milk and water admixture (pouring between two vessels)

[defeated_soldier]

I have a formula given in my book

If a vessel contains 'a' litres of liquid A, and if 'b' litres be withdrawn...
then if 'b' litres of mixture be withdrawn and replaced by liquid B, and the...
times in all, then:

. . .\(\displaystyle \L \frac{\mbox{Liquid A left in vessel after}\,n^{th}\,\mbox{operation}}{\mbox{initial quantity of liquid A in vessel}}\, =\, \left(\frac{a\, -\, b}{a}\right)^n\)

. . .\(\displaystyle \L \frac{\mbox{Liquid A left after}\, n^{th} \,\mbox{operation}}{\mbox{Liquid B left after}\, n^{th} \,\mbox{operation}}\, =\, \frac{\left(\frac{a\, -\, b}{a}\right)^n}{1\, -\, \left(\frac{a\, -\, b}{a}\right)^n}\)

i want to solve this question:

A container contains 200L solution of milk and water in the ratio 1:3 . You take out 50L of solution and replace it with milk and do this operation twice . At the end of the second operation , find the ratio of milk to water in the solution.

Can I use the formula given in my book? I tried that, but the answer I found did not match.

Liquid A after nth operation
------------------------------- =
Liquid B after nth operation

[(200-50)/200 ]^2
-------------------- = 9/7
1-[(200-50)/200 ]^2

so, M+W/M = 9/7

M/W=7/2

 

[galactus]

Since the ratio is 1 part milk to 3 parts water, that means the mixture is 25% milk.

\(\displaystyle \L\\0.25(200-50)+50=87.5\)

There is now 87.5 liters of milk in the 200 Liters of mixture. That's 43.75% milk.

Do it again:

\(\displaystyle \L\\0.4375(200-50)+50=115.625\)

After both operations there is now 115.625 liters of milk in the 200 liter mixture. That is 37/64 parts milk and 27/64 parts water.

57.8125% milk and 42.1875% water.

 

[defeated_soldier]

Since the ratio is 1 part milk to 3 parts water, that means the mixture is 25% milk.

\(\displaystyle \L\\0.25(200-50)+50=87.5\)

There is now 87.5 liters of milk in the 200 Liters of mixture. That's 43.75% milk.

Do it again:

\(\displaystyle \L\\0.4375(200-50)+50=115.625\)

After both operations there is now 115.625 liters of milk in the 200 liter mixture. That is 37/64 parts milk and 27/64 parts water.

57.8125% milk and 42.1875% water.

Okay, but I can't do it if there are multiple operations, so I would like to apply the formula.

Is there something wrong with the formula, or have I applied it incorrectly?

Thank you!

 

[galactus]

Use n=3 in your formula, instead of n=2.

\(\displaystyle \L\\\frac{(\frac{3}{4})^{3}}{1-(\frac{3}{4})^{3}}=\frac{27}{37}\)

 

[stapel]

I can't do it if there are multiple operations...

Do you mean that there were instructions for this exercise, not included, which specified that you must answer this question using the given formula once...?

Thank you.

Eliz.

 

[defeated_soldier]

Use n=3 in your formula, instead of n=2.

\(\displaystyle \L\\\frac{(\frac{3}{4})^{3}}{1-(\frac{3}{4})^{3}}=\frac{27}{37}\)

Hi, galactus,
thank you for the answer.

But i dont understand some points in your answer and hence i want to ask you 2 questions on your answer.

Q1 : why you used n=3 ? see, the question mentioned "At the end of the second operation" and so here nth operation is n=2 but NOT 3

Q2: According to the formula we have

Liquid A left after nth operation
------------------------------------ = formula
Liquid B left after nth operation

is not it m liquid A =solution itself i.e Milk + Water = M+W ?
and liquid B (which is replaced by Milk ) = Milk

so, that should have been M+W/M


please post your comment[/tex]

 

[Denis]

Mathsoldier, your problem is weirdly worded...

The 1st operation is removing 50 from A (which starts with 200)
and replacing it with 50 of B; it is THEN that you have 150:50

Following this, there is 2 more operations.

Look at the formula CAREFULLY: it clearly (or unclearly!) treats
the initial stage as being ALL WATER in one container:
"If a vessel contains 'a' litres of liquid A, and if 'b' litres be withdrawn..."
So in your problem, that means:
If a vessel contains 200 litres of water, and 50 litres be withdrawn

Your calculation:
[(200-50)/200 ]^2
-------------------- = 9/7
1-[(200-50)/200 ]^2
clearly starts with 200 less 50; there is 2 more "less 50" following that;
so that means n=3, so you need ^3.

Keeping A and B separate makes it clear:

A: 200 -50 150     150 -37.5 112.5     112.5 -28.125  84.375      84.375
B:   0       0 +50  50 -12.5  37.5 +50  87.5 -21.875  65.625 +50 115.625

84.375 / 115.625 = 27 / 37