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?


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 = \(\frac{(2N÷x)}{[2N/x + 3N(1 - 1/x)]}\)
= \(\frac{(2÷x)}{[2/x + 3(1 - 1/x)]}\)
= \(\frac{(2÷x)}{[3 - 1/x)]}\)
= \(\frac{(2÷x)}{[(3x - 1)/x)]}\)
= \(\frac{2}{(3x - 1)}\)

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

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