Asking Price of a House

Date: 10/14/2001 at 20:32:03
From: Helen Carman
Subject: Profit on real estate

If a seller wants to clear $50,000 for his house and the real estate 
commission is 6%, what is the formula for the asking price for the 
house? And what should the asking price for the house be?

Date: 10/15/2001 at 02:01:33
From: Doctor Jeremiah
Subject: Re: Profit on real estate

Hi Helen,

Maybe you just want the answer, but I want to show you how to think about 
questions like this, so if you will humor me...

The asking price equals $50000 + 6% of the asking price.

If we write that down in mathematics we get:

   price = $50000 + 6% of price

The tricky part is combining the two "price" terms. 

We can subtract things from or add things to both sides of the 
equation, and if we do that with "6% of price" we get:

   price - 6% of price = $50000 + 6% of price - 6% of price

And since "6% of price - 6% of price" equals "0" we get:

   price - 6% of price = $50000 + 0
   price - 6% of price = $50000

But "price" is the same as "100% of price" so we can do this:

   100% of price - 6% of price = $50000

And "100% of price - 6% of price" is obviously "94% of price"

   94% of price = $50000

Now we need to know the secret of percentages! All percentages work 
this way: percent means per hundred or "out of 100." Remember that 
there are 100 cents in a dollar, so 94 percent of a dollar is 0.94, 
and 94 percent of two dollars is 0.94 times 2. So 94 percent of the 
asking price is 0.94 times the price:

   0.94 x price = $50000

Which we can write as:

   price x 0.94 = $50000

The tricky part now is combining the two numbers. We can multiply or 
divide by the same thing on both sides of the equation, and if we do 
that with 0.94 we get:

   price x 0.94 / 0.94 = $50000 / 0.94

But 0.94 / 0.94 equals 1, so we get:

   price x 1 = $50000 / 0.94

And of course, any time we multiply by 1, the 1 just goes away:

   price = $50000 / 0.94

And that is the price you should ask. 

Here is how to prove it:

   price = $53191.49

If the price is $53191.49 and we use our original equation:

   price = $50000 + 6% of price

We get:

   $53191.49 = $50000 + 6% of $53191.49

We use our percentage secret:

   $53191.49 = $50000 + 0.06 x $53191.49

And we get:

   $53191.49 = $50000 + $3191.49

So if you ask $53191.49, then the commission will be $3191.49 and you 
will end up with $50000 left over.

Personally I would ask for $53900 (53.9 thousand) because it's a bit 
more but it's not quite 54 thousand, which would sound bigger. You 
know what I mean: everything for sale has a nine at the end of the 
price. By making it a bit bigger you don't have to get your exact 
asking price, so if your potential buyers want to haggle you could 
drop your price by $700 and still get 50 thousand out of it.

How much would you get if it did sell for $53900?

The formula we ended up with was:

   asking price = amount you get / 1 minus the commission

If the commission is still 6%, then the formula is:

   asking price = amount you get / .94

If your asking price is $53900, then:

   $53900 = amount you get / .94

Now remember that we can multiply both sides by anything we want (.94 
in this case):

   .94 x $53900 = .94 x amount you get / .94

And the two .94's on the right side cancel to give:

   .94 x $53900 = amount you get

Which is equal to:

   $50666 = amount you get

Which is a bonus, but most buyers want to get a bargain so don't be 
surprised if they want to haggle. That's the advantage of setting the 
price slightly higher.

- Doctor Jeremiah, The Math Forum
  http://mathforum.org/dr.math/