Percent Change: Markup and Margin

We have looked at how to calculate, apply, and undo a percent increase or decrease. Here we will look at some special terms used in business for percent increases, which have been a source of many questions over the years.

Finding the markup or margin

Markup and margin refer to the profit (difference between cost and selling price) as a percentage of either cost or price, respectively. These are closely related to percent change. Consider this question from 2007:

Profit Margin and Percentage Markup

I have a small business and need to be able to compute profit margins and percentage increases.  There seems to be a difference of semantics when I try to discuss this with customers.  

Specifically, suppose I have an item that my customer can buy at the wholesale price of $60.00, and then they can sell it at the retail price of $125.00.  What is the percentage mark-up and is this a different number than what they would consider their profit margin?

I believe to find the increase I would just take 60 divided by 125 which would equal .48 which would mean a 48% increase on price.  Is this also considered the profit margin?  I don't think so!  I could say to the client that they have made a profit of what percent?

Evelyn has actually calculated what percent the cost is of the price. Doctor Ian responded, first giving a (now broken) link to a source of definitions, then continuing:

Following these definitions, 

                        gross profit
  gross profit margin = ------------
                           sales

                        125 - 60
                      = --------
                          125

                         65
                      = ---
                        125

                      = 0.52 
     
or 52 percent.  And

           sales price - cost
  markup = ------------------
                  cost

           125 - 60
         = --------
              60

           65
         = --
           60

         = 1.08

or 108 percent.

The difference is what you're comparing the profit to:  If you express the profit as a percentage of the sale price, you're computing the profit margin.  If you express the profit as a percentage of the cost, you're computing the markup.

In other words, the markup is the percent increase from the cost to the selling price; the margin is the percent decrease from the selling price to the cost, because the selling price is taken as the base for the percentage.

Here are the formulas, again: $$\text{markup} = \frac{\text{selling price}- \text{cost}}{\text{cost}}\cdot100\%$$ $$\text{margin} = \frac{\text{selling price}- \text{cost}}{\text{selling price}}\cdot100\%.$$

Applying a given margin to a cost

Applying a markup, when you know the cost, is easy; you are just applying a percent increase to the cost. But when it is the margin you know, finding the price is the same as undoing a percent decrease.

Here is a question from 2006:

The Difference between Calculating Markup and Profit Margin

I have a question about how to calculate [selling price from] profit margin.  We have been calculating this as fixed costs multiplied by whatever the margin percentage is.  For example, with costs of $50,000 and a desired margin of 25% we do $50,000 x 1.25 = $62,500, so we would sell the product for $62,500.

But I am told that the correct way to calculate this is to take the fixed costs and divide by the desired margin percentage subtracted from 100%.  So that would be $50,000 / .75 = $66,667.  I am having some trouble understanding how this works.  Can you explain this is simple layman's terms?

I answered, emphasizing the distinction:

You have no idea how common this question is!

Profit margin means "what percentage OF THE PRICE is profit?"  That is different from "what percentage OF COST is the profit?"--that's the "markup".  When we mark up, we add some percentage of what it costs us.  Margin is seen from the customer's perspective: how much of what I'm paying is going into their pockets?

So you want to increase your fixed cost of $50,000 by some amount that will be 25% of the amount you'll be charging--which you don't know yet!  You can't just add on 25% of the fixed cost, because that's the right percentage of the wrong thing.  (You're actually calculating a 25% markup rather than a 25% margin.)

When you have to work with a number you don’t know yet, that is where algebra is useful!

So what can we do?  Well, we think backward.  Suppose we knew the price we're going to charge; I'll call it X.  Then the margin would be 25% of that, or 0.25 times X.  The cost would be that much less than X:

  cost = X - 0.25X

But 1 times X, minus 0.25 times X, is 0.75 times X.  That is, 100% - 25% is 75%.  So

  cost = 0.75X

But now we can find X, because we know the cost is $50,000!

  50,000 = 0.75X

  X = 50,000 / 0.75

That is, we undo the multiplication by 0.75, by dividing 50,000 by 0.75. So

  X = $66,666.67

This is just what we did to reverse a percent decrease: divide by the complement of the percentage.

We can check the result, of course. A 25% margin would mean that the profit is 25% of this, namely \(0.25\times \$66,666.67 = \$16,666.67\). And, in fact, the profit is the selling price minus cost, which is \(\$66,666.67 – \$50,000 = \$16,666.67\).

Note that this is more than what you were calculating, because you were adding 25% of a smaller number--here we're adding 25% of $66,667 rather than only 25% of $50,000.  By charging only $62,500, you had a 25% markup but only a 20% margin (12,500/62,500).

