Although this question can be solved in the conventional method using expressions, a faster approach in this question would be to observe the numbers and pick options accordingly.
The withdrawal and replacement operations are being carried out four times. Each time, 8 litres are withdrawn and replaced. 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 = \(3^4\).
So, let’s pick up those options which are multiples of 3.
Let’s start with option A. Out of 18 litres, if 8 litres is removed, the concentration of wine left after 4 operations = \(\frac{5}{9} * \frac{5}{9} * \frac{5}{9} * \frac{5}{9}\). This will definitely not give us \(\frac{16}{65}\). Option A can be eliminated.
Let’s try option B. Out of 24 litres, if 8 litres is removed, the concentration of wine left after 4 operations = \(\frac{2}{3} * \frac{2}{3} * \frac{2}{3} * \frac{2}{3}\) = \(\frac{16}{81}\). Clearly, this is,
\(\frac{Wine}{Total}\) = \(\frac{16}{81}\) i.e. \(\frac{Wine}{Wine + Water}\) = \(\frac{16}{16 + 65}\). Therefore, the ratio of wine to water is 16:65, which is what is mentioned in the question.
So, the correct answer option is B.
As you can see, this approach where we try out options based on a certain logic worked really well and did not take too much time as well. So, in such questions, instead of always relying on an expression to help you solve the question, try using logic and the answer options instead.
Hope this helps!
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