discount of
5% for end-of-season purchases. How much would you pay for an item with a list price of $800 if
you qualify for both discounts? What is the amount of the trade discount?
Solution
Net price after the 1st trade discount = 800(1 - 0.10) = 800 # 0.90 = $720.00
Similarly,
Net price after the 2nd trade discount = 720(1 - 0.05) = 720 # 0.95 = $684.00
Chapter 5 | Mathematics of Merchandising
Solution
continued
Therefore,
Amount of the trade discount = List price - Net price after 2nd discount
= 800.00 - 684.00 = $116.00
Therefore, you would pay $684.00 for the item and the amount of trade discount is $116.00.
Calculating the Net Price of an Item Using the Formula for
Series of Trade Discounts
Let the list price of an item be 'L' and the series of trade discounts be d1 , d2 , d3, ... dn .
Net price after the 1st trade discount = L(1 - d1)
Similarly,
Net price after the 2nd trade discount = L(1 - d1)(1 - d2)
and,
Net price after the 3rd trade discount = L(1 - d1)(1 - d2)(1 - d3)
and,
Net price after the nth trade discount = L(1 - d1)(1 - d2)(1 - d3) ... (1 - dn)
Therefore, the formula for the net price after a series of trade discounts is shown below:
Formula 5.2(c)
Net Price After a Series of Trade Discounts
N = L(1 - d 1)(1 - d 2)(1 - d 3) ... (1 - d n)
Example 5.2(f)
Calculating the Net Price of an Item After a Series of Trade Discounts
A wholesaler gives a retailer the following three discounts on the $1250 list price of an item: 15%
for an end-of-season sale, 10% for a large quantity purchase, and 5% for paying cash on delivery.
Calculate the net price of the item if a retailer qualifies for all three discounts.
Solution
Using Formula 5.2(c),
N = L(1 - d1)(1 - d2)(1 - d3)
Substituting values,
N = 1250(1 - 0.15)(1 - 0.10)(1 - 0.05)
N = 1250(0.85)(0.90)(0.95) = $908.44
$1062.50
$956.25
$908.4375
Therefore, the net price of the item is $908.44.
Example 5.2(g)
Comparing the Net Price of Similar Items that Have Different List Prices and Trade Discounts
An electronics manufacturer, Rudolph Electronics, offers HDTVs for a list price of $3500 each,
offering trade discounts of 30% and 5%. Another manufacturer, Best Tech Electronics, offers a
similar model of HDTVs for $3430 each, offering trade discounts of 20% and 15%. Which offer
is cheaper and by how much?
Solution
Using Formula 5.2(c),
Substituting values for Rudolph Electronics,
N = L(1 - d 1)(1 - d2)(1 - d3)
NRudolph = 3500.00(1 - 0.30)(1 - 0.05)
= 3500.00(0.70)(0.95)
= $2327.50
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Solution
continued
Substituting values for Best Tech Electronics,
NBest Tech = 3430.00(1 - 0.20)(1 - 0.15)
= 3430.00(0.80)(0.85)
The difference in price = 2332.40 - 2327.50 = $4.90
= $2332.40
Therefore, Rudolph Electronic's offer is $4.90 cheaper than that of Best Tech Electronics.
Example 5.2(h)
Calculating the Additional Discount Rate to Match the Net Price of Similar Items
Company A sells a product at a list price of $300 with trade discounts of 20% and 5%. Company
B sells a similar product at a list price of $375 with a trade discount of 10%. What further trade
discount must Company B offer to match the price of the product offered by Company A?
Solution
Method 1:
Using Formula 5.2(c),
Substituting values for Company A,
NA = 300.00(1 - 0.20)(1 - 0.05)
= 300.00(0.80)(0.95) = $228.00
Substituting values for Company B,
N = L(1 - d1)(1 - d2)(1 - d3)
NB = 375.00(1 - 0.10)
= 375.00(0.90) = $337.50
Company B has to provide an additional trade discount of 337.50 - 228.00 = $109.50
Using rearranged Formula 5.2(a),
Amount of trade discount
Trade discount rate =
List price
109
.
50
Additional discount rate =
#
#100%
100%==32.44%
0.324444... # 100% = 32.44%
337.50
Therefore, Company B should provide an additional trade discount rate of 32.44% to
match the price of the product available at Company A.
