91Ó°ÊÓ

In algebra, mixture problems involve combining different substances to create a blend with specific properties, such as concentration of a particular component. In this exercise, we are working with soybean meal and corn meal to create a protein mixture. The soybean meal is 16% protein and the corn meal is 9% protein. Our goal is to find how many pounds of each type of meal are needed to create 350 pounds of mixture with 12% protein.

To solve mixture problems, it is vital to understand how to set up expressions that represent the properties of the substances being combined. For example, the weight of soybean meal is denoted as x and the weight of corn meal is 350 - x pounds. By setting up correct expressions for the amounts, we can form an equation that helps us find the solution.

Breaking down mixture problems into smaller, manageable steps simplifies the solving process, making it easier to understand how different components impact the final blend.

Percentage problems relate to finding parts of a whole expressed as percentages. They are common in mixture problems, finance, and everyday scenarios where proportions need to be calculated. In our exercise, the soybean and corn meals' protein percentages are key to creating the final mixture's desired protein content.

When dealing with percentages, it is essential to convert percentages to their decimal form before performing calculations. For instance, 16% becomes 0.16, 9% becomes 0.09, and 12% becomes 0.12. Using these decimal forms, we can set up equations that represent the total amount of protein contributed by each component:

• Total protein from soybean meal: 0.16x
• Total protein from corn meal: 0.09(350 - x)

Adding these gives the equation for the overall protein content: 0.16x + 0.09(350 - x) = 0.12 × 350. Breaking this equation down further helps solve for the unknowns.

A system of equations involves solving multiple equations simultaneously to find the values of unknown variables. In this mixture problem, we deal with one primary equation derived from the total protein content requirement. This system in the form of one equation arises when we express the total weight and protein contributions in terms of the same variables.

The equation 0.16x + 0.09(350 - x) = 0.12 × 350 is simplified through steps:

• Expand 0.09(350 - x) to get 31.5 - 0.09x.
• Combine like terms: 0.16x - 0.09x + 31.5 = 42.
• Simplify to find 0.07x = 10.5.
• Solve for x: x = 150 pounds.

This shows that we need 150 pounds of soybean meal. The remaining calculation gives us 200 pounds of corn meal.

Verifying the solution involves substituting these values back into the initial context to ensure the final mixture meets the problem's requirement. Systems of equations like this one can provide efficient solutions to mixing problems by representing constraints mathematically.