Question

The diluted wine contains only 8 litres of wine and the rest is water. A new mixture whose concentration is \(30 \%\), is to be formed by replacing wine. How many litres of mixture shall be replaced with pure wine if there was initially 32 litres of water in the mixture ? (a) 4 (b) 5 (c) 8 (d) none of these

Short answer

(Assume that the options are (a) 1/2 litres, (b) 1/4 litres, (c) 1/8 litres, and (d) none of these.) Answer: (d) none of these.

Step-by-step solution

Find the initial concentration of wine

Firstly, we need to find the initial concentration of wine in the mixture. We are given the total volume of the mixture, which is 8 litres of wine, and 32 litres of water. The initial concentration of wine can be found using the formula: \(\text{Initial Concentration} (\%) = \frac{\text{Wine Volume}}{\text{Total Volume}} \times 100\)

Set up the equation to find the amount of mixture to be replaced

Let's denote the volume of the mixture to be replaced with pure wine as x litres. After replacing x litres of the mixture, the final volume of wine in the mixture will be (8 - x + x) litres, and the final volume of water in the mixture will be (32 - x) litres. The new concentration of wine in the mixture is given to be 30%. We can set up the following equation: \(\frac{\text{Final Wine Volume}}{\text{Total Volume}} = 0.3\)

Solve the equation

Now substitute the expressions for the final wine and water volumes, and the total volume into the equation from step 2: \(\frac{8 - x + x}{40 - x} = 0.3\) Next, solve the equation for x: 40 - x = 3(8 - x + x) 40 - x = 24 x = 16

Check the answer and choose the correct option

x=16 indicates that we need to replace 16 litres of the initial mixture with pure wine. However, this answer does not match any of the given options. So, the correct answer is (d) none of these.

Key concepts

Concentration of Mixtures

Understanding the concentration of mixtures is crucial for various disciplines, including chemistry and mathematics. In the context of our example involving the diluted wine, concentration refers to the amount of a substance (in this case, wine) present in a mixture relative to the total amount of the mixture. The concentration is typically expressed as a percentage and is calculated using the formula:
\[ \text{Concentration} (\%) = \frac{\text{Substance Volume}}{\text{Total Volume}} \times 100 \]
In the given exercise, the initial mixture contains 8 litres of wine and 32 litres of water, making a total of 40 litres. To solve the problem, we must calculate the new concentration after replacing a certain volume of this mixture with pure wine to achieve a 30% concentration. It's crucial when solving such problems to keep consistent units and carefully track the volumes of each component in the mixture.

Quantitative Aptitude

Quantitative aptitude encompasses the ability to deal with numerical and mathematical calculations. This skill is essential in various competitive exams and problem-solving scenarios. The core aspect of quantitative aptitude is the application of mathematical concepts to solve problems efficiently. For instance, in our exercise on mixtures and alligations, we used mathematical formulas to calculate concentration percentages and replacement volumes.
It's important to approach such problems systematically: identify the relevant information, set up equations based on the known values, and solve for the unknown. To improve quantitative aptitude, practice is key. Tackle a wide range of problems, learn from mistakes, and solidify understanding of basic mathematical operations. By doing so, calculations become more intuitive and less time-consuming, ultimately enhancing one's problem-solving capabilities.

Problem Solving

Problem-solving skills enable one to find solutions to complex or tricky situations. In mathematics, problem-solving usually involves dissecting the problem into smaller, more manageable parts, formulating equations, and applying logical reasoning. For our mixture problem, we followed specific steps, starting with the calculation of the initial wine concentration, then setting up an equation to find the amount of mixture to be replaced.

One critical factor in problem solving is verifying that the solution makes sense in the context of the problem. In this case, we discovered that the calculated volume to be replaced did not match any of the answer options, leading us to select (d) none of these. This highlights the importance of double-checking and considering all possibilities before finalizing an answer. The key to developing problem-solving skills is to work through diverse problems and practice various strategies, such as drawing diagrams, making lists, or working backward from the desired conclusion.