Here are the formulas for applying a markup and a margin: $$\text{selling price} = \frac{100 +\text{markup}}{100}\cdot\text{cost}$$ $$\text{selling price} = \frac{100}{100 – \text{margin}}\cdot\text{cost}.$$ Or, using decimals rather than percent values, $$\text{selling price} = \left(1 +\text{markup}\right)\cdot\text{cost}$$ $$\text{selling price} = \frac{1}{1 – \text{margin}}\cdot\text{cost}.$$

Markup or margin, combined with tax

Here is a 2004 question that initially dealt with the common confusion between markup and margin, then added a twist:

Calculating Percentage Markup Versus Profit

In order to figure a profit of 10% I would normally take the amount and then multiply by 1.1 to find the selling price.  For example, 800 x 1.1 = 880 so I say we should charge $880 to make a 10% profit. However my employer tells me that in order to find a 10% profit I should divide by .9 to get 800/.9 = $888.89.  I do not understand how this is 10%.  Can you please explain?  Thank you very much.

They are both calculating the selling price to yield a given percent profit, but they are talking past one another by not using clear terms. I replied (somehow neglecting to use the term “margin”, which I am adding here for clarity):

You and your employer are calculating two subtly different things: a markup versus a profit [margin].

Your calculation is correct when you are given the cost and want to find a sale price that will give you a profit of 10% OF THE COST TO YOU.  If the cost to you is C and you sell it for 1.1C, your profit is 

  1.1C - C = 0.1C, or 10% of C.

which is 10% of what you paid for the item, or a 10% markup. 

But your employer is calculating the price that will give you a profitof 10% OF THE SELLING PRICE.  If you sell the item for C/0.9, then your profit is

  C/0.9 - C = (C - 0.9C)/0.9 = 0.1 * C/0.9, or 10% of C/0.9.

which is 10% of the selling price, C/0.9.  What he is doing is solving the equation

  P - C = 0.1P

which says the profit (price minus cost) is 10% of the price, this way:

  P - 0.1P = C

  0.9P = C

  P = C/0.9

You method solves the equation

  P - C = 0.1C

Your employer's method is correct for a 10% profit margin on the sale.  Your method is correct for a 10% markup over the cost.  It's all a matter of words.

We have found that this sort of miscommunication is common; that’s why we have different words for the two cases, and why the base must always be specified when we talk about percentages.

Lisa had another question:

Thanks!  I can follow this now that I see the mathematical equations written out.  This has been a great help.

I have another similar question for you, if I could, again involving us buying and then selling an item, but now figuring tax in as well.

For example, the price we pay is $250.00 (tax not included).  We would then figure in 7% tax (250 * 1.07 = 267.50) to find our total cost.  We then proceed to figure out our selling price by calculating 267.50/0.9 = 297.22.  This selling price will give us 10% profit.

Other people believe that we should do it the following way.  Take 250.00 (our cost)/0.93 = 268.82 (price with tax) and then calculate 268.82/0.9 = 298.69 for the selling price with 10% profit. 

I have always figured sales tax as a multiplication factor but then again I had never thought about the difference between a markup of 10% and a selling price with a 10% profit either, so I wanted to verify that I truly am understanding everything correctly.

Lisa is calculating tax on their cost as a markup, and profit as a margin. Others are treating both like margins. I answered:

There could be tax issues that are beyond my knowledge, but the math aspect seems straightforward.

You would divide by 0.93 if, as in the case of the profit, you wanted 7% of the price you charge to go for tax.  But that isn't how tax works.  The tax part of the transaction, as I understand it, is simply your payment when you buy the item, based on the amount charged to you, so your calculation is clearly correct, and the fact that the other version gives a different answer shows that it is wrong.

You can always check something like this (assuming you have defined clearly what you want to do) by seeing how much the actual tax and profit are based on the number you calculate.  If you charge $297.22; 10% of that is $29.72, which leaves $267.50.  That is $250 plus the 7% tax.  The other number will not check out this way.

To put it simply, you multiply by 1.07 because the tax is calculated forward, from the cost to you; you divide by 0.9 because the profit margin is calculated backward, from the price to your customer.

Now, this question was significantly edited from the original longer discussion, and looking back at that, I suspect I may have been wrong. In my answer I emphasized my understanding that the tax is on something they are buying (paying the tax themselves), and then reselling to the customer, who may then also have tax to pay. If the tax involved was paid by the customer as a percentage of their cost, things might be different.

