A large tank contains 400-kg mixture of water and alcohol. The mixture is 64% alcohol by weight. At each step, 100 kg of the mixture will be drained from the tank, replaced with 100 kg of water and then stirred. After three steps, what percent of the final solution will be alchohol?

thanks
~zach

At the start we had >64(400) kg of alcohol

100 kg were removed,
so we had .64(300) left, adding 100 kg of water does not change that, so
after 1st step, alcohol left is .64(300) of the 400 kg

in the 2nd step the same argument can be applied
so the amount of alcohol left is .64(.64(300))

after the 3rd step, amount left is .64(.64(.64(300)))
= 300(.64)^3 = 78.64 kg

so the percentage of the tank that is alcohol is 78.64/400 = 19.66%

Thank you

To find out the percent of alcohol in the final solution after three steps, we need to calculate the amount of alcohol left in the tank after each step.

First, we know that the initial tank contains 400 kg of a mixture that is 64% alcohol by weight. This means that there are 400 kg * (64/100) = 256 kg of alcohol in the tank.

At each step, 100 kg of the mixture is drained from the tank and replaced with 100 kg of water. After the first step, the tank contains 400 kg - 100 kg = 300 kg of mixture, but the alcohol content remains the same since no alcohol was removed or added.

After stirring, the mixture will be thoroughly mixed, so the alcohol is evenly distributed. Thus, the percentage of alcohol by weight remains the same as before (64%).

After the second step, another 100 kg of the mixture is drained, leaving 300 kg - 100 kg = 200 kg of mixture in the tank. Again, the alcohol content remains the same at 64%.

After stirring, the alcohol is evenly distributed, so the percentage of alcohol by weight remains at 64%.

After the third step, another 100 kg of the mixture is drained, leaving 200 kg - 100 kg = 100 kg of mixture in the tank. Still, the alcohol content remains at 64%.

After stirring, the alcohol is evenly distributed, keeping the percentage of alcohol by weight at 64%.

Therefore, after three steps, the final solution will contain 100 kg * (64/100) = 64 kg of alcohol.

To find the percentage of alcohol in the final solution, divide the weight of alcohol by the weight of the final solution and multiply by 100:

Percentage of alcohol in final solution = (64 kg / 100 kg) * 100% = 64%.

Therefore, after three steps, the final solution will be 64% alcohol.