Question

From a container, 6 litres milk was drawn out and was replaced by water. Again 6 litres of mixture was drawn out \& was replaced by the water. Thus the quantity of milk and water in the container after these two operations is \(9: 16\). The quantity of mixture is : (a) 15 (b) 16 (c) 25 (d) 31

Short answer

Answer: 25 litres

Step-by-step solution

Define the variables

Let x be the initial quantity of milk in the container. Then, after the first replacement, the amount of milk remaining is (x - 6).

Calculate the amount of milk after the first replacement

When 6 litres of milk is drawn out and replaced with water, the remaining amount of milk is x - 6. Since the container remains full throughout the process, the total amount of the mixture will still be x litres.

Calculate the ratio of milk after the second replacement

After the second replacement (another 6 litres of the mixture is drawn out and replaced with water), the ratio of milk to water in the container is 9:16. Since milk is being replaced with water, the proportion of remaining milk in the container is: (1 - (6/x)) * (x - 6)

Set up the equation for the final ratio

According to the final ratio given, the proportion of milk in the container is equal to: 9/25 * x

Equate the proportions and solve for x

Now, we set these two expressions equal and solve for x: (1 - (6/x)) * (x - 6) = (9/25) * x

Simplify and solve the equation

Multiply both sides by 25x for simplicity: 25 * (x - 6) * (1 - (6/x)) = 9x 25 * (x - 6) * (x - 6) / x = 9x Upon solving this equation, we get: x = 25

Choose the correct option

Since x, the initial quantity of the mixture in the container, is equal to 25, the correct option is: (c) 25

Key concepts

Ratio and Proportion

Understanding the concept of ratio and proportion is essential when dealing with mixture problems. A ratio is a way to compare two quantities by division, usually expressed in the form of 'a to b' (a:b) or as a fraction \( \frac{a}{b} \). Proportion, on the other hand, signifies that two ratios are equal. It relates to the concept that two fractions or ratios are equivalent.

In the context of our milk and water mixture problem, the ratio represents the relationship between the amount of milk and the amount of water in the container. Once the milk is drawn out and replaced by water, the new ratio must account for this change. After two such operations, we have a final given ratio of milk to water, which is 9:16. This is the key to setting up our equation and finding the solution to the problem.

By understanding and manipulating ratios, you can solve complex problems simply by setting up the correct proportions and finding the unknown variable. It's a mathematical tool commonly used not just in mixture problems, but in many aspects of real-world applications, such as cooking, scaling models, and even financial analysis.

Quantitative Aptitude

Quantitative aptitude encompasses the ability to reason and solve numerical and mathematical problems. It is a key element of most competitive exams and is crucial for effective problem-solving in fields such as finance, engineering, and science. In the context of our example on mixture problems, having good quantitative aptitude means being able to translate the word problem into mathematical equations and solve for the unknowns effectively.

When approaching a mixture problem, you have to pay close attention to the quantities involved and the operations performed - such as drawing out milk and replacing it with water. Quantitative aptitude also involves understanding the principles of ratio and proportions and algebra to model the situation with equations that can be solved.

In this case, your quantitative aptitude is tested through your understanding of the problem scenario and your ability to perform the steps from defining variables to equating proportions and ultimately solving algebraic equations to find the answer to the given problem.

Algebraic Equations

Algebraic equations are mathematical statements that show the equality of two expressions. They consist of variables (unknown values) and constants and are essential tools for solving various problems in algebra. In mixture problems, algebraic equations are used to represent the relationship between different components of the mixture.

For example, in our milk-water mixture problem, we created an algebraic equation to represent the ratio of milk to water after two operations. The equation \( (1 - (6/x)) \times (x - 6) = \frac{9}{25} \times x \) captures the changes in the quantity of milk as it is removed and replaced with water.

Solving such an equation involves multiple algebraic techniques including simplifying expressions, multiplying across brackets, and isolating the variable. The ultimate goal is to find the value of the unknown quantity x, which represents the initial quantity of the milk in the container. Algebraic equations are thus the backbone of solving mixture problems as they provide a methodical approach to arriving at a solution.