Markup with a fee added

Here’s a longer question about a similar issue, from 2011:

Mark-ups, Merchant Fees, and Multiplication ... or Division?

How do you calculate a 30% mark-up on an item that normally retails for $300?

If I take $300.00 and multiply it by 30% (= .30), this equals $90.00; and adding that mark-up to the original price makes it $390.00.

On the other hand, 100% - 30% = 70%, and if I take $300.00 and divide it by 70% (= 0.70), this equals $428.57.

Which is the proper 30% mark-up?

By now, you know the answer. What would you say?

But there’s more, similar to the case above including tax:

Similar question for an item that includes the 3% merchant fee imposed by credit card companies for all purchases made by plastic.

In this case, I think the purchase becomes

   100.00 x 0.03 = 3.00
   100.00 + 3.00 = 103.00
   
The credit card company charges the merchant 3% of $103.00:

   103.00 x 0.03 = 3.09
   103.00 - 3.09 = 99.91
   
So the store would lose nine cents on this transaction?

However, you could also calculate the mark-up this way:

   100.00 / 0.97 = 103.09

The credit card company charges 3%:

   103.09 * 0.03 = 3.09
   103.09 - 3.09 = 100.00
   
With this method, the store does not incur a loss on the sale.

Why does one have to divide the $100.00 by .97 to make the credit card charge equal out? Does this type of calculation have a name?

And why does dividing by .70 add more to the $300.00 sale than does the amount derived from multiplying by 1.30? Does that calculation have a name?

The fact that the store would not get the desired profit means that the first calculation is wrong!

Doctor Wallace answered:

What a great question!

The correct way to mark something up by a percentage of its value is the first way you mentioned. That is, an item costing $300, marked up 30%, would be $390.

With the credit card example, there wouldn't be a problem if the credit card company actually charged 3% of the selling price -- but they don't. They charge 3% of whatever the transaction amount is that was placed on the credit card. So, trying to account for the credit card company's merchant fee by offsetting it with a 3% markup is not going to work out.

The reason is that the credit card company is taking 3% of the 3% the store is adding on, which results in a little extra.

The fee has to be treated as a margin, a percentage of the total after adding the fee, not a markup.

For example, on a $100 item, the store adds 3% to the purchase price, giving

            $100 + .03 ($100)  = $103

Now when the credit card company takes 3% again, they're taking 3% of the 3%. This amounts to 9 hundredths of a percent (.03 * .03 = .0009)

     .03 * [$100 +  .03($100)] 
         .03*100 +  .03(.03)(100)
               3 +  .0009(100)
               3 +  .09
                 = 3.09

In order for the store to not incur a loss of 9 cents on the transaction, they need to charge 3.09%, not 3%.

            $100 + .0309 (100) = $103.09

Now when the credit card company takes their 3%, 3% of $103.09 is $3.09, which was the full amount passed on to the customer, who paid $103.09, leaving the store with the full purchase price.

This is why the first method didn’t work.

The division calculation you mentioned does have a name. It is called the margin, not the markup. These are easy to confuse.
Markup is the percentage gain based on COST, while margin is the percentage gain based on PRICE.

(I have corrected a typo, where “profit” was written instead of “price”.)

For example, returning to the example of an item that usually retails for $100:

If you mark it up 30%, you would calculate 30% of 100, which is $30, so the selling price is $130.

The markup is 30/100 = 30%.

The MARGIN, however, is 30/130 = 23%. This is because selling the item for $130 results in a $30 profit, and 30/130 means that 23% of the money the store took in was profit. We say their margin was 23%.

In fact, a 30% markup will always result in a 23% profit margin.

To calculate the selling price at a given margin, you do what you said: divide the cost by (1 - margin percent). So if you want a 30% profit on selling this item, then your cost will be 70% of the sales price:

   70% of sales price = $100
          sales price = $100/.70
                      = $142.86

This means that you will make $42.86 profit, giving you a margin of

         42.86/142.86 = 0.30, or 30%
   
Your cost ($100) is 100/142.86 = 0.70, or 70% of the sales price.

This last calculation confirmed that the sales price did lead to the required margin. And I imagine 30% rather than 3% was used in this example in order to make differences more visible. Now back to the actual question:

So when the store divides the purchase price by 0.97, it is a quick way to calculate the sales price (not the markup) in a way that ensures that they get a margin of 3% (extra) profit, all of which will be passed on to the credit card company, leaving the store with no loss on the sale.

The hard thing is to explain to a customer how this all works, and why they are getting charged $3.09 instead of $3.00 for a 3% fee!

I imagine the customer wouldn’t normally see how much the fee is …

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