BM2.2 – Merchandising | For the Love of Maths

Chapter 2, Lesson 2

In this lesson students will:

calculate the cost price, selling price, or trade discount of an item
calculate the markup or markdown of an item in dollars
use markups and markdowns, or their respective rates, to help calculate selling price, cost price or sale price
understand the difference between the rate of markup and gross profit margin, and be able to calculate given some financial information

Trade Discounts

When a retailer brings in a good to sell to the consumer, the retailer typically will get the item at a discounted price. For example, suppose the consumer pays $\$20$ for a t-shirt. It wouldn’t make sense for the retailer to purchase the t-shirt for $\$20$ from their distributor, otherwise they would not make a profit selling the shirt! What happens instead, is that the distributor applies a trade discount to the t-shirt so that the retailer can purchase it at a lower price and still make a profit selling the shirt.

A trade discount is a percentage applied to the selling price to obtain the buying price for the retailer. Let $S$ represent the selling price of an item in dollars, $C$ represent the cost for the retailer (in dollars) to purchase the item after discount, and $d$ represent the percentage trade discount. Then we can write the following:

$\begin{aligned} C &= S - S \times d \\ C&= S(1-d). \end{aligned}$

For example, if the selling price of a t-shirt was $\$20$ , yet the retailer obtained a $20\%$ trade discount on this item, the cost to bring in the t-shirt for the retailer is $C = \$20(1-0.2)=\$20(0.8)=\$16$ . The video below will work through a couple more examples.

Markups and Markdowns

Markups

A markup, denoted by $M_{U}$ , is a dollar amount added to the cost of an item to arrive at the selling price. In our t-shirt example above, we had $S = \$20$ and $C = \$16$ . The difference, in dollars, between the selling price and the cost is $M_{U} = \$20-\$16 = \$4$ , so there is a $\$4$ markup on the t-shirt. An equation connecting these three concepts is:

$\begin{aligned} S &= C + M_{U}. \end{aligned}$

Since the markup often accounts for overhead expenses ( $E$ ) and profit ( $P$ ), also in dollars, we have

$\begin{aligned} M_{U} &= E + P. \end{aligned}$

Finally, substituting the previous equation into the selling price equation above, gives an alternate expression for the selling price

$\begin{aligned} S &= C + M_{U} \\ S &= C + E + P. \end{aligned}$

Let’s work through a couple examples using these concepts.

Markdowns

Similarly, a retailer may decide to markdown an item if they intend to liquidate some of their stock. A markdown is a dollar amount taken off the original selling price to obtain a sale price. Denoting the markdown as $M_{D}$ , and the sale price as $S_{sale}$ , we have the following relationship:

$\begin{aligned} S_{sale} &= S - M_{D}. \end{aligned}$

Quite often markups and markdowns are better understood as percentages of the selling price or cost, which is what we will discuss next.

Selling Price and Cost Relationships

Rate of Markdown

We already know the relationship between the sale price, selling price and markdown (in dollars) to be $S_{sale} = S - M_{D}$ . The markdown can also be found by applying a percent discount to the selling price, very much like a trade discount. To differentiate between the markdown in dollars and the markdown as a percentage, we will call the latter the rate of markdown and write as $ROMD$ . The relationship is summarized here:

$\begin{aligned} S_{sale} &= S - S \times ROMD \\ S_{sale} &= S(1 - ROMD). \end{aligned}$

The following video will guide you through a markdown example.

Rate of Markup

Similarly, it is often beneficial to think about the markup as a percentage value, and quite often what we do is compare it to the cost. The ratio comparing the markup (in dollars) to the cost is called the rate of markup, and is denoted by $ROMU$ . Having a predetermined rate of markup on an item ensures that you are always making a profit, even as cost fluctuates.

In our t-shirt example, we have $C = \$16$ and $M_{U} = \$4$ . We might wonder what percentage of the cost is equivalent to the markup? Since $\frac{\$4}{\$16} = 0.25$ , we would say that the rate of markup is $25\%$ . In general, we have

$\begin{aligned} ROMU &= \frac{M_{U}}{C} \end{aligned}$

and

$\begin{aligned} S = C + C \times ROMU \\ S = C(1 + ROMU). \end{aligned}$

Let’s work through a couple examples using this concept.

Gross Profit Margin

The gross profit margin, or $GPM$ , is a ratio comparing the markup to the selling price. The gross profit margin gives us an indication of the markup in dollars as a percentage of total sales. In our t-shirt example, we had $S=\$20$ and $M_{U}=\$4$ . Since $\frac{\$4}{\$20} = 0.2$ , we might say the gross profit margin is $20\%$ . In other words, $20\%$ of our revenue was profit – $\$4$ of the $\$20$ revenue was profit, with the remaining $\$16$ going toward the cost of the t-shirt. The overall relationship can be summarized as follows: