# Incorrectly Computed Discount Followed By Markup

Quote of the Day

Doubt is not a pleasant condition, but certainty is an absurd one.

— Voltaire. I have to agree – it seems that the most dangerous people are often those who are absolutely certain of something, but should not be.

## Introduction

Figure 1: Tile Looks Great, But I Do Not Like to Install It (Source).

A few years ago, I hired a tile installer to tile a bathroom I was remodeling. He was a talkative guy, and he casually mentioned that he had chosen not to make any money on his material – he would make his money on labor alone. He said that most of the tile shops he works with give him a 15% discount on material, which he passes on to his customers. Other contractors markup their material charge by 15% to bring the cost back to retail price – this statement bothered me because the markup should be larger than the discount. The math did not seem right to me at the time, but I did not raise any questions because I wanted my bathroom done. I did not think much about this math error at the time until I saw an article in the Journal of Light Construction (JLC) telling contractors that this is a math error that is costing them money. As I read the article, I realized that this is exactly the same error I heard my tile contractor make.

I have seen a variation of this error when engineers will use conversion cost (which is really a margin) as a markup (see this post for more information). The key is to understand that a percentage discount on a product's cost does not equal the markup required to bring the discounted product price back to its original value.

## Analysis

### Example Error

Figure 2 shows the math error that my tile installer would have made if he chose to markup his cost by the supplier's discount percentage. The contractor thought that the cost reduction due to a percentage discount would be cancelled out by a markup of the same percentage – it is not.

Figure 2: Tile Installer Math Error.

### Reason For The Error

Figure 3 shows why the error occurs. Essentially, there is an error term equal to the square of the discount rate.

Figure 3: Reason For The Discount/Markup Error.

### Correct Markup Derivation

Figure 4 shows the equation for the markup value (highlighted) that the contractor should have used. Note that computer algebra systems often throw an arbitrary minus sign onto the front of their simplified expressions. I do not know why – however, the expressions they derive are correct.

Figure 4: Derivation of the Correct Markup Percentage.

I summarize this result in Equation 1.

 Eq. 1 $\displaystyle {{K}_{{Markup}}}=\frac{{{{K}_{{Discount}}}}}{{1-{{K}_{{Discount}}}}}$

where

• KDiscount is the discount from the tile vendor.
• KMarkup is the markup the tile installer must apply to his tile cost to get back to the retail cost

### Correct Markup Example

Figure 5 shows how the markup should have been computed for the tile contractor to get back to the original price.

Figure 5: Example of a Correctly Computed Markup.

## Conclusion

I see people make these percentage calculation errors all time. The most common one I see is to use conversion cost as markup rather than a margin. I will provide a quick example of this error here:

• Assume I am buying an assembly from a contract manufacturer for $100. • Assume the contract manufacturer is charging me 12% of this$100 for his assembly charge, which is called the conversion cost.
• This means that the material cost (i.e. BOM cost) is $88. • Engineers normally know the BOM cost and try to calculate the assembly charge by multiplying$88·12%=$10.56, which is incorrect. • The correct way is determine the markup using Equation 1, 12%/(1-12%)=13.6%. Now we can take$88·13.6%=\$12, which is the correct answer.

When it comes to calculation errors, none are more common than those that involve dB.  The most common dB calculation error I see is to use 10 log () instead of 20 log() when working with voltage.

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