BUSINESS

QUARTER 1 Week 6
MATHEMATICS

NAME: _______________________________ GRD. & SEC.: _____________________

Competencies:
The learner differentiates mark-up from margins; describes how gross margin is used in sales; and computes single trade
discounts and discount series.

To the Learners:
Before engaging with the module, I want you to set aside other tasks or activities that may take away your attention while
enjoying the lessons contained in this module. Read the simple instructions below to successfully accomplish the objectives of this
kit. Have fun!
1. Follow carefully all the instructions indicated in every page of this module.
2. Writing enhances learning. Keep this in mind and take note of important concepts in your
notebook.
3. Perform all the provided activities in the module. As much as possible, answer the activities
in succession as they are designed connectedly with each other to prepare you gradually to a more challenging
activities at the later section of this module.
4. Let your facilitator/guardian assess your answers using the answer key card.
5. Analyze and answer the posttest and apply what you have learned.
6. Enjoy studying!

Expectations
This module is designed to help you understand the concepts and practical applications of the mark-up, margins, gross
margins in sales, single trade discounts, and discount series in business settings. The mathematical principles applied to each concept
and the arithmetic and algebraic techniques can be helpful in simplifying these into bite piece concepts.
After going through this module, you are expected to:
1. Differentiate the mark-up from the margin;
2. Describe how gross margins are used in sales;
3. Be familiar with the existing trade discount types used in sales and businesses;
4. Compute single trade discount and discount series; and
5. Convert a discount series into the equivalent single-trade discount.

Pre-test
Choose the letter of the correct answer. Write the chosen letter on a separate sheet of paper.
1. Find the cost of a box of pencils being sold for Php.135 with a 25% margin rate.
A. Php. 101.25 C. Php. 101.52
B. Php. 110.25 D. Php. 110.52
2. A jacket which costs Php.1 350 is being sold at Php. 2 700. What is the margin rate?
A. 30% C. 40%
B. 50% D. 60%
3. A Department Store paid Php.15 000 for a dining set. Expenses are 18% of the selling price while the net profit is 15% of the
selling price. What was the regular selling price?

A. Php. 22 088.06 C. Php. 22 188.06

B. Php. 22 288.06 D. Php. 22 388.06
1 1
4. What is the equivalent single trade discount for a discount series of 40%, 12 2%, 8 3 %, and 2%?
A. 52.84%. C. 50.85%
B. 60.84% D. 60.85%
HENRY E. PERALTA
BUSINESS MATHEMATICS QUARTER 1 WEEK 6 P a g e 1 | 10
5. A distributor was able to buy an item for Php. 736 after a trade discount series of 15%, 10%, and 4%. How much was the
original selling price of this item?
A. Php. 902.18 C. Php.1 002.18
B. Php.1 102.18 D. Php.1 202.18
Looking Back at your Lesson
From your previous lesson, you have learned about the business terms in buying and selling of goods and services and how
to calculate for these in a given business situation. These terms are Cost Price, Operating Cost, Net Profit, Selling Price, Mark-Up,
Mark-On, and Mark-Down. You will; however, focus on the definitions and formulas for some of these key terms to help you walk
through the entire track of this module as follows:

Cost Price: the price that a company or store has to pay for the goods it is going to sell the price that
has to be spent to produce goods or services before any profit is added
Operating Cost: the price (per unit) incurred relative to the production and sale of a commodity
Net Profit: money earned after the cost price and the operating costs are accounted for after the sale of a commodity
Selling Price: the price at which the commodity is sold per unit
Selling Price = Cost Price + Operating Cost + Profit
S=C+E+P
where S = Selling Price, C = Cost Price, E = Operating Expenses, P = Profit
Mark-Up: the difference between the selling price and the cost price; sometimes referred to as
Margin or Gross Profit.
Mark-Up = Selling Price – Cost Price
MU = S – C
where MU = Mark-up, S = Selling Price, C = Cost Price
Introduction to the Topic
The success of a business in selling and buying goods or services depends highly on the knowledge of the persons engaged
in the business. Fundamental to business engagement is the familiarity and clarity to the basic terms being used in the daily operation
of the business establishments. Among these terms that need distinction and clarity between each other are the Mark-Up and the
Margin. It was previously stated in the Looking Back at Your Lesson section that the two terms are similar. This section of the
module will provide you the clear understanding of how the Mark-up is different from a Margin at some very important business
perspectives. The distinct features of each term can guide the business person on how to project a reasonable profit and a competitive
selling price of the goods and services he/she offers to the costumers.

A mistake in the use of these terms may lead to a price setting that is either substantially too low or too high, which could
result to lost profits or lost sales, respectively. There can also be an unwanted impact on the market share, due to the excessively high
or low prices which may be well outside the prices charged by other business competitors

Lesson 1: Mark-Up Versus Margin

Based on the previous definition, a Mark-Up is defined to be the difference between the selling price and the cost price;
alternatively, you can also look at other definition for the Mark-Up as the amount by which the cost price of a product or service is
increased in order to derive the selling price. This alternative definition of the Mark-Up will be consistently used in the entirety of
this module. The definition of the Margin still refers to the selling price minus the cost price of the goods or services. In short, the
previous definition of Mark-Up is now being used as the definition for Margin in the subsections of this module.

Suppose a product sells for Php. 1 000 and the cost price is Php. 700, its Margin is Php. 300. On the context of the Mark-
up, it will also be equivalent to Php. 300 since it is the amount needed to raise the cost price of Php. 700 to reach the selling price of
Php. 1 000. Somehow, the Mark-Up is just equivalent to the Margin. But if the two will be viewed in terms of percentages, their
difference becomes apparent and vivid between each other since the Mark-Up rate equals to the Mark-Up amount divided by the
cost price of the product. On the other hand, the Margin rate is equals to the Margin amount divided by the selling price of the
product. As a result, the Mark- Up rate and the Margin rate can be calculated as follows:

