Solving Ratio Problems in Mixture and Alligation

Introduction

Understanding how to solve problems involving ratios in mixtures and alligation is essential for anyone working in chemistry, food science, or other fields where precise mixing ratios are important. This article will guide you through the process of solving such problems, using specific examples to illustrate the steps involved.

Problem Statement

Let's start with a real-world problem: In a mixture of 45 liters, the ratio of sugar solution to salt solution is 1:2. We want to determine how much additional sugar solution should be added to achieve a new ratio of 2:1. Similarly, a related problem involves a mixture of 90 liters with a ratio of sugar solution to salt solution of 1:2 that we want to adjust to a 2:1 ratio.

Step-by-Step Solution

Problem 1: Mixture of 45 Liters

Step 1: Determine the initial amounts of each solution.

Lets denote the amount of sugar solution by S and the amount of salt solution by A.

Given the ratio S:A 1:2, we can express the amounts in terms of a variable x as follows:

S x (sugar solution) A 2x (salt solution)

The total mixture is 45 liters, so:

S A 45

x 2x 45

3x 45

x 15

Thus, the amounts are:

Sugar solution: 15 liters Salt solution: 30 liters

Step 2: Set up the new ratio.

We want the new ratio of sugar solution to salt solution to be 2:1. Let y be the amount of sugar solution to be added. The new amount of sugar solution will be 15 y liters, and the amount of salt solution remains 30 liters.

Step 3: Set up the equation for the new ratio.

The new ratio of sugar solution to salt solution is:

[ frac{15 y}{30} frac{2}{1} ]

Step 4: Solve for y.

Cross-multiplying gives:

[ 15 y 60 ]

Now solving for y:

[ y 60 - 15 ]

[ y 45 ]

Conclusion

To achieve the desired ratio of 2:1, 45 liters of sugar solution must be added to the mixture.

Problem 2: Mixture of 90 Liters

We start by determining the initial amounts of each solution:

Sugar solution 1/3 of 90 liters 30 liters Salt solution 2/3 of 90 liters 60 liters

To make the ratio 2:1, the total sugar solution should be double the salt solution. Therefore, 120 liters are required.

How much needs to be added later? Since 30 liters were already there, 120 - 30 90 liters of sugar solution need to be added.