There are two containers A and B filled with mentha oil with different

Question

There are two containers A and B filled with mentha oil with different prices and with volumes 140 and 60 liters respectively. Equal quantities are drawn from both A and B in such a manner that the oil drawn from A is poured in into B and oil drawn from B is poured into A. After doing so, the price per liter becomes equal in both containers. What is the (equal) quantity that was drawn?

A. 21 liters
B. 27 liters
C. 35 liters
D. 42 liters
E. 45 liters

A. 21 liters
B. 27 liters
C. 35 liters
D. 42 liters
E. 45 liters

sumitkrocks · Accepted Answer

Let us assume price per liter of the mentha oil in container A and B be a and b respectively 
From the given information we can say that quantity of oil in the container A and B will be 140 and 60 each respectively; However, concentration or mix/composition will change to achieve an average price which will be between a and b with more aligned towards content of A as A is higher in quantity i.e 140 compared to 60

Average price of the mix =  \frac{140*a + 60*b}{ 200}  Equation ---1 which will be equal to the average price of the content in each container

Say x Ltr of oil is drawn from each container and put in the other one

Average price of content of container A =  \frac{(140-x)*a + x*b }{140}  Equation ---2

Average price of content of container A =  \frac{(60-x)*b + x*a}{ 60}  Equation----3

Equation 1 and 2 and 3 will be equal

From 2 and 3

\frac{140*a + x*(b-a) }{ 140} = \frac{60*b + x*(a-b) }{ 60}

Solving further ...  3*14(a-b) = x*(a-b)  
Since we are given a is not equal to b; a-b can not be zero 
x= 3*14 =42

(D) is the answer IMO

andreagonzalez2k · Answer

In order to have the same price ratio in both mixtures must be the same.

\frac{140-x}{x}=\frac{x}{60-x}

x^2=140*60-200x+x^2

x=42

zanto · Answer

I don't know if my approach is right. I probably got a lucky guess.  Bunuel , do you mind giving input / providing the official solution?

Price / Liter = P(total) / Q(total)

A container: P  / (Qtotal - a + b)
a= mixture leaving a
b = mixture added from b

B container: P  / (Qtotal -b + a)
b = mixture leaving b
a = mixture added from a

set both equations equal to each other:
P  / (Qtotal - a + b) = P  / (Qtotal -b + a)

The Ps cancel.

Plug in 140 and 60.

Side note: notice both are integers that are even, all the answers are odd except for 42. That was my thinking..

Thunderbird03 · Answer

P is the quantity removed.
Q is the cost of the first liquid.
R is the cost of the second liquid.

(140-P)Q+PR/140= (60-P)R+PQ/40

On solving the above equation you get => 42Q-PR=42R-PQ

42-P*Q-R=0

Either P has to be 0 or A & B should be equal. But it's already said that prices are different, so P is 42

vipulgoel · Answer

can some one please elaborate this ??

andreagonzalez2k · Answer

You remove the same amount of liters from A container and B container. Let it be 'x'.

After that you have:
In A container:
- '140 - x' of A mentha oil
- 'x' of B mentha oil
In B container:
- 'x' of A mentha oil
- '60 - x' of B mentha oil

Ratio of A mentha oil to B mentha oil must be the same in both containers in order that the price is the same.

\frac{140-x}{x}=\frac{x}{60-x}

x^2=140*60-200x+x^2

x=42

KarishmaB · Answer

Say the oil in 140 lts container is cheaper and the oil in 60 ltr container is more expensive. 
After mutual replacement, overall price becomes the same. This means that the concentration of either ingredient (one kind of oil) is the same in both mixtures.

Say, we remove x litres from each and replace in the other.

Concentration of expensive oil in both containers should be equal. So 
 \frac{x}{140} = \frac{(60 - x)}{60}  
x = 42 liters

Answer (D)

Note that we could have done the same thing with the concentration of cheaper oil and we would have got the same answer. In mixtures, we need to deal with the concentration of any one ingredient e.g. with concentration of sugar in sugar solution.

SoHaaM · Answer

Ok, there are two containers A and B. Suppose there are two types(price) of mentha oil:

- X is in A, which is 140 L.
- Y is in B, which is 60 L.

Since they became equal in price, we can assume that both have the same proportion of X and Y in them. we would option test

------------ A ------------

option(a)

x=140-21=119
y=21

now, x/y = 119/21

------------ B ------------

x=21
y=60-21=39

≠ 21/39

(b)(C)(E) options don't work too.

----------------------------------------------------------------

test option(d)

------------ A ------------

x=140-42=98
y=42

x/y=98/42=7/3

------------ B ------------

x=42
y=60-42=18

= 42/18=7/3

----------------------------------------------------------------

so answer is (D)

Gmatiseasyforall · Answer

Hi Bunuel! Thanks for posting this question! This is one of those mixture problems that confuse me a lot!

The Answer provided by andreogon is right on he dots! The answer to this type of mixture problems is always 
  \frac{ ab}{a+b }   where   a   and   b   are the respective volumes of the two containers. Notice that this is the HALF of the Harmonic Mean formula i.e  \frac{ Harmonic Mean}{2 }  . My question is why is Harmonic mean is applicable to this type of mixture problems? Is there any specific structure of the question that make Harmonic mean appropriate for this type of question? Or is there any specific property of Harmonic Mean which makes it a fit for this type of question?

Will much appreciate your insights in this regard! Thanks in advance!

Bunuel · Answer

I’d say there is a real reason, but it is not a general rule that mixture problems use the harmonic mean.

If x liters are exchanged, then x/a of container A is replaced and x/b of container B is replaced. For the final prices to become equal, these two fractions must together account for the whole original difference, so:

x/a + x/b = 1

Solving gives:

x = ab/(a+b)

This expression is half the harmonic mean of a and b. So the harmonic mean appears because this problem naturally creates a sum of reciprocals, not because harmonic mean is generally used in mixture problems.

There are two containers A and B filled with mentha oil with different

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There are two containers A and B filled with mentha oil with different

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