Mark-Up rate Margin rate

𝑀𝑈𝑟 = 𝐶
𝑀𝑈 𝑀
𝑀𝑟 =
Where: 𝑀𝑈𝑟 = Mark-Up rate 𝑆
Where: 𝑀𝑟 = Margin rate
𝑀𝑈= Mark-Up amount
𝑀= Margin amount
𝐶= Cost Price
𝑃ℎ𝑝.300 300 3 𝑆= Selling Price
Thus; 𝑀𝑈𝑟 = 𝑃ℎ𝑝.700 = 700 = 7 ≈ 43% 𝑃ℎ𝑝.300 300 3
Thus; 𝑀𝑟 = 𝑃ℎ𝑝.1 000 = 1 000 = 10 = 30 %

It is noticeable that in the above equations, the two yield different percentages to come up with the same amount for both
the Mark-Up and Margin. Contextually, you can put the Mark-Up as the amount to be raised to the cost price to achieve the selling
price in relation to the cost price itself or simply the percentage of the cost price while the Margin is the amount in relation to the
selling price of the product or services or simply the percentage of the selling price. For comparison purposes, the matrix below will
show the differences in rate for Mark-Up compared to the Margin rate. Relatively, the Mark-Up rates are greater than the Margin

HENRY E. PERALTA
BUSINESS MATHEMATICS QUARTER 1 WEEK 6 P a g e 2 | 10
rates provided that the Selling price remains constant and having the cost price is an independent variable while the gross profit
(Mark-Up and Margin) is a dependent variable. If you would follow the same formula and calculations for the Mark-Up rate and
Margin rate previously discussed, you would be able to complete a table similar to the matrix shown below. Note that, it does not
necessarily mean that the Mark-Up rate is better than the Margin rate simply because the former is relative to the amount of the cost
price and the latter to the selling price. Having the cost price always less than the amount of the selling price.
Selling Price Cost Price Gross Profit Mark-Up Rate Margin Rate
Php. 1 000 Php. 900 Php. 100 11% 10%
Php. 1 000 Php. 800 Php. 200 25% 20%
Php. 1 000 Php. 700 Php. 300 43% 30%
Php. 1 000 Php. 600 Php. 400 67% 40%
Php. 1 000 Php. 500 Php. 500 100% 50%
Php. 1 000 Php. 400 Php. 600 150% 60%
Php. 1 000 Php. 300 Php. 700 233% 70%

Normally in business, both the Mark-Up approach and the Margin approach are useful in determining the desired reasonable
profit and competitive pricing for the goods and services being sold. It is being emphasized that the use of Mark-Up is a more value
or amount-oriented approach than that of using the margin which is a percentage or rate-oriented approach. So, when a business
person would prefer to think of a specific amount of money to add up to the cost price of the product to arrive at the selling price, it
is more indicative of the Mark-Up Method. In a similar manner that when a business person thinks of relating his gross profit as a
percentage of the selling price; then, it is indicative of a Margin Method.
The Margin calculation has a slight advantage in using it as basis for calculating the selling price as it is more stable
compared to the Mark-Up as the basis of the selling price. The Mark-up calculation is more likely to result in changes of the selling
price over time as it is calculated based on the cost price of the product or the services which over time could also vary that results
to different calculations of the selling price. A raw product used in producing the goods could be priced differently in different times
in our country, such as petroleum. As a result, this may lead to different price hikes and price rollbacks.

Lesson 2: Gross Margins in Sales

The vast knowledge of business persons in projecting the intended reasonable profits and competitive pricing of the goods
or services of his/her business is a key to a successful business. The examples below show the extensive applications of the Margin
Method in determining the intended profit and pricing for the goods and services. Although you cannot discount that other business
persons may use or may have been using the Mark-Up method more conveniently in their business endeavor as efficiently and as
effectively as the Margin Method; here, we will limit the discussion in the use of the Margin Method instead.
Example 1. A retailer bought 13dozens of eggs where each egg cost Php. 6.75. The owner was able to sell all the eggs at
Php. 1 980. Determine the Margin rate of the retailer.
Solution: Let 𝐶𝑇 be the total cost price of all the eggs and 𝑀𝑟 the Margin rate.

𝐶𝑇 = (13)(12)(𝑃ℎ𝑝. 6.75)  Calculate the total cost price of all the eggs using the equation.
𝑪𝑻 = 𝟏 𝟎𝟓𝟑
𝑀 = 𝑆 − 𝐶𝑇  Using the equation for the margin, you can directly substitute the total cost
𝑀 = 𝑃ℎ𝑝. 1 980 − 𝑃ℎ𝑝. 1 053 price; then, perform subtraction to determine the margin.
𝑴 = 𝑷𝒉𝒑. 𝟗𝟐𝟕
𝑀 𝑃ℎ𝑝. 927  By using the formula for the Margin rate, you can now substitute the values
𝑀𝑟 = = ≈ 𝟒𝟔. 𝟖𝟐%
𝑆 𝑃ℎ𝑝. 1 980 calculated previously.
Example 2. Find the selling price if the cost of the product is Php. 6 000 with a Margin rate of 50%.
Solution: Let 𝐶 and 𝑆 be the margin, the cost price and selling price of the product, respectively.

Equation 1: 𝑀 = 𝑆 − 𝐶  Using the formula for the Margin

Equation 2: 𝑀 = 0.5𝑆  Note that the gross profit or referred to as the Margin is equal to 50% of the selling price.
0.5𝑆 = 𝑆 − 𝑃ℎ𝑝. 6 000  By substitution of Equation 2 to Equation 1 as well as the value of the cost price.
 By the addition property of equality, the equation can be simplified. Note that the term S has
𝑆 − 0.5𝑆 = 𝑃ℎ𝑝 6 000 its coefficient to be 1.
0.5𝑆 = 𝑃ℎ𝑝 6 000  Simplifying the right side of the equation
0.5𝑆 𝑃ℎ𝑝. 6 000  Dividing the left side and the right side of the equation by the coefficient of S that is 0.5
=
0.5 0.5  The selling price of the product has a Margin rate of 50%
𝑺 = 𝑷𝒉𝒑. 𝟏𝟐 𝟎𝟎𝟎
Example 3. Suppose a construction firm wanted to bid for a government project which they assessed to have a total cost price of
materials to be around Php. 200 000. Which approach would be better in having a larger gross profit and in serving as the
basis for the amount for the bidding:
A. To double the amount of the cost price which is at Php. 400 000 as the basis to bid, or
B. To base the gross profit at 60% of the bidding cost to cover for the 20% operating cost of the entire project and to have
a 40% net profit (Gross Profit minus Operating Cost)?
HENRY E. PERALTA
BUSINESS MATHEMATICS QUARTER 1 WEEK 6 P a g e 3 | 10
Solution:

