A clothing retailer purchased a line of fall leather coats which were priced to sell at $600 each. This price reflected a markup of 45% on the selling price. At the end of the season the retailer had three coats left, which were marked down to 25% and sold. What was the retailer's actual percentage markup on the coats that were sold at 25% off? Calculate this percentage both as a percentage of the selling price, and as a percentage of the cost price.

If the markup was 45% of the selling price, the cost was 55%, so the markup based on cost was 80%.

So, if the cost was c,
1.8c = 600
c = 333.33

The sale price was 75% of 600, or 450.

So, the markup was 116.66

the sale price of 450 is thus a markup of

50% of cost
26% of selling price

To find the retailer's actual percentage markup on the coats that were sold at 25% off, we first need to calculate the original selling price and the cost price of the coats.

1. Calculate the original selling price:
The given price of $600 reflects a markup of 45% on the selling price. Let's denote the original selling price as "S". We can set up the following equation:
S + 0.45S = $600
Solving this equation, we find:
1.45S = $600
S = $600 / 1.45
S ≈ $413.79 (rounded to two decimal places)

2. Calculate the cost price:
Let's denote the cost price as "C" and the retailer's actual percentage markup on the cost price as "M".
The selling price can be expressed in terms of the cost price and markup as follows:
S = C + MC
Rearranging the equation to solve for the cost price:
C = S / (1 + M)
Plugging in the values we know:
C = $413.79 / (1 + M)

Now let's calculate the retailer's actual percentage markup on the coats that were sold at 25% off, expressed as a percentage of the selling price:

3. Calculate the selling price after the 25% markdown:
Since the coats were sold at 25% off, the selling price after the markdown can be calculated as:
S' = S - (0.25S)
S' = $413.79 - ($413.79 * 0.25)
S' ≈ $310.34 (rounded to two decimal places)

4. Calculate the actual percentage markup on the selling price:
The actual markup on selling price is the difference between the original selling price and the selling price after the markdown, divided by the selling price after the markdown, expressed as a percentage:
Markup on selling price = ((S - S') / S') * 100%
Markup on selling price = (($413.79 - $310.34) / $310.34) * 100%
Markup on selling price ≈ 33.37% (rounded to two decimal places)

Now let's calculate the retailer's actual percentage markup on the coats that were sold at 25% off, expressed as a percentage of the cost price:

5. Calculate the actual percentage markup on the cost price:
Again, the actual markup on the cost price is the difference between the original selling price and the selling price after the markdown, divided by the cost price, expressed as a percentage:
Markup on cost price = ((S - S') / C) * 100%
Markup on cost price ≈ (($413.79 - $310.34) / ($413.79 / (1 + M))) * 100%

Substituting the value of C, we get:
Markup on cost price ≈ (($413.79 - $310.34) / ($413.79 / (1 + M))) * 100%
Markup on cost price ≈ (($413.79 - $310.34) * (1 + M) / $413.79) * 100%

Since we don't know the value of M, we can't calculate an exact percentage. However, we can observe that the actual markup on the selling price will always be greater than the markup on the cost price, as the selling price is higher than the cost price.

I hope this helps! Let me know if you have any further questions.