Helena's Garden Mix Calculating Topsoil And Compost Quantities

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Introduction

In this article, we will delve into a practical mathematical problem involving Helena's gardening endeavors. Helena requires a blend of compost and topsoil for her garden and has made a purchase of both. The goal is to determine the exact amount of topsoil Helena acquired, given the total volume of the mixture, the overall cost, and the individual prices of compost and topsoil. This problem is a classic example of a system of equations, where we use multiple variables and equations to represent the given information and solve for the unknowns. Let’s embark on this mathematical journey to unravel the solution, step by step, ensuring clarity and comprehension for all readers. Understanding how to solve such problems is not only beneficial for academic purposes but also for real-life scenarios involving resource allocation and budgeting.

Problem Statement

Helena uses a mixture of compost and topsoil in her garden. She purchased a total of 10 cubic yards of compost and topsoil for $180. If compost costs $25 per cubic yard and topsoil costs $15 per cubic yard, how many cubic yards of topsoil did she purchase?

Breaking Down the Problem

To effectively solve this problem, we need to dissect it into smaller, manageable parts. The problem provides us with the following key pieces of information:

  1. The total volume of the mixture (compost and topsoil) is 10 cubic yards.
  2. The total cost of the purchase is $180.
  3. Compost costs $25 per cubic yard.
  4. Topsoil costs $15 per cubic yard.

Our objective is to find the number of cubic yards of topsoil Helena purchased. To achieve this, we will use a systematic approach involving algebraic equations. By translating the given information into mathematical expressions, we can then apply algebraic techniques to solve for the unknown variable, which represents the quantity of topsoil.

Defining Variables

Before we set up the equations, it’s crucial to define our variables clearly. Let’s use the following:

  • Let x represent the number of cubic yards of compost Helena purchased.
  • Let y represent the number of cubic yards of topsoil Helena purchased.

By defining these variables, we create a foundation for constructing our equations and moving towards a solution. Clear variable definitions are paramount in mathematical problem-solving, as they provide a structured framework for the subsequent steps.

Setting Up the Equations

Now that we have defined our variables, we can translate the given information into mathematical equations. This is a critical step in solving the problem, as the equations will serve as the foundation for our algebraic manipulations. We have two primary pieces of information that can be converted into equations:

  1. The total volume of the mixture:

The total volume of compost and topsoil is 10 cubic yards. This can be represented by the equation:

x + y = 10

This equation states that the sum of the cubic yards of compost (x) and the cubic yards of topsoil (y) equals 10.

  1. The total cost of the purchase:

The total cost of the compost and topsoil is $180. We know the cost per cubic yard for each material, so we can create an equation representing the total cost:

25x + 15y = 180

This equation signifies that the cost of compost ($25 per cubic yard times x cubic yards) plus the cost of topsoil ($15 per cubic yard times y cubic yards) equals the total cost of $180.

With these two equations, we have a system of linear equations that we can solve to find the values of x and y. The next step involves choosing an appropriate method to solve this system.

Solving the System of Equations

We have established the following system of equations:

  1. x + y = 10
  2. 25x + 15y = 180

There are several methods to solve a system of linear equations, including substitution, elimination, and graphical methods. For this problem, we will use the substitution method, as it is a straightforward approach for this particular system. The substitution method involves solving one equation for one variable and then substituting that expression into the other equation.

Step 1: Solve the first equation for x

From the first equation, x + y = 10, we can isolate x by subtracting y from both sides:

x = 10 - y

This gives us an expression for x in terms of y. Now we can substitute this expression into the second equation.

Step 2: Substitute the expression for x into the second equation

Substitute x = 10 - y into the second equation, 25x + 15y = 180:

25(10 - y) + 15y = 180

This substitution eliminates x from the second equation, leaving us with an equation in terms of y only.

Step 3: Simplify and solve for y

Expand and simplify the equation:

250 - 25y + 15y = 180

Combine like terms:

250 - 10y = 180

Subtract 250 from both sides:

-10y = 180 - 250

-10y = -70

Divide both sides by -10:

y = 7

Thus, we have found that y = 7, which represents the number of cubic yards of topsoil Helena purchased.

Step 4: Substitute the value of y back into the equation for x

Now that we know y = 7, we can substitute this value back into the equation x = 10 - y to find the value of x:

x = 10 - 7

x = 3

This tells us that Helena purchased 3 cubic yards of compost.

Solution and Interpretation

We have successfully solved the system of equations and found the values of x and y:

  • x = 3 cubic yards of compost
  • y = 7 cubic yards of topsoil

The question asked for the amount of topsoil Helena purchased, which we found to be 7 cubic yards. Therefore, Helena purchased 7 cubic yards of topsoil for her garden mix. This solution aligns with the problem's constraints: the total volume is 10 cubic yards (3 cubic yards of compost + 7 cubic yards of topsoil), and the total cost is $180 (25 * 3 + 15 * 7 = 75 + 105 = 180).

Interpretation

Understanding the solution in the context of the problem is crucial. Helena needed a specific blend of compost and topsoil for her garden. By purchasing 7 cubic yards of topsoil and 3 cubic yards of compost, she met her volume requirement of 10 cubic yards while staying within her budget of $180. This problem exemplifies how mathematical problem-solving can be applied to practical situations, such as gardening and landscaping.

Conclusion

In conclusion, by applying algebraic techniques to the information provided, we determined that Helena purchased 7 cubic yards of topsoil. This problem demonstrated the use of a system of linear equations to solve a real-world scenario involving resource allocation and budgeting. The steps involved defining variables, setting up equations, solving the system of equations, and interpreting the solution in the context of the problem.

This methodical approach to problem-solving can be applied to various other situations, highlighting the practical relevance of mathematical skills. Whether it's calculating costs, determining quantities, or optimizing resources, the ability to translate real-world problems into mathematical models and solve them is an invaluable asset.

Through this exercise, we have not only found the answer to a specific question but also reinforced the broader applicability of mathematical thinking in everyday life. The clarity and precision required in solving such problems underscore the importance of a structured approach, ensuring accuracy and understanding. As we continue to explore and apply these skills, we become more adept at navigating the quantitative aspects of our world, making informed decisions, and achieving our goals effectively.