A. It may be haphazardly construed that option A would be the better choice since it will double the amount of the cost
price of Php. 200 000 to Php 400, 000.This simply implies that the Gross Profit (Mark- Up) is Php. 200 000.
B. Since the Margin bears the formula: 𝑀𝑎𝑟𝑔𝑖𝑛 = 𝑆𝑒𝑙𝑙𝑖𝑛𝑔 𝑃𝑟𝑖𝑐𝑒 − 𝐶𝑜𝑠𝑡 𝑃𝑟𝑖𝑐𝑒 or 𝑀 = 𝑆 − 𝐶 thus,
Equation 1: 𝑀 = 𝑆 − 𝐶  Using the formula for the Margin
Equation 2: 𝑀 = 0.6𝑆  Note that the gross profit or referred to as the Margin is equal to 60% of the
0.6𝑆 = 𝑆 − 𝑃ℎ𝑝. 200 000 selling price.
𝑆 − 0.6𝑆 = 𝑃ℎ𝑝 200 000  By substitution of Equation 2 to Equation 1 as well as the value of the cost
0.4𝑆 = 𝑃ℎ𝑝 200 000 price.
0.4𝑆 𝑃ℎ𝑝. 200 000  By the addition property of equality, the equation can be simplified accordingly.
=
0.4 0.4  Simplifying the left side of the equation
𝑺 = 𝑷𝒉𝒑. 𝟓𝟎𝟎 𝟎𝟎𝟎  Dividing the left side and the right side of the equation by the coefficient of S
Using Equation 1: that is 0.4
𝑀 = 𝑃ℎ𝑝 500 000 − 𝑃ℎ𝑝. 200 000  The bidding price for the construction project which is Php. 100 000 greater
𝑴 = 𝑷𝒉𝒑. 𝟑𝟎𝟎 𝟎𝟎𝟎 than that of option A.
Alternatively using Equation 2:
 Using Equation 1 you can now calculate the margin by subtracting the cost price
𝑀 = 0.6𝑆
from the selling price in this case the bidding price.
𝑀 = 0.6(𝑃ℎ𝑝. 500 000)
𝑴 = 𝑷𝒉𝒑. 𝟑𝟎𝟎 𝟎𝟎𝟎
 Also using Equation 2, the Margin can also be calculated having the same value
Mathematically, it implies that Option B is the better option for the bidding. The gross profit is 50% better than
Option A, having the gross profits of Php. 300 000 and Php. 200 000 respectively for Option B and Option A. This means
that the operational cost of 20% which sums up to Php. 100 000 can be easily covered. There is still a relatively higher net
profit of 40% at the amount of Php. 200 000. Option B is also the more strategic choice since this is a bidding process where
it is better to bid high and adjust the amount of the bid lower, if necessary, while the bidding is in progress. The higher
bidding amount provides more room for gross profit which guarantees coverage of the operational expenses and a reasonable
net profit as soon as the bidding has been won in the favor of the construction firm.

Lesson 3: Single Trade discount and Discount Series

The manufacturers or producers buy raw materials which they will use to produce finished products. The buying habits of
these manufacturers are to buy merchandises or raw materials at the lowest possible cost in order to maximize their profits. In the
trading arena, the trader or the buyer should determine the goods needed by the costumers and the price the costumers are willing to
pay for these merchandises. Knowing where to obtain the goods at the lowest possible price minimizes cost and maximizes profit. If
the buyers fail to account the demand of the goods, they will be stuck with the goods they have paid for and will fail to sell and
therefore will lose profit. This is a highly similar case for the manufacturers.
In general, a manufacturer sells their products in bulk to a wholesaler. On a similar manner, the wholesaler sells a lesser
number of the goods or finished products to the retailers, who in turn, sell the product to the end users or consumers like you and me.
To drive more consumers to purchase goods, discounts are being incorporated to the strategies. Trade discount simply
refers to the reduction of price or cost of goods and services while businesses are still able to profit from the sale of goods and
rendering of services at more sustainable and regular manner. The amount paid after the discount has been deducted to the list price
(selling price) is called the net price.

Types of Single Trade Discounts.

1. Incentive or Loyalty Discount: A discount provided by the seller for repeatedly and regularly patronizing the
products and business establishment. An example would be when you go to the market and buy meat products from a
regular vendor, you may be considered a “suki”; thus, a discount may be provided.
2. Cash Discount: A reduction in price for cash purchases as full payment of the merchandise or goods. Buying a
merchandise in cash is usually better than buying it through installments, as a higher discount is offered to cash
purchases.
3. Volume Discount: A reduction to the price of a product based on the quantity in the quote line or order line item.
Buying in volume provides an opportunity for discounted price per item rather than buying in small quantities.
However, caution should be considered because buying merchandise in order to obtain a discount is not worthwhile if
the products cannot be consumed or disposed within a short period of time. The money could have been used for
other more important purposes.

To compute for the TRADE DISCOUNT and the NET PRICE:

𝑻𝑹𝑨𝑫𝑬 𝑫𝑰𝑺𝑪𝑶𝑼𝑵𝑻 = ( 𝑇𝑅𝐴𝐷𝐸 𝐷𝐼𝑆𝐶𝑂𝑈𝑁𝑇 𝑅𝐴𝑇𝐸 ) (𝑆𝐸𝐿𝐿𝐼𝑁𝐺 𝑃𝑅𝐼𝐶𝐸 )

