Ratio of Alcohol to Water

[baily106]

Hello,

This question is from NOVA's math book.

It states, "A cask initially contains pure alcohol up to the brim. The cask can be emptied by removing exactly 5 liters at a time. Each time this is done, the cask must be filled back to the brim with water. The capacity of the cask is 15 liters. When the cask is emptied and filled back to the brim two times, what is the ratio of alcohol to water in the cask?"

A) 1:2
B) 2:1
C) 1:1
D) 4:5
E) 5:4

I understand that the first time you remove 5 liters of alcohol, the ratio of alcohol to water becomes 2:1. However, I don't understand how to obtain the ration the second time.

Thanks.

[tommywallach]

What a poorly written question: you don't empty something if you remove 5 liters. Anyway, I still understand it. They're saying it goes like this:

Remove 5 liters pure alcohol and replace:

Now it's 10 liters alcohol, 5 liters water.

Remove 5 liters and replace:

The 5 liters you remove are 1/3rd water and 2/3rds alcohol, which means 5/3 liters of water and 10/3 liters of alcohol.

10 liters alcohol - 10/3 = 20/3 alcohol remaining
5 liters water - 5/3 = 10/3 water

The you add back 5 liters of water, ending up with

20/3 alcohol
25/3 water

that's a ratio of 20:25, or 4:5.

-t

[baily106]

Got it; thanks.

[tommywallach]

Glad to help!

[arjunturakhia]

I still didn't get the second part - could you please elaborate? Thanks!

[tommywallach]

Hey Arjun,

Not sure what you mean by second half. Let me know exactly where it stops making sense. (And keeping in mind it's a very badly written question!).

-t