# Solve Water Mixing Problem: Volume K Tank of Alcohol

• adi11235
In summary, the problem involves a tank with a volume of K liters of alcohol, where one liter of alcohol is removed and one liter of water is added. This process is repeated two more times. At the end, the ratio of water to alcohol in the tank should be 7:1. The solution involves multiplying the water and alcohol components by (K-1)/K and adding 1 to the water component for each iteration. The initial state is {K, 0} and the final state is {1/8K, 7/8K}.
adi11235
I've struggled with this problem for a bit and I ran out of ideas.

"We have a tank of volume K liters of alcohol. We remove one liter from the tank and add one liter of water. From the mix, we remove one liter and add one liter of water. We do this one more time.

At the end of the process, there should be 7 times more water than alcohol in the tank.

What is the volume of the tank?"

My attempt at a solution is attached below. I hope it's legible.

The idea is that adding one liter of alcohol means multiplying the water and the alcohol components by (k-1)/k and adding 1 to the water component.

The alcohol component has been written down as a_i, the water component is b_i, where:
i={0,1,2,3} is the step
{a0, b0} is the initial state (a0=k, b0=0)
{a1,b1} is the state after adding one to b
{a3,b3} is the final state.

#### Attachments

• water_alcohol.jpg
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Last edited:
Mixture(0) = alcohol.
After the first iteration, mixture(1) = (K-1)/K alcohol + 1/K water
After the second, mixture(2) = (K-1)/K mixture(1) + 1/K water
After the third, mixture(3) = (K-1)/K mixture(2) + 1/K water, and alcohol(3)/water(3) = 1/7.

Thanks. I mixed up stage one :)

## 1. What is the best method for solving a water mixing problem in a tank of alcohol?

The best method for solving a water mixing problem in a tank of alcohol is to use the principle of concentration and volume. This involves finding the initial concentration of alcohol in the tank, the desired concentration after mixing, and the final volume of the mixture. By using these values, the amount of water needed to achieve the desired concentration can be calculated.

## 2. How do you calculate the initial concentration of alcohol in the tank?

The initial concentration of alcohol in the tank can be calculated by dividing the volume of alcohol in the tank by the total volume of the mixture. This will give you the percentage of alcohol in the tank before any water is added.

## 3. What formula should be used to determine the final volume of the mixture?

The final volume of the mixture can be calculated by using the formula: Vf = Vi + Vw, where Vf is the final volume, Vi is the initial volume of alcohol, and Vw is the volume of water added.

## 4. Can the water mixing problem be solved for any tank size?

Yes, the water mixing problem can be solved for any tank size as long as the initial concentration of alcohol, desired concentration after mixing, and final volume of the mixture are known. However, it may be more challenging for larger tanks due to the larger volume of alcohol and water involved.

## 5. Are there any factors that can affect the accuracy of the calculation?

Yes, there are some factors that can affect the accuracy of the calculation. These include any inaccuracies in measuring the initial concentration of alcohol, any evaporation or spillage during the mixing process, and any variations in the density of the alcohol. It is important to take these factors into account and make adjustments as needed to ensure an accurate calculation.

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