Mathematics of Merchandising
7.5 Integrated Problems
Integrated problems often involve calculations that include trade discounts, markups, and markdowns. When tackling such problems, it is beneficial to approach the problem from the perspective of a business owner, focusing on determining costs and selling prices.
As these calculations are often sequential, with one building upon the next, it is important to break down the problem into manageable steps:
- Determine the Purchase Cost ([asciimath]C[/asciimath]): Start by calculating the cost of purchase if it’s not directly provided. This may involve applying trade discounts to the list price to find the actual cost.
- Establish Selling Price ([asciimath]S[/asciimath]): Use the information given about markup (M), expenses (E), and desired profit (P) and the proper formula that connects these elements. This will help you determine the regular selling price.
- Calculate the Reduced Selling Price ([asciimath]S_(Red)[/asciimath]): If the problem includes a markdown, apply the markdown rate to the original selling price to find the sale price or reduced selling price.
A store purchased drawers for $90 less 10% trade discount and marked them up 40% of the selling price. Operating expenses for each drawer accounted for 15% of the selling price. Later, when the drawer was discontinued, the store marked it down by 30%. a) What was the cost of each drawer? b) What was the regular selling price? c) What was the break-even price? d) What was the sale price? e) What was the profit or loss at the sale price?
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Show/Hide Solution
- List price: [asciimath]L=90[/asciimath]
- Trade discount: [asciimath]d=10%[/asciimath]
- Markup amount: [asciimath]M = 40%*S[/asciimath]
- Expenses are 12% of the selling price: [asciimath]E=15%*S[/asciimath]
- Markdown rate: [asciimath]MD\ Rate=30%[/asciimath]
a) [asciimath]C=?[/asciimath]
The purchase cost for each drawer is the net price that the retailer pays after applying the trade discount to the list price. Using Formula 7.3a, we can find the net price:
[asciimath]C=L(1-d)[/asciimath]
[asciimath]C=90(1-0.1)[/asciimath]
[asciimath]=$81[/asciimath]
b) [asciimath]S=?[/asciimath]
Since we know the percentages for markup relative to the selling price for each drawer (but not the markup exact amount), we can apply Formula 7.6 to calculate the selling price. However, as the actual value of the markup is not provided, we need to express it in terms of the unknown selling price. By substituting the markup into the formula, we can solve for the selling price algebraically.
[asciimath]S=C+M[/asciimath]
[asciimath]1S=81+0.40S[/asciimath]
[asciimath]1S-0.40S=81[/asciimath]
[asciimath](1-0.40)S=81[/asciimath]
[asciimath]0.60S=81[/asciimath]
[asciimath]S=81/0.60[/asciimath]
[asciimath]=$135[/asciimath]
c) [asciimath]S_(BE)=?[/asciimath]
To find the break-even selling price we need to know the expenses.
[asciimath]E=15%*S=15%(135)=$20.25[/asciimath]
The break-even selling price is given by Formula 7.11:
[asciimath]S_(BE)=C+E[/asciimath]
[asciimath]=81+20.25[/asciimath]
[asciimath]=$101.25[/asciimath]
d) [asciimath]S_(Red)=?[/asciimath]
To directly find the reduced selling price, we can apply Formula 7.14:
[asciimath]S_(Red)=S(1-"MD Rate)"[/asciimath]
[asciimath]S_(Red)=135(1-0.30)[/asciimath]
[asciimath]=135(0.70)[/asciimath]
[asciimath]=$94.50[/asciimath]
e) [asciimath]P_(Red)=?[/asciimath]
Approach 1:
Since the cost of the product and overhead expenses are fixed, reducing the selling price will also decrease the operating profit. Consequently, we can use Formula 7.8 with the reduced selling price to calculate the profit during the sale period.
[asciimath]P_(Red)=S_(Red)-C-E[/asciimath]
[asciimath]P_(Red) =94.50-81-20.25[/asciimath]
[asciimath]=-$6.75[/asciimath] Loss
Approach 2:
Once the break-even price is known, the profit or loss during the sale can be determined by comparing the sale price to the break-even price. If the sale price exceeds the break-even price, the retailer realizes a profit; conversely, if the sale price is below the break-even price, the retailer incurs a loss.