𝑵𝑬𝑻 𝑷𝑹𝑰𝑪𝑬 = 𝑆𝐸𝐿𝐿𝐼𝑁𝐺 𝑃𝑅𝐼𝐶𝐸 − 𝑇𝑅𝐴𝐷𝐸 𝐷𝐼𝑆𝐶𝑂𝑈𝑁𝑇

Example 1. Quartz’s Auto Supply purchased 80 tires from a local tire wholesaler at Php. 4 350 each. This purchase was subjected
to a trade discount of 30%. How much was saved by the client?
Solution: Assume the following: 𝐷 be the trade discount; 𝑑 be the discount rate; and 𝑆𝑇 as the total selling price of the 80 tires.
HENRY E. PERALTA
BUSINESS MATHEMATICS QUARTER 1 WEEK 6 P a g e 4 | 10
 𝑆𝑇 = (80)(𝑃ℎ𝑝. 4 350)  Calculate the total selling price of the 80 tires by multiplying the list price to the number
𝑺𝑻 = 𝑷𝒉𝒑. 𝟑𝟒𝟖 𝟎𝟎𝟎 of tires.
𝐷 = (𝑑)(𝑆)
 Using the formula for the trade discount substitute the total selling price to the equation
𝐷 = (0.3)(𝑃ℎ𝑝. 348 000)
which results to the amount of savings or discount.
𝑫 = 𝑷𝒉𝒑. 𝟏𝟎𝟒 𝟒𝟎𝟎

Example 2. Kimberly admired a pair of earrings that the jewelry store agent was offering for Php. 36,000, payable in 6 months. She
bargained to be given a discount and the agent agreed to give her 12% if she pays in cash. How much must be paid for the
pair of earrings to earn the discount?
Solution: Assume the following: 𝐷 the trade discount; 𝑑 the discount rate; 𝑆 the selling price; and 𝑁 the net price.
Equation 1: 𝐷 = (𝑑)(𝑆)  The formulas for the trade discount and the net price.
Equation 2: 𝑁 =𝑆−𝐷
Equation 3: 𝑁 = 𝑆 − (𝑑 )(𝑆) = 𝑆(1 − 𝑑 )  By substituting the value of the trade discount in formula 1 to formula 2
you will arrive at much shorter equation for the net price. Simplifying
𝑁 = 𝑃ℎ𝑝. 36 000(1 − 0.12)
further all you need to do is to subtract the discount rate to 1 and
𝑁 = 𝑃ℎ𝑝. 36 000(0.88)
multiply to the selling price or list price then the result is the net price.
𝑵 = 𝑷𝒉𝒑. 𝟑𝟏 𝟔𝟖𝟎
Example 3. A buyer was offered discounts by two suppliers of the same medical equipment. Supplier A offered a list price of Php.
32 500 with a net price of Php. 28 600. Supplier B offered a list price of Php.34 600 with the net price of Php. 28 750.
Which supplier offered a higher discount rate?
Solution: Assume the following for both suppliers: 𝑑𝐴 , 𝑑𝐵 the discount rates; 𝑆𝐴 , 𝑆𝐵 the selling prices or list prices; and 𝑁𝐴 , 𝑁𝐵 the
net prices.
Equation 1: 𝐷 = (𝑑)(𝑆)  Using the formula for the trade discount you can derive the formula for the
(𝑑)(𝑆) 𝐷 discount rate by dividing the equation by 𝑆
= =𝑑
𝑆 𝑆
Equation 2: 𝑁 =𝑆−𝐷  Using the formula for the net price you can also derive the equation for the
𝐷 =𝑆−𝑁 trade discount by adding 𝐷 and −𝑁 to both sides of the equation.
𝑆𝐴 −𝑁𝐴
Equation 3: 𝑑𝐴 = 𝑆𝐴
𝑃ℎ𝑝. 32 500 − 𝑃ℎ𝑝. 28 600  Simply substitute the value of the list price and the net price of Supplier A
𝑑𝐴 = which results to the discount rate of 12%.
𝑃ℎ𝑝 32 500
𝑃ℎ𝑝. 3 900
𝑑𝐴 = = 0.12 = 𝟏𝟐%  Simply substitute the value of the list price and the net price of Supplier B
𝑃ℎ𝑝 32 500
𝑆𝐵 −𝑁𝐵 which results to the discount rate of 17%.
Equation 4: 𝑑𝐵 =
𝑆𝐵
𝑃ℎ𝑝. 34 600 − 𝑃ℎ𝑝. 28 750
𝑑𝐵 =  Note that the higher the discount rate for any merchandise the better. Clearly
𝑃ℎ𝑝. 34 600 17% is greater than 12%. It follows that Supplier B offered a better discount
𝑃ℎ𝑝. 5 850 rate than Supplier A.
𝑑𝐴 = ≈ 0.16907 = 𝟏𝟕%
𝑃ℎ𝑝 34 600
𝒅𝑩 > 𝑑𝐴 ↔ 𝟏𝟕% > 12%

The Trade Discounts Series

In order for the business establishments to minimize its losses and maximize profits, they need to sell their respective
products as quickly as possible. They may offer successive and multiple trade discount rates to their clients to encourage purchases
and minimize the storage cost for such products. You may think that the multiple discount rates could be added to represent a
single trade discount, but this approach is highly erroneous since discount series must be applied in succession. For that, it follows
that each discount rate must be applied one after the other until all discount rates have been considered to arrive at the net price of
the product. In the latter part of this module, you will learn how to convert a given discount series into an equivalent single trade
discount.

Example 1. Trade discounts of 20%, 10%, and 8% are offered on an item listed for Php. 8 400. Find the Net Price in the nearest
centavo.
Solution: Assume the following: 𝐷 be the trade discount; 𝑑1 , 𝑑2 and 𝑑3 be the discount series rates; and 𝑆 as the selling price or the
list price. Note that trade discount series is highly similar to the previous solutions in Example 2 in page 7 and 8 that you
have encountered. The net price is calculated by applying each of the discount rates one after the other or in successions.

