A container has 3 liters of pure wine. 1 liter from the container is

Hi Bunuel and VeritasPrepKarishma,

Would it be possible to use this formula in this case?

New Concentration of wine= Old concentration of wine * (V1/V2)^n

n= number of iterations (it is 19 in this case)
v1 = volume of liquid withdrawn
v1 = initial volume of liquid

I noticed a similar formula being used here:
a-20-litre-mixture-of-milk-and-water-contains-milk-and-water-22212.html

Regards,

Let's go step by step:

First operation: 3L-1L=2=6/3L of wine left, total 4L;
#2: 6/3L-(6/3)/4=6/3-6/12=18/12=6/4L of wine left, total 5L;
#3: 6/4L-(6/4)/5=6/4-6/20=24/20=6/5L, total 6L;
#4: 6/5L-(6/5)/6=6/5-6/30=30/30=6/6L, total 7L;
....

At this point it's already possible to see the pattern: x=6/(n+2)

n=19 --> x=6/(19+2)=6/21=2/7L

Answer: A.

The question can be solved in under a minute if you understand the concept of concentration and volume.

Removal and addition happen 19 times so:

\(C_f = 1 * (\frac{2}{4}) * (\frac{3}{5}) * (\frac{4}{6}) * (\frac{5}{7}) * .......* (\frac{19}{21}) * (\frac{20}{22})\)
All terms get canceled (4 in num with 4 in den, 5 in num with 5 in den etc) and you are left with \(C_f = \frac{1}{77}\)
Since Volume now is 22 lt, Volume of wine = \(22*(\frac{1}{77}) = \frac{2}{7}\)

Theory:
1. When a fraction of a solution is removed, the percentage of either part does not change. If milk:water = 1:1 in initial solution, it remains 1:1 in final solution.

2. When you add one component to a solution, the amount of other component does not change. In milk and water solution, if you add water, amount of milk is the same (not percentage but amount)

3.
Amount of A = Concentration of A * Volume of mixture
Amount = C*V
( e.g. In a 10 lt mixture of milk and water, if milk is 50%, Amount of milk = 50%*10 = 5 lt)
When you add water to this solution, the amount of milk does not change.
So Initial Conc * Initial Volume = Final Conc * Final Volume
\(C_i * V_i = C_f * V_f\)
\(C_f = C_i * (V_i/V_f)\)
In the question above, we find the final concentration of wine. Initial concentration \(C_i\) = 1 (because it is pure wine)
When you remove 1 lt out of 3 lt, the volume becomes 2 lt which is your initial volume for the addition step. When you add 2 lts, final volume becomes 4 lt.
So \(C_f = 1 * 2/4\)
Since it is done 19 times, \(C_f = 1 * (\frac{2}{4}) * (\frac{3}{5}) * (\frac{4}{6}) * (\frac{5}{7}) * .......* (\frac{19}{21}) * (\frac{20}{22})\)

The concentration of wine is 1/77 and since the final volume is 22 lt (the last term has \(V_f\) as 22, you get amount of wine = 1/77 * 22 = 2/7 lt

It would have taken forever!

\(C_f = 1 * (\frac{2}{4}) * (\frac{3}{5}) * (\frac{4}{6}) * (\frac{5}{7}) * ....... (\frac{18}{20}) * (\frac{19}{21}) * (\frac{20}{22})\)

You need to observe here that other than first two numerators and last two denominators, all other terms will cancel out.
First term's denominator will cancel out third term's numerator.
Second term's denominator will cancel out fourth term's numerator.
The last two denominators will have no numerators to cancel them out.
The first two numerators have no denominators to cancel them out.
Usually, in such expressions (where terms have a pattern), things simplify easily. You just need to observe the pattern.

Hi VeritasKarishma

I'm stuck with the last bit here, it'll be great if you can help

I've come to the part where I have (2*3)/(21*22) after all cancellations which ultimately gives me 1/77. In your blog there was another sum where we started with a 50% concentration solution. So the final concentration was 50 * 9/10 * 9/10. In this case, since it's pure lemon juice, shouldn't it be 100% * 1/77 ? Where am I going wrong? And how do I get the answer after this?

The question can be solved in under a minute if you understand the concept of concentration and volume.

Removal and addition happen 19 times so:

\(C_f = 1 * (\frac{2}{4}) * (\frac{3}{5}) * (\frac{4}{6}) * (\frac{5}{7}) * .......* (\frac{19}{21}) * (\frac{20}{22})\)
All terms get canceled (4 in num with 4 in den, 5 in num with 5 in den etc) and you are left with \(C_f = \frac{1}{77}\)
Since Volume now is 22 lt, Volume of wine = \(22*(\frac{1}{77}) = \frac{2}{7}\)

Theory:
1. When a fraction of a solution is removed, the percentage of either part does not change. If milk:water = 1:1 in initial solution, it remains 1:1 in final solution.

2. When you add one component to a solution, the amount of other component does not change. In milk and water solution, if you add water, amount of milk is the same (not percentage but amount)

3.
Amount of A = Concentration of A * Volume of mixture
Amount = C*V
( e.g. In a 10 lt mixture of milk and water, if milk is 50%, Amount of milk = 50%*10 = 5 lt)
When you add water to this solution, the amount of milk does not change.
So Initial Conc * Initial Volume = Final Conc * Final Volume
\(C_i * V_i = C_f * V_f\)
\(C_f = C_i * (V_i/V_f)\)
In the question above, we find the final concentration of wine. Initial concentration \(C_i\) = 1 (because it is pure wine)
When you remove 1 lt out of 3 lt, the volume becomes 2 lt which is your initial volume for the addition step. When you add 2 lts, final volume becomes 4 lt.
So \(C_f = 1 * 2/4\)
Since it is done 19 times, \(C_f = 1 * (\frac{2}{4}) * (\frac{3}{5}) * (\frac{4}{6}) * (\frac{5}{7}) * .......* (\frac{19}{21}) * (\frac{20}{22})\)

The concentration of wine is 1/77 and since the final volume is 22 lt (the last term has \(V_f\) as 22, you get amount of wine = 1/77 * 22 = 2/7 lt

Hi VeritasKarishma,

Can you please share first 4 iteration with formula:\(C_i * V_i = C_f * V_f\).

I get how you got:\(C_f = 1 * 2/4\)

Similarly can you explain using this formula how you got 3/5, 4/6, 5/7?

This will help in clearly understanding the operation of the formula.

Thanks.

If the operation is only done 19 times then where and why does "22" Lt pop up in the final volume of mixture I was following how the demoninators increased but dont understand the "22".

Also if 1 L of wine is removed every operation how is the concentration of the wine mixture go up since part of it is being removed...only thing that is increasing the total volume of the solution..


Thanks a lot.

Karishma how did you arrive as in 1/77 ? Please explain