23% of Something is Better Than 100% of Nothing! - The Solutions
08/02/2004

Remember that the rate of commission for the first $200 is 38% and the rate of commission for the next $300 is 30%. The commission on the final $2500 – 500 = $2000 is only 20%. This amounts to .38 × $200 + .30 × $300 + .20 × $2000 = $76 + $90 + $400 = $566.

We figured that Auction Mills is keeping $566 of the $2500. That is 566 ÷ 2500 = 22.64% to the nearest hundredth.

We know that the commission for the first $200 is at a rate of 38% and the next $300 is at a rate of 30%. Therefore, the commission rate for the first $500 is somewhere between 38% and 30%. We can see from the first problem’s solution that it was actually $166, which is 33.2% of $500. In order to get the effective percentage rate down to 25%, the selling price is going to have to be greater than $500 so that we have a significant portion being charged at the 20% level. We’ll let the selling price be x. We know that the commission on the first $500 is $166. The remaining $x – 500 will have a commission rate of 20%. This leads to the equation 166 + .20(x – 500) = .25x. If we distribute the .20, we have 166 + .20x – 100 = .25x, which is 66 + .20x = .25x. Subtracting .20x from both sides yields 66 = .05x. Finally, dividing by .05 leads us to x = 1320. We can check our answer. If an item sells for $1320, the commission is .38(200) + .30(300) + .20(820) = 76 + 90 + 164 = 330. This $330 is 330 ÷ 1320 = 25% of $1320.

You can get very close! If you sell an item for a million dollars, the effective percentage rate of commission is 20.0066%. If you sell an item for a billion dollars, the effective percentage rate of commission is 20.0000066%. We are getting very close to 20%, but we’ll never get down to it exactly. Notice that no matter how much of the selling price will be considered at the 20% level, there will always be $500 that is considered at a 33.2% level, and this will not allow the average rate of the commission to reach 20%.