Equation 1: 𝐷 = (𝑑)(𝑆)  Using the formula for the trade discount rate and the net price you can
Equation 2: 𝑁 =𝑆−𝐷 derive a shorter equation for determining the net price for each stage of
Equation 3: 𝑁 = 𝑆 − (𝑑 )(𝑆) = 𝑆(1 − 𝑑 ) the discount series.
Equation 4: 𝑁1 = 𝑆1 (1 − 𝑑1 )

HENRY E. PERALTA
BUSINESS MATHEMATICS QUARTER 1 WEEK 6 P a g e 5 | 10
 𝑁1 = 𝑃ℎ𝑝. 8 400(1 − 0.2)  Remember that there will be three stages of your solution by applying
𝑁1 = 𝑃ℎ𝑝. 8 400(0.8) = 𝑷𝒉𝒑. 𝟔 𝟕𝟐𝟎 each of the discount rates one after the other. Simply substitute the list
Equation 5: 𝑁2 = 𝑆2 (1 − 𝑑2 ) price as 𝑆1 and the first discount rate of 20% to Equation 4 and arrive at
the initial net price 𝑁1 = 𝑃ℎ𝑝 6 720. In Equation 5 you assume that
𝑁2 = 𝑃ℎ𝑝. 6 720(1 − 0.1)
𝑁1 = 𝑆2 since it is just the initial net price after the first discount thus
𝑁2 = 𝑃ℎ𝑝. 6 720(0.9) = 𝑷𝒉𝒑. 𝟔 𝟎𝟒𝟖
you can substitute 𝑁1 = 𝑃ℎ𝑝 6 720 to 𝑆2 . Consequently, you arrive at
the second initial net price of 𝑁2 = 𝑃ℎ𝑝 6 048. For the last stage you
Equation 6: 𝑁3 = 𝑆3 (1 − 𝑑3 ) can repeat the similar process applied in Equation 5 to Equation 6 this
𝑁3 = 𝑃ℎ𝑝. 6 048(1 − 0.08) time assuming 𝑁2 = 𝑆3 . You will arrive at the amount of 𝑁3 which is
𝑁3 = 𝑃ℎ𝑝. 6 048(0.92) = 𝑷𝒉𝒑. 𝟓 𝟓𝟔𝟒. 𝟏𝟔 the net price after applying the 3 different discount rates in succession.

Equivalent Single Discount Rate

Alternatively, the trade discount series above can be converted into an equivalent single discount rate. Subsequently, this
equivalent single discount rate can be solved similarly with the previous examples in the trade discount topic. Using Algebra, a
simplified or shortened formula for the gross discount and the net price can be derived to shorten the computing process for these
unknown variables for a certain merchandise subjected to a trade discount series. Your skills in Basic Algebra will be useful in this
endeavor.
To derive the Net Price (The price of the merchandise after the trade discount series has been deducted to the selling price
or the list price)
Equation 6: 𝑁3 = 𝑆3 (1 − 𝑑3 )  In deriving the net price for the trade discount series, you can use
Equation 7: 𝑁3 = 𝑁2 (1 − 𝑑3 ) Eq. 6 which was previously discussed in this section. Since 𝑁2 =
𝑁3 = 𝑆(1 − 𝑑2 )(1 − 𝑑3 ) 𝑆3 you can directly substitute 𝑁2 for 𝑆3 shown in Eq. 7. The value
Equation 8: 𝑁3 = 𝑁1 (1 − 𝑑2 )(1 − 𝑑3 ) of 𝑁2 = 𝑆2 (1 − 𝑑2 ) shown in Eq. 5 can also be substituted for 𝑁2
𝑁3 = 𝑆1 (1 − 𝑑1 )(1 − 𝑑2 )(1 − 𝑑3 ) thus, the simplified form of Eq. 7.
𝑵𝟑 = 𝑺(𝟏 − 𝒅𝟏 )(𝟏 − 𝒅𝟐 )(𝟏 − 𝒅𝟑 )
Therefore:  Similarly, since 𝑁1 = 𝑆2 = 𝑆1 (1 − 𝑑1 ), it can again be substituted
𝑁3 = 𝑃ℎ𝑝. 8 400(1 − 0.2)(1 − 0.1)(1 − 0.08) to Eq. 7 thus, the resulting Eq. 8. By using Eq. 8 you can just
𝑁3 = 𝑃ℎ𝑝. 8 400(0.8)(0.9)(0.92) substitute the list price and the corresponding discount series rates
𝑁3 = 𝑃ℎ𝑝. 8 400(𝟎. 𝟔𝟔𝟐𝟒) in Example 1, which results to the same net price of 𝑵𝟑 =
𝑁3 = 𝑃ℎ𝑝. 8 400(𝟔𝟔. 𝟐𝟒%) 𝑷𝒉𝒑. 𝟓 𝟓𝟔𝟒. 𝟏𝟔
𝑵𝟑 = 𝑷𝒉𝒑. 𝟓 𝟓𝟔𝟒. 𝟏𝟔
Thus, for any discount series rates 𝑑1 , 𝑑2 , 𝑑3 , ⋯ , 𝑑𝑟 applied to any discount series transaction the net price can be calculated
using the formula: 𝑵𝒓 = 𝑺(𝟏 − 𝒅𝟏 )(𝟏 − 𝒅𝟐 )(𝟏 − 𝒅𝟑 ) ⋯ (𝟏 − 𝒅𝒓 ). Where 𝑟 is the number of the discount rates, 𝑆 is the selling price
or list price, and 𝑁𝑟 is the net price.

