Solving the Milk and Water Mixture Problem with Proportions and Ratios

Introduction

Mixtures of milk and water involve basic concepts in proportions and ratios. This article will guide you through the steps to solve a problem involving a milk-water mixture. We will discuss how to use proportions and ratios to find the amount of milk in a mixture when given the initial ratio and an additional quantity of water.

The Problem Statement

Suppose we have a mixture of milk and water in a ratio of 7 to 9. This means for every 7 parts of milk, there are 9 parts of water. When we add 27 liters of water, the new ratio becomes 7 to 12. The question is: what is the amount of milk in the mixture?

Mathematical Approach

Let's start by setting up the problem. We can represent the original amount of milk and water as 7x and 9x liters, respectively, where x is a common multiplier.

Step 1: Initial Setup

Initial Ratio: 7x/9x 7/9

Initial Mixture: 7x 9x 16x

After adding 27 liters of water, the new ratio becomes 7:12:

New Ratio: (7x) / (9x 27) 7/12

Step 2: Setting Up the Equation

From the new ratio, we can write the equation:

7x / (9x 27) 7/12

By cross-multiplying, we get:

12 * 7x 7 * (9x 27)

84x 63x 189

Subtracting 63x from both sides:

21x 189

Solving for x:

x 189 / 21

x 9

Therefore, the original amounts are:

Milk: 7x 7 * 9 63 liters

Water: 9x 9 * 9 81 liters

Validation

Let's validate the solution:

Initial mixture: 63 liters milk 81 liters water 144 liters

After adding 27 liters of water: 81 27 108 liters water

New ratio: 63 / 108 7 / 12

Thus, the solution is correct.

Conclusion

Solving the milk and water mixture problem involves understanding the initial ratio, setting up the appropriate equation, and solving for the unknown variables. The method used here is applicable to similar problems involving proportions and ratios in mathematics.

Related Keywords

Mixture problem Proportion and ratio Mathematics problem

Further Reading

For more detailed problem-solving techniques and additional practice, you can refer to the following articles and resources:

Math Is Fun - Proportions and Ratios Khan Academy - Proportions and Ratios Purplemath - Solving Mixture Problems with Proportions