[asciimath]P_(Red)=S_(Red)-S_(BE)[/asciimath]
[asciimath]P_(Red)=94.50-101.25[/asciimath]
[asciimath]=-$6.75[/asciimath] Loss
Try an Example
A retailer purchased tablets at a list price of $420.00 each less trade discounts of 35% and 22%. The retailer’s overhead expenses amounted to 20% of the cost, and they had a profit of 30% of the cost of each tablet. During a holiday sale, the tablets were offered at a 25% markdown. a) What was the cost of each tablet? b) What was the regular selling price? c) What was the sale price? d) What was the profit or loss at the sale price?
Show/Hide Solution
- Tablet’s list price: [asciimath]L=420[/asciimath]
- Trade discount rate 1: [asciimath]d_1=35%[/asciimath]
- Trade discount rate 2: [asciimath]d_2=22%[/asciimath]
- Expense: [asciimath]E=20%*C[/asciimath]
- Profit: [asciimath]P=30%*C[/asciimath]
- Markdown rate was 25% of the regular selling price: [asciimath]"MD Rate"=25%[/asciimath]
a) [asciimath]C=?[/asciimath]
The purchase cost for each tablet is the net price that the retailer pays after applying the trade discount series to the list price. Using Formula 7.4, we can find the net price:
[asciimath]C=L(1-d_1)(1-d_2)[/asciimath]
[asciimath]C=420(1-0.35)(1-0.22)[/asciimath]
[asciimath]=420(0.65)(0.78)[/asciimath]
[asciimath]=$212.94[/asciimath]
b) [asciimath]S=?[/asciimath]
Applying Formula 7.8 gives
[asciimath]S=C+E+P[/asciimath]
[asciimath]S=1C+0.2C+0.3C[/asciimath]
[asciimath]=1.5C[/asciimath]
[asciimath]=(1.5)(212.94)[/asciimath]
[asciimath]=$319.41[/asciimath]
c) [asciimath]S_(Red)=?[/asciimath]
To find the reduced selling price, we can use Formula 7.14:
[asciimath]S_(Red)=S(1-"MD Rate)"[/asciimath]
[asciimath]S_(Red)=319.41(1-0.25)[/asciimath]
[asciimath]=319.41(0.75)[/asciimath]
[asciimath]=$239.56[/asciimath]
d) [asciimath]P_(Red)=?[/asciimath]
To find the profit during the sale, we need to know the expenses.
[asciimath]E=20%*C=20%(212.94)=$42.59[/asciimath]
Since the cost of the product and overhead expenses are fixed, reducing the selling price will also decrease the operating profit. Rearranging Formula 7.8 and plugging in the reduced selling price gives the profit during the sale:
[asciimath]P_(Red)=S_(Red)-C-E[/asciimath]
[asciimath]P_(Red) =239.56-212.94-42.59[/asciimath]
[asciimath]=-$15.97[/asciimath] Loss
Since [asciimath]P_(Red)[/asciimath] is a negative value, the retailer incurred a loss of $15.97 on each tablet.
Try an Example
Section 7.5 Exercises
- A store purchased laptops for $1,050 less 15% trade discount and marked them up 50% of the selling price. Operating expenses for each laptop accounted for 10% of the selling price. Later, when the laptop was discontinued, the store marked it down by 10%. a) What is the cost of purchasing each laptop? b) What was the regular selling price? c) What was the break-even price? d) What was the sale price? e) What was the profit or loss at the sale price?
Show/Hide Answer
a) C= $892.50
b) S = $1785
c) BE = $1071
d) Sale = $1606.50
e) Profit of $535.50
- A phone costs a retailer $1,400 less 29%, 14%. The retailer’s overhead expenses are 35% of the cost and its profit is 25% of the cost. During a sale, the phone is marked down 20%. a) What is the cost of purchasing each phone? b) What was the regular selling price? c) What was the amount of markdown? d) What was the sale price? e) What was the profit or loss at the sale price?
Show/Hide Answer
a) C = 854.84
b) S = $1367.74
c) MD = $273.55
d) Sale = $1094.19
e) Loss of $59.84
