A lemonade stand sells cups of Yellow Lemonade and Pink Lemonade. Pink

Question

A lemonade stand sells cups of Yellow Lemonade and Pink Lemonade. Pink Lemonade sells for 50% more than Yellow Lemonade. On a certain day, 1/x of all the cups of lemonade sold was Yellow Lemonade. In terms of x, what fraction of the total revenue from lemonade sales came from sales of Yellow Lemonade?

A. $\frac{x − 1}{x}$

B. $\frac{2}{3x − 1}$

C. $\frac{3x − 3}{3x − 1}$

D. $\frac{3}{2x + 1}$

E. $\frac{2x − 1}{2x + 1}$

A.  \frac{x − 1}{x}

B.  \frac{2}{3x − 1}

C.  \frac{3x − 3}{3x − 1}

D.  \frac{3}{2x + 1}

E.  \frac{2x − 1}{2x + 1}

GMATinsight · Answer

Pink Price = 50% more than Yellow

Let, Yellow price = p
Pink price = 1.5p

Yellow sold = 1/x of all cups
Pink sold = 1- (1/x) of all cups

i.e. Price of Yellow = p*(1/x) 
i.e. Price of Pink = 1.5p*(1 - (1/x)) = 1.5p*(x-1)/x

Total Revenue = p*(1/x) + 1.5p*(x-1)/x = p(1+1.5x-1.5)/x = p(1.5x-0.5)/x

Yellow / Total revenue =   / p(1.5x-0.5)/x = 1/(1.5x-0.5) = 2 / (3x-1)

Answer: Option B

IanStewart · Answer

I'd do the problem algebraically, but if someone wanted to avoid algebra, an alternative: if x = 1, that would mean 1/1 = 100% of the lemonade sold was yellow, so naturally 100% of the revenue would come from yellow lemonade. Plugging x=1 into each answer choice, only B and D give an answer of 1. To decide between those, if x = 2, then 1/2 of the lemonade sold is yellow, but yellow is cheaper than pink, so less than half the revenue would then come from yellow lemonade sales. Plugging x = 2 into the remaining answers B and D, only answer B gives a value less than 1/2, so it must be right.

sujoykrdatta · Answer

Question:  
A lemonade stand sells cups of Yellow Lemonade and Pink Lemonade. Pink Lemonade sells for 50% more than Yellow Lemonade. On a certain day, 1/x of all the cups of lemonade sold was Yellow Lemonade. In terms of x, what fraction of the total revenue from lemonade sales came from sales of Yellow Lemonade?

Solution:

Pink Lemonade sells for 50% more than Yellow Lemonade
=> If we assume the price of Yellow Lemonade (YL) as $2, the the price of Pink Lemonade (PL) is $3

On a certain day, 1/x of all the cups of lemonade sold was Yellow Lemonade
Thus, if total number of cups sold were N:

Number of cups of YL = N/x => Revenue = $ 2N/x
Number of cups of PL = N - N/x = N(1 - 1/x) => Revenue = $ 3N(1 - 1/x)

Thus, fraction of revenue from YL =  \frac{(2N÷x)}{ } 
=  \frac{(2÷x)}{ } 
=  \frac{(2÷x)}{ } 
=  \frac{(2÷x)}{ } 
=  \frac{2}{(3x - 1)}

Answer B

Alternate approach:

Observe that the options are in terms of x. Thus, we can simply replace x by a easy-to-use value and compute the options as well. 
Thus, since 1/x of all the cups of lemonade sold was Yellow Lemonade, we can assume that x = 1, i.e. ALL (i.e. 100%) the cups were YL
Thus, the fraction of revenue from YL should also be 1, or 100% (since no cups of PL were sold)

Plugging in x = 1 in the options: 
Option A: (x - 1)/x = 0, hence incorrect
Option B: 2/(3x - 1) = 1, maybe correct
Option C: (3x - 3)/(3x - 1) = 0, hence incorrect
Option D: 3/(2x + 1) = 1, maybe correct
Option E: (2x - 1)/(2x + 1) = 1/3, hence incorrect

Thus, we have either Option B or Option D is correct.

Thus, we take another value of x, say x = 2
=> 1/2 of all the cups of lemonade sold were YL
Pink Lemonade sells for 50% more than Yellow Lemonade
Thus, the revenue from PL should be more than 1/2 or 50% (since equal quantities of YL and PL were sold while the PL was priced higher)

Plugging in x = 2 in the options B and D: 
Option B: 2/(3x - 1) = 2/5 < 1/2 correct
Option D: 3/(2x + 1) = 3/5 > 1/2 incorrect

Answer B

manas.cl · Answer

Hi,

I am unsure what's wrong with my setup as I am getting the incorrect answer. any help would be greatly appreciated.

Price of pink lemonade = P(P)
Price of Y lemonade = P(Y)

P(p) = P(y)*3/2. ------> Given
can be re-written as P(y)/P(p) = 2/3 --- (1)
Y and P are the quantity of cups
Y = (Y+P)*1/x
this can be re-written as x-1 = P/Y or 1/(x-1) = Y/P --- (2)

Yellow cups share of revenue = P(Y)*Y/(P(Y)*Y + P(P)*P)
= 1 + P(Y)*Y/P(P)*P
From (1)
= 1 + 2/3*Y/P
from (2)
= 1 + 2/3*1/(x-1)
Solving this gives 3x-1/3x-3

I understand maybe the way I have setup my equations is wrong (particularly 2) but I don't get it intuitively.

Regards,
Manas

IanStewart · Answer

I'd suggest you try to use simpler notation, since that makes algebra much easier -- you're using capital P for two different things, and you're writing simple unknowns as if they were functions. I didn't check all of your work, but there's a mistake in what I've quoted above; essentially you have a fraction like x/(x + y), and you have rewritten it as 1 + (x/y), which you can't do (if the sum were in the numerator, so if you had (x+y)/x, then you can rewrite it as (x/x) + (y/x) = 1 + (y/x), but you can't rewrite fractions in that way if the sum is in the denominator).

A lemonade stand sells cups of Yellow Lemonade and Pink Lemonade. Pink

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A lemonade stand sells cups of Yellow Lemonade and Pink Lemonade. Pink

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