Method 2:
Let the additional discount offered by Company B be d%.
NA after trade discounts of 20% and 5% = NB after trade discounts of 10% and d%
300.00(1 - 0.20)(1 - 0.05) = 375.00(1 - 0.10)(1 - d)
Rearranging and solving for d,
228.00 = 337.50(1 - d)
228.00 = 337.50 - 337.50d
= 337.50 - 228.00
dd =
337.50
= 0.324444... = 32.44%
Therefore, Company B should provide an additional trade discount rate of 32.44% to
match the price of the product offered by Company A.
Single Equivalent Trade Discount Rate for a Series of
Trade Discount Rates
As explained in the previous section on series of trade discount rates, we cannot simply add trade
discount rates to get the single equivalent trade discount rate. We can, however, find a formula to
calculate the single equivalent trade discount rate for a series of trade discount rates.
Let de be the single equivalent trade discount rate for a series of trade discount rates d1, d2, d3, ... dn .
Chapter 5 | Mathematics of Merchandising
From Formula 5.2(c),
N = L(1 - de)
From Formula 5.2(d),
N = L(1 - d1)(1 - d2)(1 - d3) ... (1 - dn)
Equating the above 2 formulas,
L(1 - de ) = L(1 - d1)(1 - d2)(1 - d3) ... (1 - dn)
Dividing by L on both sides,
(1 - de ) = (1 - d1)(1 - d2)(1 - d3) ... (1 - dn)
Rearranging the above will result in the formula for a single equivalent trade discount rate to a
series of trade discount rates:
Formula 5.2(d)
Single Equivalent Trade Discount Rate
de = 1 - [(1 - d1 )(1 - d2 )(1 - d3 ) ... (1 - d n)]
Example 5.2(i)
Calculating a Single Equivalent Trade Discount Rate for a Series of Trade Discount Rates
If an item with a list price of $1000 is discounted under a series of trade discounts of 10%, 5%, and
2%, what is the single equivalent trade discount rate that represents all these three trade discount
rates?
Solution
Method 1: Using the Single Equivalent Trade Discount Rate Formula
From Formula 5.2(d)
de = 1 - [(1 - d1)(1 - d2)(1 - d3)]
Substituting values,
= 1 - [(1 - 0.10)(1 - 0.05)(1 - 0.02)]
= 0.1621 = 16.21%
Therefore, the single equivalent trade discount rate that represents the series of three trade
discount rates is 16.21%.
Method 2: Using the Amount of Trade Discount And List Price
L = $1000.00
From Formula 5.2(c),
Substituting values,
So,
N = L(1 - d1)(1 - d2)(1 - d3)
= 1000.00(1 - 0.10)(1 - 0.05)(1 - 0.02)
= $837.90
Amount of trade discount = L - N
Trade discount rate, = 1000.00 - 837.90 = $162.10
d = Amount of trade discount (d # L)
List price (L)
162.10
de =
= 0.1621 = 16.21%
1000.00
Therefore, the single equivalent trade discount rate that represents the series of three trade
discount rates is 16.21%.
Method 3: U
sing the Net Price Formula for a Series of Trade Discounts and the Equivalent
Single Trade Discount Rate
L = $1000.00
For a series of trade discounts:
From Formula 5.2(c),
N = L(1 - d1)(1 - d2)(1 - d3)
Substituting values,
= 1000.00(1 - 0.10)(1 - 0.05)(1 - 0.02)
= $837.90
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Chapter 5 | Mathematics of Merchandising
Solution
continued
For the equivalent trade discount de :
From Formula 5.2(b)
N = L(1 - de )
= 1000.00(1 - de )
Equating both equations,
837.90 = 1000.00(1 - de)
Solving for de ,
837.90 = 1000.00 - 1000.00(de)
1000.00 de = 1000.00 - 837.90
de = 1000.00 - 837.90 = 162.10 # 100% = 16.21%
1000.00
1000.00
Therefore, the single equivalent trade discount rate that represents the series of three trade
discount rates is 16.21%.
