M37-60

Question

At a market, two merchants sold a combined total of 100 pineapples, and each earned the same amount of money. If the first merchant had sold the same number of pineapples as the second merchant but at his own price per pineapple, he would have earned $$15$. Conversely, if the second merchant had sold the same number of pineapples as the first merchant but at his own price, he would have earned $$6\frac{2}{3}$. How many pineapples did the second merchant sell?

A. $20$
B. $30$
C. $40$
D. $60$
E. $80$

A.  20 
B.  30 
C.  40 
D.  60 
E.  80

Bunuel · Answer

Official Solution:

At a market, two merchants sold a combined total of 100 pineapples, and each earned the same amount of money. If the first merchant had sold the same number of pineapples as the second merchant but at his own price per pineapple, he would have earned $$15$. Conversely, if the second merchant had sold the same number of pineapples as the first merchant but at his own price, he would have earned $$6\frac{2}{3}$. How many pineapples did the second merchant sell?

A. $20$
B. $30$
C. $40$
D. $60$
E. $80$

A.  20 
B.  30 
C.  40 
D.  60 
E.  80

Let's say the first merchant sold  x  pineapples at  $a  each, and the second merchant sold  y  pineapples at  $b  each.
 
Since they earned the same amount, we have:  xa = yb . From this, we can deduce:  \frac{a}{b} = \frac{y}{x} .
 
If the first merchant had sold the same number of pineapples as the second merchant, but at his own price per pineapple, he would have earned  $15 . This gives us the equation  ya=15 .
 
Similarly, if the second merchant had sold the same number of pineapples as the first merchant, but at his own price, he would have earned  $6\frac{2}{3} . This provides the equation  xb=\frac{20}{3} .
 
Now, dividing the equation  ya=15  by the equation  xb=\frac{20}{3} , we obtain:  \frac{y}{x}*\frac{a}{b}=\frac{9}{4} .
 
Since we know  \frac{a}{b} = \frac{y}{x} , substitute this into the equation  \frac{y}{x}*\frac{a}{b}=\frac{9}{4}  to get:  (\frac{y}{x})^2=\frac{9}{4} .
 
Thus,  \frac{y}{x}=\frac{3}{2} .
 
Given that  x+y=100 , we can determine that  x=40  and  y=60 .
 
Therefore, the second merchant sold 60 pineapples.

Answer: D

Mdabhi1 · Answer

Hi,

For the stem "If the first merchant had sold the second merchant's pineapples at the first merchant's price per pineapple, he would have earned an additional $15", I wrote the equation ya= xa + 15

so ya = yb + 15......(xa=yb, both the merchants have earned equal amount of money)

Y(a-b) = 15......(1)

and

for the stem "if the second merchant had sold the first merchant's pineapples at the second merchant's price per pineapple, he would have earned an additional $6 2/3" I wrote the equation xb = yb + 20/3

xb= xa + 20/3

x(b-a) = 20/3

-x(a-b) = -20/3.....(2)

dividing equation (2) by equation (1), I got x/y = 4/9

Please help me to understand where I am making a mistake.

Bunuel · Answer

You're correct that the word "additional" in the original text created some ambiguity. I've revised the question to clarify the intended meaning. Now, instead of saying the merchants would have earned an 'additional' amount, the question states exactly how much each merchant would have earned if they had sold the same number of pineapples as the other at their own prices. I hope this revised wording makes the scenario clearer and removes any ambiguity. Thanks for bringing this to my attention.

nivivacious · Answer

Let merchant A sell x pineapples and merchant B sell y pineapples
x+y = 100 (given)
Price of pineapples sold by A = a
Price of pineapples sold by B = b
xa = yb ... (1)

(100-x)a = 15 (given)
xb = 20/3 (given)
Substituting this in (1)
x (15/(100-x)) = (100-x) 20/3x
this results in x^2-160x-8000=0
if stressed we can use -b+/- \sqrt{b^2-4ac}/2a formula
x= 40 or -200
we need to find y
100-40 = 60
Hence D

Shivang29 · Answer

This is a great question that’s helpful for learning and I like the solution - it’s helpful.

Kashli · Answer

I like the solution - it’s helpful. Earned here means revenue generated and not profit.

ElCanalero · Answer

A method to guess in 1 minute:

First Merchant: taking the other's quantity results in $15
Second Merchant: taking the other's quantity results in $6.33

Thus, you know that that Second merchant has higher quantity, but lower price vs. First. 
This narrows down your guess to either 60 or 80 (i.e., more than 50).

I guessed 60. A split of 80/20 I feel like should correspond to a sharper drop in revenue (though this is just gut feeling)

kevincan · Answer

Suppose x > 1
I will call the first merchant A and the second B
If B sold x times as many pineapples as A , then merchant A charged x times as much per pineapple as B

If A had sold x times as many and charged x times as much per pineapple as B , A’s revenue would be x^2 as high as that of the B .

Since 15 divided by 20/3 is 9/4 , x = 3/2

Thus merchant B sold 3/2 as many pineapples as A , 3/5 of the 100 pineapples

nikitathegreat · Answer

Earned here means revenue generated? Even when I assumed CP to be same for both these merchants, my answer is coming correct 
ax-px= by-qy where A and B is the SP for merchant 1 and 2 respectively. X and Y are no. of apples. P and Q are CP for merchant 1 and 2
With this equation also, I was able to derive the ans

ax-px=15
ay-py=20/3
x+y =100
Can someone check this and confirm and how do we understand that earn means Revenue and not profit?

Bunuel · Answer

Here, “earned” means the money received from selling the pineapples, so it means revenue.

M37-60

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