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

Date: 01/06/2011 at 17:08:40
From: Rodney
Subject: The price of an item is 300.00 and needs to be marked up 30%

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?

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?



Date: 01/06/2011 at 19:04:56
From: Doctor Wallace
Subject: Re: The price of an item is 300.00 and needs to be marked up 30%

Dear Rodney,

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.

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.

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 PROFIT.

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.

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!

Thanks for writing to Dr. Math!

Don't hesitate to write again if you need further help with this or
another question.

- Doctor Wallace, The Math Forum
  http://mathforum.org/dr.math/ 
Associated Topics:
Middle School Fractions

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