Eight Litres of Wine is Drawn From a Cask Full of Wine and is Replaced GMAT Problem Solving

Question: Eight litres of wine is drawn from a cask full of wine and is replaced with water. The operation is repeated three more times. The ratio of the quantity of wine left in cask to that of water is 16:65. How much wine did the cask originally hold?

1. 18 litres
2. 24 litres
3. 32 litres
4. 42 litres
5. 60 litres

Correct Answer: B

Solution and Explanation
Approach Solution 1:

Given:
Eight litres of wine is taken out from a cask full of wine and is replaced with water.
The process is repeated three more times making total number of times to four.
The ratio of the quantity of wine left in cask to that of water is 16:65.
Find out:
How much wine was the cask originally holding or how much wine it can hold?
After the replacement process has been performed 4 TIMES, the anmount left in the casket is the following fraction:
wine/total =\(\frac{16}{16+65}\)
=\(\frac{16}{81}\)
Now, if we break 16/85, we get
\((⅔)^4= \frac{16}{81}\)

We can conclude that, each of the 4 replacement processes must leave behind \(\frac{2}{3}\)of the wine.
There is also another way to look at it. As the replacement process leaves behind \(\frac{2}{3}\)of the wine, each replacement process must reduce the amount of wine in the casket by ⅓ rd.
Thus, the 8 liters drawn off in the first replacement process must constitute
\(\frac{1}{3}\)of the original amount of wine:
Hence, 8=\(\frac{1}{3}w\)
w= 24
Hence, the answer is B.

Approach Solution 2:
Given:
Eight litres of wine is taken out from a cask full of wine and is replaced with water.

The process is repeated three more times making total number of times to four.
The ratio of the quantity of wine left in cask to that of water is 16:65.
Find out:
How much wine was the cask originally holding or how much wine it can hold?
As per the problem statement, the withdrawal and replacement process are carried out four times. Each time, 8 litres are withdrawn and replaced with water. At the end of four operations, the ratio of wine to
water is 16:65.
This means that the total volume left in the cask is a multiple of 81 (16+65). We know 81 = 34
So, we will pick up those options which are multiples of 3.
If we consider option A, out of 18 litres, if 8 litres is removed, the concentration of wine left after 4 operations:

\(\frac{5}{9}*\frac{5}{9}*\frac{5}{9}*\frac{5}{9}*\)
The result will not definitely be \(\frac{16}{65}\). Hence, we can eliminate option A.
Now, let’s try option B. Out of 24 litres, if 8 litres is removed, the concentration of wine left after 4 operations

\(\frac{2}{3}*\frac{2}{3}*\frac{2}{3}*\frac{2}{3}*=\frac{16}{81}\)
Wine/ Total = \(\frac{16}{81}\)
Wine/ Wine + Water = \(\frac{16}{16+65}\).
Therefore, the ratio of wine to water is 16:65. This is the same ratio as mentioned in the question.
Hence, the correct answer is B.

Approach Solution 3:
Let the quantity of the wine in the cask originally be x litres.
After first operation, wine left in the cask =(x−8) litres.
So, the ratio of the quantity of wine now left in cask to that of filled cask= x−8/x
This operation is performed three more times.
Then, Wine : Water =16:65
or, Wine : Total =16:81
⇒(x−8)^4/x= 16/81
⇒x−8/x= 2/3
⇒3x−24= 2x
⇒x= 24 litres

“Eight litres of wine is drawn from a cask full of wine and is replaced”- is a topic of the GMAT Quantitative reasoning section of GMAT. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. GMAT Quant practice papers improve the mathematical knowledge of the candidates as it represents multiple sorts of quantitative problems.

 

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