Solving Mixture Ratio Problems: A Comprehensive Guide

Understanding and solving mixture ratio problems is a crucial skill in both school and real-world applications, particularly in chemistry, cooking, and manufacturing. In this article, we will explore how to solve these problems using practical examples. We'll cover a detailed step-by-step explanation for two different scenarios, providing you with a solid foundation to tackle similar problems on your own.

Scenario 1: Initial Mixture with a 3:2 Milk and Water Ratio

Let's start by solving a common problem: finding the new ratio when a certain amount of water is added to an existing mixture with a known ratio.

Problem Statement

A 40 liters mixture contains milk and water in the ratio 3:2. If 8 liters of water is added to it, what will be the new ratio of milk and water in the final mixture?

Solution

First, we need to determine the initial quantities of milk and water in the 40 liters mixture.

Step 1: Determine Initial Quantities

The mixture has milk and water in a ratio of 3:2. This means for every 3 parts of milk, there are 2 parts of water. The total parts in the mixture are 3 2 5 parts.

Using this information:

Quantity of Milk: (3/5) * 40 24 liters Quantity of Water: (2/5) * 40 16 liters

Step 2: Add Water and Calculate the New Quantity

Now, we add 8 liters of water to the mixture:

New Quantity of Water: 16 8 24 liters

Final Quantities:

- Milk: 24 liters

- Water: 24 liters

Step 3: Determine the New Ratio

The new ratio of milk to water:

Milk:Water 24:24 1:1

Therefore, the new ratio of milk to water in the final mixture is 1:1.

Scenario 2: Sequential Dilution in a Mixture

Now let's explore a more complex scenario involving the sequential dilution of a mixture.

Problem Statement

A 40 liters mixture of milk and water contains milk and water in a 3:2 ratio. 10 liters of the mixture is removed and replaced with pure milk, and this operation is repeated once more. What will be the ratio of milk and water after the second operation?

Solution

We will solve this step-by-step, using the given information and relations.

Step 1: Determine Initial Quantities

Initial Ratio: Milk to Water 3:2 Total Parts: 5

Using this information:

Milk: (3/5) * 40 24 liters Water: (2/5) * 40 16 liters

Step 2: First Extraction and Replacement

10 liters of the original mixture is removed and replaced with pure milk.

Milk in Extracted Fluid (MS1): (3/5) * 10 6 liters Water in Extracted Fluid (WS1): (2/5) * 10 4 liters

New Quantities:

Milk: 24 - 6 18 liters Water: 16 - 4 12 liters

New Ratio: M1/W1 18/12 3/2 7/3

Step 3: Second Extraction and Replacement

10 liters of the new mixture (M1:18, W1:12) is extracted and replaced with pure milk.

Milk in Extracted Fluid (MS2): (7/10) * 10 7 liters Water in Extracted Fluid (WS2): (3/10) * 10 3 liters

New Quantities:

Milk: 18 - 7 10 21 liters Water: 12 - 3 9 liters

New Ratio: M2/W2 21/9 7/3 31/9

Therefore, the final ratio of milk to water after the second operation is 31:9.

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