What is the profit percent if an article is sold without any discount

Question

What is the profit percent if an article is sold without any discount on the original selling price?

(1) If the article is sold at x% discount on the original selling price, there is x% profit.
(2) If the article is sold at  \frac{5x}{4} % discount on the original selling price, there is  \frac{x}{2} % profit.

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GMATinsight · Answer

Question: Profit Percentage = ?

Statement 1:  If the article is sold at x% discount on the original selling price, there is x% profit.
SP*(1- x/100) = CP (1+ x/100)----(1)
x is unknown hence 
NOT SUFFICIENT

Statement 2:  If the article is sold at  \frac{5x}{4} % discount on the original selling price, there is  \frac{x}{2} % profit.
SP*(1- 5x/400) = CP (1+ x/200)----(2)
x is unknown hence 
NOT SUFFICIENT

Combining the statements

Divide equation (1) by Equation (2)
We are left with only one equation in terms of x hence we have the value of x
hence
SUFFICIENT

Answer: Option C

itsyodaa · Answer

isn't we can assume the value of x like 10 or 20 and say 10% discount on selling price and 10% profit,, so basically we can find the profit percentage clearly with this help.
we are getting 10% profit and even we don't consider 10 or 20, we are still getting x% profit,
so statement 1 alone is sufficient to answer the question.

gullyboy09 · Answer

Hi  GMATinsight , it will be a quadratic in x, shouldn't we solve the equation to see if there is one value of x exist, or two, or zero? How did we arrive at that it will have unique solution.

egmat · Answer

Hi gullyboy09,

You're absolutely right to push on this. GMATinsight's line "we're left with one equation in x,  hence  we have the value" skips a real step, and your instinct is correct: combining the statements  does  produce a quadratic in x. So let's actually solve it and see what happens.

Writing a = x/100, the two statements give two expressions for the markup S/C:

- From (1): S/C = (1+a)/(1-a)
- From (2): S/C = (1 + a/2)/(1 - 5a/4)

Setting them equal and cross-multiplying collapses to:

3a2 - a = 0 -> a(3a - 1) = 0

So the roots are a =  0  and a =  1/3 . Two roots - exactly as you suspected.

Why only one survives

Here a =  0  means x =  0 , which is a degenerate case: a " 0 % discount giving  0 % profit" describes no actual discount scenario at all, so x =  0  isn't a valid value for the situation the problem sets up. We discard it.

That leaves the single valid root a =  1/3 , i.e. x =  100/3 . Plugging back:

S/C = (1 +  1/3 )/(1 -  1/3 ) = ( 4/3 )/( 2/3 ) =  2

So the no-discount profit is a definite  100% .

The takeaway

Your worry was the right one - a quadratic  can  leave you with  0 ,  1 , or  2  valid answers, and you should always check. The reason this question is  sufficient  (answer  C ) isn't that the equation is automatically linear; it's that after solving, only  one  root is admissible. Combining the statements pins x down to a unique meaningful value, which is exactly what sufficiency requires.

Answer: C

What is the profit percent if an article is sold without any discount

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GMAT Data Sufficiency (DS) Questions

What is the profit percent if an article is sold without any discount

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