Solving Complex Mixture Problems: A Comprehensive Guide

Mixture problems are a common type of word problem that require the application of algebraic techniques. One such problem involves combining two different solutions to achieve a desired concentration. This article will explore the detailed solution to a specific mixture problem that involves adding liters of a 7% salt solution to a 28% salt solution to obtain 903 liters of a 24% salt solution. We will break down the algebraic steps and the logic behind the calculations using both a single equation method and a step-by-step approach.

Understanding the Problem:

The goal is to determine how many liters of a 7% salt solution should be added to a 28% salt solution to achieve a final mixture of 903 liters with a 24% salt concentration. Let's denote the volume of the 7% solution as x liters and the volume of the 28% solution as y liters.

Setting Up the Equations:

First, we establish the relationship between x and y based on the total volume of the final mixture:

x y 903

Next, we set up the equation for the concentration of salt in the final mixture. The total amount of salt from the 7% and 28% solutions added should equal the amount of salt in the final 24% solution:

0.07x 0.28y 0.24 times 903

Step-by-Step Solution:

We will solve this system of equations step-by-step:

From the first equation, we can express y in terms of x:
y 903 - x

Substituting y 903 - x into the second equation, we get:
0.07x 0.28(903 - x) 0.24 times 903
0.07x 252.84 - 0.28x 216.72

Combining like terms: -0.21x 252.84 216.72 -0.21x 216.72 - 252.84 -0.21x -36.12 x frac{-36.12}{-0.21} 172

Thus, we need 172 liters of the 7% salt solution.

Alternative Method Using Ratios:

We can also solve this problem using the concept of ratios by considering the additional steps when adding solutions. Let's assume we start with 10 liters of the 28% salt solution and need to determine how much of the 7% salt solution to add to achieve a 24% solution.

Let x be the additional volume of the 7% salt solution that we add. The total volume becomes 10 x liters, and the total amount of salt is:

0.28 times 10 0.07x 0.24 times (10 x)

Expanding and simplifying:

2.8 0.07x 2.4 0.24x

2.8 - 2.4 0.24x - 0.07x 0.4 0.17x x frac{0.4}{0.17} 2.352941176

This means we need to add approximately 2.353 liters of the 7% salt solution to 10 liters of the 28% solution to get a 24% solution.

To generalize this result, we multiply the amount of 7% solution we need to add to 10 liters of 28% solution by 73.1 (since 903 liters is 73.1 times 12.352941176 liters):

172 73.1 times 2.352941176

This confirms our previous result, that we need 172 liters of the 7% salt solution to achieve the desired mixture.

Conclusion:

Mixture problems are fundamental in algebra and have practical applications in various fields such as chemistry, engineering, and everyday life. Whether you're dealing with salt solutions or other mixtures, the key is to set up the correct equations and solve them step-by-step. Using the methods demonstrated in this article, you can approach similar problems with confidence and accuracy.