Example 5.2(j)
Calculating the List Price, Given the Net Price and a Series of Trade Discount Rates
Jason, a retailer, paid $1349.46 for a flat screen TV at a Boxing Day sale after receiving three successive
discounts of 15%, 10%, and 2% on the TV. What was the list price of the TV?
Using Formula 5.2(c), N = L(1 - d1)(1 - d2)(1 - d3)
N
L=
Rearranging
(1 - d1 )(1 - d2 )(1 - d3 )
Solution
=
=
1349.46
(1 – 0.15)(1 – 0.10)(1 – 0.02)
1349.46
0.7497
= $1800.00
Example 5.2(k)
Therefore, the list price of the TV was $1800.00.
Calculating the Additional Trade Discount Rate Required
An item is listed for $3000 less discounts of 15%, 10%, and 5%. What further rate of discount should
be given to bring the net price to $2000?
Solution
Using Formula 5.2(c),
N = L(1 - d1)(1 - d2 )(1 - d3)
= 3000.00(1 - 0.15)(1 - 0.10)(1 - 0.05)
= $2180.25
F
urther discount of $2180.25 - $2000.00 = $180.25 should be given to bring the net price
to $2000.
Trade discount rate, Amount of trade discount
List price
180.25
d=
2180.25
d=
= 0.082674...
= 8.27%
Therefore, a further discount of 8.27% should be given to bring the net price to $2000.00.
Chapter 5 | Mathematics of Merchandising
5.2 |
Exercises Answers to the odd-numbered problems are available at the end of the textbook
For the following problems, express the answers rounded to two decimal places, wherever applicable.
Calculate the missing values for Problems 1 to 4.
1.
List Price
(L)
Single Trade Discount Rate
(d)
Amount of Trade Discount
(d # L)
Net Price
(N)
a.
?
20%
$375.00
?
b.
?
?
$27.60
$202.40
c.
$800.00
?
?
$368.00
d.
$500.00
?
$10.00
?
List Price
(L)
Single Trade Discount Rate
(d)
Amount of Trade Discount
(d # L)
Net Price
(N)
a.
?
5%
$50.50
?
b.
?
?
$217.50
$652.50
c.
$12,600.00
?
?
$630.00
d.
$30,750.00
?
2.
3.
a.
$3843.75
?
Equivalent Trade
Discount Rate
(de )
Net Price
(N)
Amount of
Trade Discount
(d # L )
20%, 10%
?
?
?
List Price
(L)
Series of Trade
Discount Rates
$540.00
b.
?
22%, 12.5%, 10%
?
?
$760.00
c.
?
5 12 %, 3%, 1%
?
$2512.00
?
d.
$1200.00
10%, 6%, d%
?
$984.74
?
List Price
(L)
Series of Trade
Discount Rates
4.
a.
$850.00
Equivalent Trade
Discount Rate
(de )
Net Price
(N)
Amount of
Trade Discount
(d # L )
10%, 5%
?
?
?
b.
?
5%, 4%, 3%
?
?
$80.50
c.
?
5 12 %, 2%, 1%
?
$1854.50
?
d.
$625
5%, 4%, d%
?
$558.60
?
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Chapter 5 | Mathematics of Merchandising
5. The list price of a fan is $165. During a sale, a trade discount rate of 8% was offered.
a. What is the amount of the trade discount?
b. What is the net price of each fan?
6. A new model of golf shoes are listed at $110 per pair. During a sale, they are offered at a trade discount rate of 15%.
a. What is the amount of trade discount offered per pair of shoes?
b. What is the net price of a pair of shoes?
7. An accounting software was sold at $780 per package after a trade discount rate of 10.5%.
a. What was the list price of the software?
b. What was the amount of discount?
8. A distributor of batteries received complete payment of $8560.50 for ten automobile batteries that it sold to a
wholesaler after a trade discount of 12%.
a. What was the list price of each battery?
b. What was the amount of discount offered for each battery?
9. A trade discount rate of 5% on a shipment of designer clothes resulted in a trade discount amount of $20,400.
a. What was the list price of the shipment of designer clothes?
b. What was the net price of the shipment of designer clothes?
10. A trade discount rate of 15% on a product resulted in a trade discount amount of $18,200.
a. What was the list price of the product?
b. What was the net price of the product?
