M04-08

Question

In a 10-liter mixture of cranberry juice and water, the ratio of juice to water is 3 to 2. Five liters of the mixture are removed and replaced with pure cranberry juice, and this process is repeated once more. What is the new ratio of juice to water in the mixture after the two removals and replacements?

A. 5 : 3
B. 6 : 4
C. 8 : 2
D. 17 : 3
E. 9 : 1

Bunuel · Accepted Answer

Official Solution:

In a 10-liter mixture of cranberry juice and water, the ratio of juice to water is 3 to 2. Five liters of the mixture are removed and replaced with pure cranberry juice, and this process is repeated once more. What is the new ratio of juice to water in the mixture after the two removals and replacements?

A. 5 : 3
B. 6 : 4
C. 8 : 2
D. 17 : 3
E. 9 : 1

We are given a 10-liter mixture of cranberry juice and water with a juice-to-water ratio of 3:2, which means it contains 6 liters of juice and 4 liters of water.
 
To solve the problem, we need to perform two rounds of removal and replacement. In each round, we remove 50% of the mixture, which is 5 liters.
 
During the first round, we remove 50% of the juice and 50% of the water, leaving us with a mixture that contains 3 liters of juice and 2 liters of water. We then add 5 liters of pure juice to this mixture, which gives us a new mixture with 8 liters of juice and 2 liters of water. 
 
During the second round, we again remove 50% of the juice and 50% of the water, leaving us with a mixture that contains 4 liters of juice and 1 liters of water. We then add 5 liters of pure juice to this mixture, which gives us a new mixture with 9 liters of juice and 1 liters of water. Therefore, the resulting mixture after the two removals and replacements has a juice-to-water ratio of 9:1.

Answer: E

manishtank1988 · Answer

My 2 cents:

rickyric395 · Answer

We are given 10 litres of juice mixture with juice:water= \frac{3}{2}  i.e juice is 6 litre and water is 4 litre. It can also be said mixture has  60 % juice.

Now only 5 litres of this mixture is kept and pure 5 litre juice is mixed with it.
We can equate concentrations of juice before and after. i.e  5*0.6 + 5*1 = 10*x  { using volume of juice*concentration. 5 litre of old juice mixture*it had 60% concentration + 5 litre of new juice with 100% concentration = 10 litre of mixture* unknown concentration. }
x=0.8 i.e this mixture has 80% juice concentration.

Again 5 litre of this new mixture is mixed with 5 litre of pure juice. Again doing the same thing as above.
 5*0.8+5*1=10*x  We get x=0.9 , in 10 litre we have 90% juice then remaining 10% is water so juice:water =  \frac{9}{1} .

Please give Kudos if it helps.

Bunuel · Answer

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.

bumpbot · Answer

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M04-08

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