## Sunday, August 7, 2011

### eBay Math Problem Image Source: http://topnews.net.nz

A friend recently asked me how to set up a "Buy it Now" price for an item on eBay, so that they would not lose any money selling the item.

They did not care if they made any money on the sale, they just wanted to unload their unwanted item without incurring a loss.

Here is their original question:

"Okay Mathematics Brainiac - Here is a real life example of why we should pay attention to our maths teachers at school.......

I have something I want to sell on ebay but am not looking to make a profit (or a loss!). The item cost me \$117.79, and ebay charge a flat-rate "insertion fee" of \$2.50 and also a "final value fee" (based on the closing price) which is 5.25% of the first \$75 and 2.75% for every dollar over \$75. I have run this so many times and keep getting the incorrect answer.... it's the rolling "final value fee" that's getting me. Any help would be greatly appreciated!"

THE FINAL ANSWER IS TO SELL THE ITEM FOR \$125.62 .

Here is how to work it out.

To make zero profit, Sell Price = Cost of Selling Price

Let d = the missing dollar amount of mark up that we need to determine.

Sell Price = 117.79 + 2.50 + (5.25% of 75) + d
SP = 124.23 + d

Cost Price = 124.23 + (2.75% of (124.23 + d - 75)
Cost Price = 124.23 + 0.0275x49.3 + 0.0275d

CP = 125.59 + 0.0275d

Now set SP = CP for zero profit and solve:

124.23 + d = 125.59 + 0.0275d (Now subtract 0.0275d both sides)
0.9725d = 125.59 - 124.23
0.9725d = 1.36 (Now divide by 0.9725 both sides)

d = \$1.40

So SP = 124.23 + d (Now substitute in our answer for d)
SP = 124.23 + 1.40
SP = \$125.63

Check this Answer:

The Original 117.79 should be left when we take away all the charges
125.63 - 2.50 - (5.25% of 75) - (2.75% of (125.63 - 75) = \$117.80

So due to rounding off errors introduced while doing this problem, there will be a 1 cent profit.

If we try to sell for 125.62 we will have:
125.62 - 2.50 - (5.25% of 75) - (2.75% of (125.63 - 75) = \$117.79.

\$117.79 is how much the item cost, and so the cost of buying the item will be recovered and no loss incurred.

We did do some Googling to see if we could find an Online Calculator that could work all of this out in a computer program, but could not find anything useful.

So there you go. There is not always on Online Calculator for everything. Learning High School Algebra can be useful in later life!

Enjoy,
Big Passy Wasabi