To derive the Gross Discount (The total discount applied to a transaction using the discount rates) and the Equivalent
Single Discount Rate.
Equation 9: 𝑁𝑟 = 𝑆 − 𝐷𝑟  In deriving the gross discount and equivalent single
𝐷𝑟 = 𝑆 − 𝑁𝑟 discount rate all you need to do is use the Eq. 2 explained
Equation10: previously, however there will be changes in notation to
indicate that you are dealing with the trade discount
𝐷𝑟 = 𝑆 − 𝑆(1 − 𝑑1 )(1 − 𝑑2 )(1 − 𝑑3 ) ⋯ (1 − 𝑑𝑟 )
series. Thus in Eq. 9 you can arrive at basic formula for
𝑫𝒓 = 𝑺{𝟏 − [(𝟏 − 𝒅𝟏 )(𝟏 − 𝒅𝟐 )(𝟏 − 𝒅𝟑 ) ⋯ (𝟏 − 𝒅𝒓 )]} the gross discount.
 Substitute the value of 𝑁𝑟 from Eq. 8 to Eq. 9 thus the
𝒅𝒔 resulting Eq. 10, then by factoring 𝑆 will represent the
gross discount series.
Equation 11: 𝐷𝑟 = (𝑑 )(𝑆) = (𝑆)(𝑑)  In deriving the equivalent single discount rate, you can
Equation 12: use Eq. 1 as iterated in Eq. 11. Clearly in comparing Eq.10
and Eq. 11 you can deduce that the equivalent single
𝒅𝒔 = 𝟏 − [(𝟏 − 𝒅𝟏 )(𝟏 − 𝒅𝟐 )(𝟏 − 𝒅𝟑 ) ⋯ (𝟏 − 𝒅𝒓 )]
discount rate is shown in Eq. 12
Thus, for any discount series rates 𝑑1 , 𝑑2 , 𝑑3 , ⋯ , 𝑑𝑟 applied to any discount series transaction the gross discount and the
equivalent single discount rate can be calculated using the formulas:
𝑫𝒓 = 𝑺{𝟏 − [(𝟏 − 𝒅𝟏 )(𝟏 − 𝒅𝟐 )(𝟏 − 𝒅𝟑 ) ⋯ (𝟏 − 𝒅𝒓 )]} and
𝒅𝒔 = 𝟏 − [(𝟏 − 𝒅𝟏 )(𝟏 − 𝒅𝟐 )(𝟏 − 𝒅𝟑 ) ⋯ (𝟏 − 𝒅𝒓 )].
Where 𝑟 is the number of the discount rates, 𝑆 is the selling price or list price, 𝐷𝑟 is the gross discount and 𝑑𝑠 is the
equivalent single discount rate.
Example 2. A distributor was able to buy an item for Php. 18 950 after a trade discount series of 18%, 12% and 7%. How much
was the original selling price of this item?
Solution: Assume the following: 𝑁𝑟 the net price; 𝑑1 , 𝑑2 and 𝑑3 be the discount series rates; 𝑑𝑠 the equivalent single discount rate;
and 𝑆 as the selling price or the list price.

HENRY E. PERALTA
BUSINESS MATHEMATICS QUARTER 1 WEEK 6 P a g e 6 | 10
 Equation 1: 𝐷𝑟 = (𝑑𝑠 )(𝑆)  Using the formula for the gross discount you can derive the
𝑫𝒓 equation for the selling price 𝑆 seen in Eq. 1.
𝑺=
𝒅𝒔  Using the formula for the net price you can likewise derive the
Equation 2: 𝑁𝑟 = 𝑆 − 𝐷𝑟 equation for the selling price 𝑆 as seen in Eq. 2. Since Eq.1 is
𝑺 = 𝑵𝒓 + 𝑫𝒓 equivalent to Eq. 2 thus by transitive property of equations
𝐷𝑟 you have Eq. 3.
Equation 3: = 𝑁𝑟 + 𝐷𝑟
𝑑𝑠

𝐷𝑟 = 𝑑𝑠 (𝑁𝑟 + 𝐷𝑟 )  By cross multiplication

𝐷𝑟 = 𝑑𝑠 𝑁𝑟 + 𝑑𝑠 𝐷𝑟  By Distributive Property of Multiplication
𝐷𝑟 − 𝑑𝑠 𝐷𝑟 = 𝑑𝑠 𝑁𝑟  Adding (−𝑑𝑠 𝐷𝑟 ) to both sides of the Eq.
𝐷𝑟 (1 − 𝑑𝑠 ) = 𝑑𝑠 𝑁𝑟  Common Monomial Factoring
𝐷𝑟 (1 − 𝑑𝑠 ) 𝑑𝑠 𝑁𝑟  Dividing both sides of the Eq. by(1 − 𝑑𝑠 )
=
(1 − 𝑑𝑠 ) (1 − 𝑑𝑠 )
𝒅𝒔 𝑵𝒓  The equation for finding the gross discount in terms of
𝑫𝒓 =
(𝟏 − 𝒅𝒔 ) equivalent single discount rate and net price.
𝑑 𝑁
𝑠 3  Adjusting the equation to the problem.
Therefore: 𝐷3 = (1−𝑑 ) 𝑠

Note that: 𝑑𝑠 = 1 − [(1 − 𝑑1 )(1 − 𝑑2 )(1 − 𝑑3 )]

 Calculating separately for the value of the equivalent single
= 1 − [(1 − 0.18)(1 − 0.12)(1 − 0.07)] discount rate, all you need to do is substitute the given rates
= 1 − [(0.82)(0.88)(0.93)] provided in the problem.
= 1 − [0.671088]
𝒅𝒔 = 𝟎. 𝟑𝟐𝟖𝟗𝟏𝟐
(0.328912)(𝑃ℎ𝑝. 18 950)
𝐷3 =  Substituting the computed values of the equivalent single
(1 − 0.328912)
discount rate to Eq. 3 and the given amount of the net price
𝑃ℎ𝑝. 6 232.8824 you will arrive at the amount of the gross discount.
𝐷3 =
0.671088
𝐷3 = 𝑃ℎ𝑝. 9 287.73
 Add the net price and gross discount you will have
Using Equation 2: 𝑆 = 𝑁𝑟 + 𝐷𝑟
𝑆 = 𝑃ℎ𝑝. 28 237.73 or the original selling price of the item.
𝑆 = 𝑃ℎ𝑝. 18 950 + 𝑃ℎ𝑝. 9 287.73
𝑺 = 𝑷𝒉𝒑. 𝟐𝟖 𝟐𝟑𝟕. 𝟕𝟑
Example 3. A latest model of a front load inverter type washing machine is being offered in the market by Mall A and Mall B. The
trade discounts being offered by Mall A are 20% and 10% while that of Mall B are 15% and 15%. If you were the buyer,
which is the better offer?
Solution: Assume the following: 𝑑1𝐴 , 𝑑2𝐴 and 𝑑1𝐵 , 𝑑2𝐵 be the discount series rates for Supplier A and Supplier B respectively and
𝑑𝑠 the equivalent single discount rate.
Mall A You need to calculate the equivalent single discount rate for Mall A by using
Equation 1: 𝑑𝑠𝐴 = 1 − [(1 − 𝑑1𝐴 )(1 − 𝑑2𝐴 )] the Eq. 1. This results to 𝟐𝟖%
= 1 − [(1 − 0.20)(1 − 0.10)]
= 1 − [(0.80)(0.90)] You need to calculate the equivalent single discount rate for Mall B by using
= 1 − 0.72 = 0.28 or 𝟐𝟖% the Eq. 2. This results to 𝟐𝟕. 𝟕𝟓%
Mall B
Equation 2: 𝑑𝑠𝐵 = 1 − [(1 − 𝑑1𝐵 )(1 − 𝑑2𝐵 )]
= 1 − [(1 − 0.15)(1 − 0.15)] Note that the higher the discount rate for any merchandise the better. Clearly
= 1 − [(0.85)(0.85)] 28% is greater than 27.75%. It follows that Mall A offered a better discount
= 1 − 0.7225 rate than Mall B.
= 0.2775 or 𝟐𝟕. 𝟕𝟓%
𝑑𝑠𝐴 > 𝑑𝑠𝐵 ↔ 𝟐𝟖% > 27.75%

