Mixing Ratios to Achieve a Desired Ratio: A Comprehensive Guide

In many practical scenarios, including chemistry, cooking, and various industries, it is essential to understand how to mix different ratios to achieve a desired result. This article presents a thorough explanation of how to mix two ratios to achieve a specific target ratio. We will explore the method of using weighted averages and provide a detailed step-by-step solution.

Introduction to Ratios and Mixing

Ratios are used to compare two or more quantities relative to each other. In this context, we will discuss how to mix two ratios, 7:3 and 21:4, to achieve a target ratio of 3:1. This process involves converting the ratios into fractions, setting up an equation for the weighted average, and solving for the desired mixing ratio.

Methodology: Using Weighted Averages

To find the ratio in which to mix the two ratios 7:3 and 21:4 to achieve the target ratio 3:1, we can use the concept of weighted averages. Let's denote the ratios as follows:

A 7:3 B 21:4 C 3:1

First, convert the ratios to fractions:

A 7/3, B 21/4, C 3/1

Let x be the amount of A and y be the amount of B. The average ratio can be expressed as:

(x * A y * B) / (x y) C

Plugging in the values:

(x * 7/3 y * 21/4) / (x y) 3/1

Cross-multiplying to eliminate the fraction:

x * 7/3 y * 21/4 3 * (x y)

Multiplying through by a common denominator (12) to eliminate fractions:

12 * (x * 7/3 y * 21/4) 12 * 3 * (x y)

Simplifying:

28x 63y 36x 36y

Rearranging the equation:

28x - 36x 63y - 36y 0

Combining like terms:

-8x 27y 0

Solving for the ratio x/y:

8x 27y

x/y 27/8

Therefore, the two ratios should be mixed in the ratio of 27:8 to achieve the target ratio of 3:1.

Alternative Approaches to Solving the Problem

Another method to solve the problem involves ensuring that the total parts in all ratios are the same. Let's consider the following transformations:

X → 7:3:10 or 70:30:100 Y → 21:4:25 or 84:16:100 Desired Z → 3:1:4 or 75:25:100

Now, compare the content of the first or second component:

75 is 5 parts above 70 and 9 parts below 84.

The quantities x and y of mixture X and mixture Y to be mixed are in the ratio:

x:y :: 9:5

A similar approach can be taken by expressing the problem in terms of the liquids in the containers:

Let X liters of the first container be mixed with Y liters of the second container.

(3X/10) (4Y/25) (X Y)/4

Simplifying:

15X 8Y 5X 5Y

550Y 632Y

18Y 1

X/Y 9/5

Therefore, the same answer of 9:5 is obtained for the mixing ratio.

Conclusion

In conclusion, we have demonstrated two methods to solve the problem of mixing two ratios to achieve a target ratio. The first method uses the concept of weighted averages, while the second method ensures that the total parts in all ratios are the same. Both methods lead to the same result: a mixing ratio of 9:5.

Further Reading

If you are interested in learning more about ratios, mixing problems, and weighted averages, consider exploring the following resources:

Geometric Progression and Its Application to Sequences and Series Algebraic Equations and their Solutions Mixture Problems and Their Applications in Real Life