Question

In a 25 litre mixture of milk and water, the water is only \(20 \% .\) How many litres of water is required to increase the percentage of water to \(90 \%\) ? (a) 45 litre (b) 70 litre (c) 115 litre (d) 175 litre

Short answer

Answer: 175 litres.

Step-by-step solution

Identify the initial mixture

Originally, there are 25 litres of mixture, with 20% of water. So, the amount of milk and water in the initial mixture can be determined as follows: - Milk: \(80\%\) of 25 litres (Since the total percentage is 100%, and 20% of it is water) - Water: \(20\%\) of 25 litres

Calculate the amount of milk and water in the initial mixture

Calculate the amount of milk and water: - Milk: \(\frac{80}{100} \times 25 = 20\) litres - Water: \(\frac{20}{100} \times 25 = 5\) litres So, we have 20 litres of milk and 5 litres of water in the initial mixture.

Write the equation for the percentage of water after adding x litres of water

We want to increase the percentage of water to 90% by adding x litres of water. To do this, we'll create an equation with the percentages: \(\frac{5 + x}{25 + x} = \frac{90}{100}\) Where "5 + x" represents the final amount of water, and "25 + x" represents the final total volume of the mixture.

Solve the equation for x

To solve the equation, we'll multiply both sides by 100 and simplify: \(100(5 + x) = 90(25 + x)\) \(500 + 100x = 2250 + 90x\) \(100x - 90x = 2250 - 500\) \(10x = 1750\) \(x = \frac{1750}{10}\) \(x = 175\)

Interpret the result

Since we found that x = 175 litres, we can conclude that we need to add 175 litres of water to the initial 25 litre mixture to increase the percentage of water to 90%. Therefore, the correct answer is (d) 175 litres.

Key concepts

Understanding Percentage Increase

Understanding percentage increase is crucial in solving mixture problems like this one. When we talk about percentage increase, we refer to the rise in value from an initial percentage to a new one.
In this exercise, you start with 20% water in the mixture and want to increase it to 90%. Here, the increase is not just numeral but relative to the total mixture. To achieve this, additional water is added, thus increasing the water's percentage in the overall mixture.
To conceptualize increase:

• Identify the initial percentage (20% in this case).
• Determine the desired new percentage (90%).
• Calculate how to augment the mixture to achieve this goal (by adding more water).

Calculating Water and Milk Ratio

The water and milk ratio expresses the relationship between the volumes of water and milk in the mixture. Initially, this ratio was 20% water and 80% milk.
Here's how you determine the components:

• Find the water content by calculating 20% of 25 litres: 't Water = \(\frac{20}{100} \times 25 = 5\) litres'\"'
• Find the milk volume from the remaining percentage: 't Milk = \(\frac{80}{100} \times 25 = 20\) litres'\"'

By understanding this ratio, you see how much water and milk initially exist in the mixture before changes are made.

Performing Volume Calculation

Volume calculation involves determining the quantities of each component in the mixture after adding a new amount. In this problem, you need to calculate the total volume after adding "x" litres of water.
The initial total volume is 25 litres, with both milk and water included. When you add "x" litres of water, the new total volume becomes \(25 + x\) litres.
This calculation helps to find both the individual and total quantities after the change. This change impacts the percentage of each component (water, in this case) in the new mixture.

Setting Up and Solving Algebraic Equations

Algebra is a powerful tool for solving problems involving mixtures and percentage changes. In this case, an equation is set to solve for the unknown "x," the additional litres of water needed.
The equation formed here is:\[\frac{5 + x}{25 + x} = \frac{90}{100}\]
This represents reaching the desired percentage of water in the mixture (90%). To find "x":

• Multiply both sides by 100 to eliminate the fraction.
• Expand and simplify: \(100(5 + x) = 90(25 + x)\).
• Combine like terms to solve for "x": \(10x = 1750\).
• Solve for "x": \(x = \frac{1750}{10} = 175\).

Thus, you find that 175 litres of water must be added to achieve the mixture goal. Algebraic equations are essential in making these types of calculations straightforward and precise.