Activities

Activity 1 True or False

Write the word TRUE if the statement is correct; otherwise, write FALSE.
𝐺𝑟𝑜𝑠𝑠 𝑃𝑟𝑜𝑓𝑖𝑡
__________1. A margin rate equals 𝑆𝑒𝑙𝑙𝑖𝑛𝑔 𝑃𝑟𝑖𝑐𝑒.
__________2. The margin rate is greater than the mark-up rate when the cost price is Php. 100 and gross profit of Php. 20.
__________3. The equation: S > C + E, shows that a business establishment gains a net profit.
__________4. The gross profit or margin is the sum of the operating cost and net profit.
__________5. A Mark up method is equating the selling price by adding a percentage of the cost price to the cost price itself.
HENRY E. PERALTA
BUSINESS MATHEMATICS QUARTER 1 WEEK 6 P a g e 7 | 10
Activity 2.1 Warm Up for the Margins
Complete the table below by providing the appropriate data needed for each row. You may use a calculator or spreadsheet
for your calculations. Round your answer to the nearest centavo for each monetary value and round the rate in percentage form to
two decimal places.
Selling Price Cost Price Gross Profit Mark-Up Rate Margin Rate
Php. 2 000 Php. 900 1. 2. 3.
Php. 8 000 4. Php. 5 000 5. 6.
7. Php. 7 000 Php. 4 500 8. 9.
Php. 11 000 Php. 6 750 10. 11. 12.
13. 14. Php. 10 000 15. 75%

Activity 2.2 Maximizing Profit

Solve the following word problems involving the applications of the Margin in buying and selling. You may use a calculator
or spreadsheet for your calculations. Feel free to explore and apply other applicable alternative solutions which you may have learnt
beyond the domains of this module.
1. At what Margin rate should a pair of shoes be sold, which was bought at a price of Php 4 550 so that the seller can
earn a gross profit of Php. 950?
2. Noly, an art collector wants to sell his highly collectible Fernando Amorsolo’s self-portrait painting costing Php.
170 500 twenty years ago. He wants a Margin rate of 90%. How much should the selling price be?
3. An importer bought a brand-new Euro 4 diesel engine generator overseas worth
Php 12 850 500 including all applicable taxes, tariffs, and related fees. He wanted to sell this locally to a power
generating plant in Batangas with a margin of 20%. How much should the importer sell the generator?

Activity 3.1 Price Off Practice

Complete the table below by providing the appropriate data needed for each row. You may use a calculator or spreadsheet
for your calculations. Round your answer to the nearest centavo for each monetary value and round the rate in percentage form to
two decimal places.
List Price Discount Rate Discount Net Price
Php. 13 400 10% 1. 2.
Php. 7 500 3. 4. Php. 6 200
5. 5% 6. Php. 1 500
7. 8. Php. 125 Php. 485
9. 10. Php. 150 Php. 2060

Activity 3.2 The Single Me

Match the given trade discount series in column A to its corresponding equivalent single trade discount in Column B.
Write the letter of your choice on your answer sheet.

Column A Column B
______1. 15% and 10% A. 32.50%
______2. 10% and 5% B. 28.00%
______3. 20% and 10% C. 23.50%
______4. 25% and 10% D. 21.30%
______5. 16.5% and 5.75% E. 14.50%
______6. 10%, 5% and, 3% F. 14.28%
______7. 30%, 10%, and 5% G. 17.07%
______8. 20%, 5%, and 10% H. 19.13%
1 1 3 I. 24.10%
______9. 10 2 %, 3 2 %, and 4 %
1 J. 31.60%
______10. 10%, 8%, 5%, and 3 %
2
K. 40.15%

Activity 3.3 Bodega Sale!

Solve the following word problems involving the applications of the trade discounts in real life situations. You may use a
calculator or spreadsheet for your calculations. Feel free to explore and apply other applicable alternative solutions which you may
have learnt beyond the domains of this module.
1. Your regular meat supplier gives you an 8% discount on your meat purchases every week. If the price of chicken wings
this week is Php. 135 per kilo and you are planning to buy 3.5 kilos, how much trade discount will you receive from this
purchase?

HENRY E. PERALTA
BUSINESS MATHEMATICS QUARTER 1 WEEK 6 P a g e 8 | 10
 2. A certain refrigerator manufacturing firm sold different units of newly manufactured refrigerators to one of its dealers as
follows: 10 units (6 cu, ft,) at Php. 8 500 each; 15 units (10 cu. ft.) at Php. 14 750 each; and 20 units (12 cu. ft.) at Php. 20
750 each. The entire purchase carried a discount rate of 35%. What must be the total net price to be paid by the dealer?
3. Find the net price of a 2018 stock gaming console listed at Php 28 665 with the discount series of 20% for cash purchase,
10% for limited edition, and 5% for having an incentive card.
4. What must be the list price of a newly introduced triple-compressor inverter type air-condition unit if a discount series of
12.5% and 10% yields a net price of Php. 27 750?

