wrong arithmetic concerning addition and percentages

Question

The problem:

A realtor takes a sale commission of 6%. A buyer offered 412,250 to buy a house and the realtor said he's willing to accept only 4% commission.

The realtor says the 2% he gave up is worth 8245$, and if this amount is added to 412,250 the result is 420,495. So this transaction is the same thing as 420,495 with 6% commission. The seller ends up with the same amount of money.

My question:

The realtor says the 2% he gives up is 8245$. This number is calculated from 412,250*0.02 = 8245. If you add 8245 to 412,250 you get 420,495.

Are the transactions equal?

412,250 * 0.04 = 16,490

412,250 - 16,490 = 395,760

420,495 * 0.06 = 25,229.7

420,495 - 395,265.3

The transactions are not equal. The seller gets more money in the 4% commission deal. Now my question is there is something wrong with the realtor's reasoning. And I believe it is the fact that he said "just add the 2% back to the total and its the same deal". But I want to prove this in a mathematical way with rigor and numbers, because now I know intuitively which is part is wrong but I can't articulate that with numbers and mathematical logic/rigor. Thanks a lot!

Answer 1

If I understand it right, the seller ends up with the two different sums of money: $$412,250\cdot 0.96>412,250\cdot 1.02\cdot 0.94 \iff \\ 0.96>1.02\cdot 0.94 \iff \\ 0.96>0.9588$$ I assume the realtor actually implied "So this transaction is $\color{red}{\text{approximately}}$ the same thing as 420,495 with 6% commission. "

Answer 2

Taking a percentage (A) of an original something (X) and then adding or subtract another percentage (B) of the original something (X) is not the same thing as taking a percentage (A) of the original something (X) and then adding or subtracting another percentage (B) of the modified something (Y).

1) the first place the seller is getting $X - 6\%X + 2\% X = 96% X$. 2) The second place $Y= X+2\% Y = 1.02Y$ and the seller is getting $Y - 6\% Y$ (NOT $6\% X$) and so the seller is getting $X - 6\%Y + 2\% X= X - 6\%(1.02X) - 2\%X \ne X - 6\%X - 2\%X$.

In the second place the seller wase added $2\%$ of the base value of $X$ to get a base values of $1.02\%$ of $X$. The the seller received $94$ perceent of the modified $1.02\%$ of $X$ for $95.88\%$ of $X$.