# Ratios and Mixture

I can't seem to to find a way to solve this problem:

In a mixture $60$ litres, the ratio of alcohol and water is $2:1$. If this ratio is to be $1:2$ then the quantity of water further added is? Ans $60$ litres

Any suggestions?

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In the original 60 liter mixture, the ratio of alcohol to water was $2:1$ which means that the original mixture was two-thirds alcohol and one-third water. This means $\frac{2}{3}\cdot 60 = 40$ liters of alcohol and $\frac{1}{3}\cdot 60 = 20$ liters of water.

Now you want to change the ratio to $1:2$ by adding water. a $1:2$ ratio means one-third alcohol and two-thirds water. Since we are adding only water, the amount of alcohol in the final mixture is still going to be $40$ liters (same as in your original mixture).

Now we know that $40$ liters of alcohol is one-third of the new mixture, which means there has to be $\dfrac{2/3}{1/3} \cdot 40 = 2\cdot 40 = 80$ liters of water in the new mixture. Since we already had $20$ liters originally, we have to add $80-20 = 60$ liters.

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I cant get the final step which gave you 80 liters of water. I got 120 grams of mixture (since 1/3 of Mixture is 40 grams of Alcohol) and 2/3 of that Mixture(120g) is 80 grams of water. Is this way correct ? I appreciate it if you could also clarify your final step of 80 liters. – MistyD Jul 2 '12 at 18:11
Ok well in the final mixture you have two-thirds water and one-thirds alcohol right? This means that you have 2 times as much water as alcohol correct? This is where the (2/3)/(1/3) = 2 comes from. Now we know that we have 40 liters of alcohol (because that is the amount in the original solution), so we must have 2 * 40 = 80 liters of water. Since we started out with 20 liters of water in the original mixture, we must add another 60 to get it to 80 liters. Does that make sense? – BYS2 Jul 3 '12 at 5:11
Yes it does. Thanks. – MistyD Jul 3 '12 at 6:27

Hint: in the original mixture, how much is alcohol? How much is water? Given the amount of alcohol, which doesn't change, how much water is needed to make the solution 1:2?

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