11. A retail store purchased a shipment of coffee at a net price of $787.20. If this shipment was listed at $960.00,
calculate the trade discount rate.
12. The list price of canned foods at a wholesaler's outlet is $50 and the net price is $46. What is the trade discount rate?
13. Annabel's store is selling a newly released camera for $465.50 each. Vashti's store is selling the same model
for $490. What rate of discount should Vashti's store offer to match the price of the camera at Annabel's
store?
14. Muskoka Tools Depot is selling snow blowers for $725 each. The same model blowers are being offered by Dixie
Appliances for $667 each. What rate of discount should Muskoka Tools Depot offer to match the lower price?
15. A clothing wholesaler listed a dress for $280 and offered a standard trade discount rate of 10%. The dress did
not sell for two months, so she offered an additional trade discount rate of 12% on the dress during a clearance
sale. Calculate the amount of discount offered if both discounts are offered during a final sale.
16. A distributor of kitchen appliances could not sell a blender that was listed at $80.50 even after a trade
discount of 25%. He decided to discount it further and offered an additional trade discount of 25%. If the
distributor sold the blender at this new price, calculate the amount of trade discount offered on the list price.
17. The owner of a manufacturing plant gives his distributors trade discounts of 5%, 5%, and 1%. He wanted to
standardize these discounts and offer just one discount that was equivalent to these three discounts. What equivalent
trade discount rate should he offer?
18. A list price is discounted under a series of three trade discount rates of 12% each. What is the single equivalent
trade discount rate that represents these three discount rates?
Chapter 5 | Mathematics of Merchandising
19. Three trade discount rates of 5% each are offered on an airplane ticket that is listed at $1200.
a. What is the net price of the ticket?
b. What is the amount of discount offered?
20. Towards the end of winter, a distributor is selling its leather jackets that were listed for $600 with the following
three discounts: 15%, 10%, and 5%.
a. What is the net price of the leather jacket?
b. What is the amount of trade discount?
21. Marissa paid $190.57 for a printer that she purchased after receiving trade discounts of 20%, 15%, and 5%.
a. What was the list price of the printer?
b. What single equivalent trade discount rate represents the series of discounts received?
22. A shoe retailer received three trade discounts of 15% each on a pair of shoes that he purchased for $42.99.
a. What was the list price of the pair of shoes?
b. What single equivalent trade discount rate represents the series of discounts received?
23. An item is listed for $280 less discounts of 25%, 15%, and 10%. What further rate of discount should be given to bring the net
price to $150?
24. Skis are listed by a manufacturer for $850, less trade discounts of 30% and 15%. What further rate of discount
should be given to bring the net price to $450?
25. Is a series of trade discount rates of 20%, 10%, and 5% equivalent to a single trade discount rate of 31.60%?
26. Is a series of trade discount rates of 50%, 40%, and 10% equal to a single trade discount rate of 85%?
27. A distributor in Northern Canada lists a product for $8700 with trade discount rates of 10%, 8%, and 4%.
Another distributor in Western Canada lists the same model of the product for $8800 with trade discount
rates of 11% and 8%. The product is also listed online for $8900 with a single trade discount rate of 22%.
a. What is the net price in each case?
b. Where is the product available the cheapest?
28. Martin, the purchasing manager at a distribution company, was asked to source steel tubes. He received
the following quotes from vendors: Vendor A: $20,600, offering trade discount rates of 10%, 8%, and 7.5%;
Vendor B: $20,600, offering trade discount rates of 11% and 10%; Vendor C: $20,600, offering a trade
discount rate of 20%.
a. What is the net price in each case?
b. Whose offer is the least expensive?
29. A pet food company in Toronto, Ontario lists a bag of food for $22 with trade discount rates of 8% and 4%. At the same
time, a company in Orillia, Ontario lists the same food for $21.50 with trade discount rates of 7% and 5%.
a. Which company is offering the food for a cheaper price?
b. What further trade discount rate must the company with the higher price provide to match the lower price?
30. A specific model of computer servers are being sold by Company A for $26,500 each, offering trade discounts of
7% and 6% and by Company B for $35,500 each, offering trade discount rates of 12% and 2%.
a. Which company is offering the servers for a cheaper price?
b. What further trade discount rate must the company with the higher price provide to match the lower price?
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