Remember
Margin rate
𝑀 Mark-Up rate
𝑀𝑟 = 𝑀𝑈𝑟 = 𝐶
𝑀𝑈
𝑆
Where: 𝑀𝑟 = Margin rate Where: 𝑀𝑈𝑟 = Mark-Up rate
𝑀= Margin amount 𝑀𝑈= Mark-Up amount
𝑆= Selling Price 𝐶= Cost Price

𝑻𝑹𝑨𝑫𝑬 𝑫𝑰𝑺𝑪𝑶𝑼𝑵𝑻 = ( 𝑇𝑅𝐴𝐷𝐸 𝐷𝐼𝑆𝐶𝑂𝑈𝑁𝑇 𝑅𝐴𝑇𝐸 ) (𝑆𝐸𝐿𝐿𝐼𝑁𝐺 𝑃𝑅𝐼𝐶𝐸 )

𝑵𝑬𝑻 𝑷𝑹𝑰𝑪𝑬 = 𝑆𝐸𝐿𝐿𝐼𝑁𝐺 𝑃𝑅𝐼𝐶𝐸 − 𝑇𝑅𝐴𝐷𝐸 𝐷𝐼𝑆𝐶𝑂𝑈𝑁𝑇
For any discount series rates 𝑑1 , 𝑑2 , 𝑑3 , ⋯ , 𝑑𝑟 applied to any discount series transaction the net price, the gross discount and
the equivalent single discount rate can be calculated using the formula: 𝑵𝒓 = 𝑺(𝟏 − 𝒅𝟏 )(𝟏 − 𝒅𝟐 )(𝟏 − 𝒅𝟑 ) ⋯ (𝟏 − 𝒅𝒓 );
𝑫𝒓 = 𝑺{𝟏 − [(𝟏 − 𝒅𝟏 )(𝟏 − 𝒅𝟐 )(𝟏 − 𝒅𝟑 ) ⋯ (𝟏 − 𝒅𝒓 )]} ; and
𝒅𝒔 = 𝟏 − [(𝟏 − 𝒅𝟏 )(𝟏 − 𝒅𝟐 )(𝟏 − 𝒅𝟑 ) ⋯ (𝟏 − 𝒅𝒓 )].
Where 𝑟 is the number of the discount rates, 𝑆 is the selling price or list price, and 𝑁𝑟 is the net price, 𝐷𝑟 is the gross
discount, and 𝑑𝑠 is the equivalent single discount rate.

Check your Understanding

Solve the following problems below. Write your respective solution and answer on a separate sheet of clean paper. You
may use a calculator or spreadsheet for your calculations; however, your solutions must be written clearly in the sheet of paper. If
there is a need to articulate a certain portion of your solution, you may write some explanation in English or Filipino.
1. Continental trading purchased chemicals worth Php. 30 650 and sold it for Php. 41 659.50. What was the Margin rate?
2. A fruit dealer bought 6 crates of pomelo or suha for Php. 2 490. The dealer has a Margin rate of 60% to cover for the
expensive transportation costs. How much was the selling price for each crate?
3. The list price of a 2019 brand new laptop computer at a certain online shopping platform is Php. 42 350. The trade discount
rate is 20%, How much should the buyer pay for the laptop computer?
4. Lake Cebu Sports Center bought three sets of scuba diving gears worth Php. 82 500 each set. If the total amount paid was
Php. 206 200, what was the discount rate?
5. Find the net price of a merchandise listed at Php. 21 500 with the discounts of 30%, 20%, and 10%. Calculate to the nearest
centavo.
6. The purchase order of Urduja Department Store is shown below. Determine the total net price. Compute to the nearest
centavo.
ITEM QUANTITY LIST PRICE PER PAIR DISCOUNT NET PRICE
Size 10 men’s shoes 25 pairs Php. 1 572 10% & 5% ?
Size 7 ladies’ shoes 12 pairs Php. 999 8% & 6% ?
Assorted socks 30 pairs Php. 120 5%, 3%, & 1% ?
TOTAL NET PRICE ?
7. Topaz’s Beauty and Health Spa was offered a variety of beauty products with trade discounts of 15% and 10% for buying
in bulk and purchasing in cash respectively. If the sum of the list price was worth Php. 85 760 how much payment must be
made to close the deal?
8. On a merchandise worth Php. 56 200, Supplier A offers a discount series of 25%, 10%, and 15%. Supplier B offers 30%
and 20%. Which is a better offer and by how much?

Post-test

Choose the letter of the correct answer. Write the chosen letter on a separate sheet of paper.
1. Find the cost of a box of pencils being sold for Php.135 with a 25% margin rate.
A. Php. 101.25 C. Php. 101.52
B. Php. 110.25 D. Php. 110.52

HENRY E. PERALTA
BUSINESS MATHEMATICS QUARTER 1 WEEK 6 P a g e 9 | 10
2. A jacket which costs Php.1 350 is being sold at Php. 2 700. What is the margin rate?
A. 30% C. 40%
B. 50% D. 60%
3. A Department Store paid Php.15 000 for a dining set. Expenses are 18% of the selling price while the net profit is 15% of the
selling price. What was the regular selling price?

A. Php. 22 088.06 C. Php. 22 188.06

B. Php. 22 288.06 D. Php. 22 388.06
1 1
4. What is the equivalent single trade discount for a discount series of 40%, 12 2%, 8 3 %, and 2%?
A. 52.84%. C. 50.85%
B. 60.84% D. 60.85%
5. A distributor was able to buy an item for Php. 736 after a trade discount series of 15%, 10%, and 4%. How much was the
original selling price of this item?
A. Php. 902.18 C. Php.1 002.18
B. Php.1 102.18 D. Php.1 202.18

Additional Activities

To better understand the lesson, watch the video lesson on https://www.youtube.com/watch?v=rD-l7kO4-F4 and
https://www.youtube.com/watch?v=tXQBU12ha3s&t=111s entitled “Margin versus Markup” and “Trade Discount
Series / Single Equivalent Rate” respectively.

References:
Bragg, S., “The Difference Between Margin and Markup”, Accounting Tools. March 4, 2014
Retrieved from: https://www.accountingtools.com/articles/what-is-the-difference-between-margin-and-
markup.html#:~:text=The%20difference%20between%20margin%20and%20markup%20is%20that%20margin%20is,to%
20derive%20the%20selling%20price.&text=For%20example%2C%20if%20a%20product,manufacture%2C%20its%20m
argin%20is%20%2430.

Department of Education, “ Business Mathematics Teaching Guide”

Pasion, D.,” Business Mathematics”, National Bookstore Inc., 1983, Pages 92-113

HENRY E. PERALTA
BUSINESS MATHEMATICS QUARTER 1 WEEK 6 P a g e 10 | 10