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David earns a commission that consists of a certain number of dollars

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abrutlag
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David earns a commission that consists of a certain number of dollars [#permalink] New post   05 Jul 2020, 10:54
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David earns a commission that consists of a certain number of dollars per sale, plus a fixed percent of the dollar amount of the sale. Is his commission on a 250,000 dollar sale at least 20,000 dollars?

(1) His commission on a 300,000 dollar sale is at least 30,000 dollars.
(2) His commission on a 100,000 dollar sale is at least 10,000 dollars.
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IanStewart
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Re: David earns a commission that consists of a certain number of dollars [#permalink] New post   06 Jul 2020, 06:22
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You can rephrase this problem as a management problem. Say you're hiring an employee, and you promise that employee they will earn $30,000 if they make a sale of $300,000, where that $30,000 is divided between an x% commission and a fixed bonus. If you knew the employee you were hiring was not very good at the job, and would sell less than $300,000, and you wanted to pay that employee as little as possible, you'd want to pay them no bonus at all and a 10% commission, because the bonus will represent a higher percentage of sales the smaller the sales are, while the commission represents a constant percentage. If instead you were hiring a great employee who would sell more than $300,000, to pay them as little as possible you'd pay entirely bonus and no commission, because the bonus will represent a smaller percentage of total sales the greater sales are, while the commission is constant.

So if, in Statement 1, we know the person earns at least 10% when selling $300,000 worth of goods, the person will necessarily earn at least 10% on a sale of less than $300,000 (and would necessarily earn at most 10% on a sale greater than $300,000). So on a $250,000 sale, the person will necessarily earn $25,000 or more, and Statement 1 is sufficient.

Statement 2 is not sufficient, because the commission could be 0% and the bonus $10,000, and then the answer is 'no', or the commission could be 10% and the bonus $0, and then the answer is 'yes' (among other possibilities). So the answer is A.
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David earns a commission that consists of a certain number of dollars [#permalink] New post   06 Jul 2020, 08:39
Nice question! For such questions, one can look at extremes.

Case 1) Such as a combination could be $1 amount of commission is fixed and the remaining is a percent (variable) or Case 2) almost the entire amount of commission is fixed and only a minuscule percent is variable. One quickly realizes that option B gives a yes and no answer hence insufficient but in option A no matter what combination you try you always get a yes answer. Hence A is sufficient.
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David earns a commission that consists of a certain number of dollars [#permalink] New post   08 Jul 2020, 13:37
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abrutlag wrote:
David earns a commission that consists of a certain number of dollars per sale, plus a fixed percent of the dollar amount of the sale. Is his commission on a 250,000 dollar sale at least 20,000 dollars?

(1) His commission on a 300,000 dollar sale is at least 30,000 dollars.
(2) His commission on a 100,000 dollar sale is at least 10,000 dollars.


x = fixed commission per sale
y = percent of sales amount

For a sale of 250,000, David earns x + y/100(250,000). The problem asks if this is at least 20,000. This simplifies to, is x+2,500y ≥ 20,000?

(1)
The problem asks if David's commission was greater than 20,000. So, try to make the commission as small as possible and see if it is still greater than 20,000.

To make the commission as small as possible, assume that his commission on a 300,000 sale was exactly 30,000. So, x + y/100(300,000) = 30,000. This simplifies to x + 3,000y = 30,000.

Therefore, y = 30,000-x/3,000. Plug this into the question.

Is x + 2,500(30,000-x/3,000) ≥ 20,000?
Is x + 2,500(10 - x/3,000) ≥ 20,000?
Is x + 25,000 - 25/30x ≥20,000?
Is 5/30x ≥ -5,000?

Because x is positive (David's commission must be positive), then this is always true and the answer is "yes"

SUFFICIENT

Note that if you substitute differently, the math becomes more complicated, but you can still find the same answer. Try substituting for x instead:

x + 3,000y = 30,000
x = 30,000-3,000y

Plug this in to the question

Is 30,000 - 3,000y + 2,500y ≥ 20,000?
Is 30,000 - 500y ≥ 20,000?
Is 10,000 ≥ 500y?
Is y ≤ 20?

This means, if y ≤ 20, the answer is "yes". But, remember what y represents. y is the percent of sale that David earns as a commission So, if y is GREATER than 20, then David earns more than 20% of 250,000, which is much greater than 20,000. So, the answer is still "yes".

SUFFICIENT

(2) This statement could mean that x = 10,000 (he earns 10,000 per sale plus 0% of the sale). So, his commission on a 250,000 sale would also be 10,000, and the answer would be "no"

Or, it could mean that x = 0, and y = 10%. So, his commission on a 250,000 sale would be 10% of 250,000, which is 25,000. The answer would be "yes"

NOT SUFFICIENT
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David earns a commission that consists of a certain number of dollars [#permalink] New post   11 Sep 2023, 13:03
Solve it through co-ordinate geometry. However, took me 2:30. Posting answer for visualization. Mixing two topics can sometimes decipher tough questions!

IanStewart couldnt think of your approach. Its really brilliant and super logical!

As you can see from the graph => line for statement 1 is always above line for prompt. Hence statement 1 is sufficient

However, line for statement 2 is not always above line for prompt. Hence, insufficient.
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