Problem 4 - Elementary Algebra

Question

Garden A produces 5.8 times as many vegetables as garden \(\mathrm{B}\). Together the gardens produce 102 pounds of vegetables. How many pounds of vegetables does garden A produce?

Step-by-Step Solution

Answer

Answer: Garden A produces approximately 87 pounds of vegetables.

Step 1: Set up an equation relating the production of garden A and B

Let x be the number of pounds of vegetables produced by garden B. Since garden A produces 5.8 times as many vegetables as garden B, garden A produces 5.8x. Now we know that the total production (garden A + garden B) is 102 pounds.

Step 2: Write the equation

Using the previous step, we can write the equation: 5.8x + x = 102

Step 3: Solve for x

Now we have an equation with one variable, x. We can solve it to find the number of pounds of vegetables produced by garden B: 6.8x = 102 x = \frac{102}{6.8} x ≈ 15

Step 4: Find the pounds of vegetables produced by garden A

Now we know that garden B produces around 15 pounds of vegetables, and we also know that garden A produces 5.8 times as many vegetables as garden B. Therefore, the number of pounds of vegetables produced by garden A is: Garden A = 5.8x Garden A ≈ 5.8 * 15 Garden A ≈ 87 So, garden A produces approximately 87 pounds of vegetables.

Key Concepts

Problem Solving

Solving word problems can sometimes feel like unraveling a mystery. The key is to break down the problem into smaller, more manageable parts. Let's consider the problem: we have two gardens producing different amounts of vegetables together totaling 102 pounds. To solve it, follow these steps:

• Understand the situation: Recognize that Garden A produces more than Garden B, specifically 5.8 times more.
• Translate words into math: Express the relationship between the two gardens with an equation.
• Solve the equation using algebraic methods to find the quantities.
• Double-check the solution to ensure you've answered the question.

Breaking problems down step by step can make even the most complex-seeming problems easier to solve.

Variables

Variables are symbols or letters that represent unknown numbers, allowing us to write general solutions in mathematics. In this problem, the variable was first assigned to Garden B's production.

Let's see how variables help:

• We used 'x' to represent the number of pounds of vegetables produced by Garden B.
• This means we could express Garden A's production in terms of 'x', as 5.8 times 'x'.
• Using a variable allows us to set up an equation that represents the entire situation, making it solvable with algebraic techniques.

Once we have the variable setup, it becomes a straightforward process of solving the equation to find the value that 'x' represents. The beauty of variables is their ability to simplify and structure solutions to seemingly tough problems.

Linear Equations

Linear equations are mathematical expressions that result in straight-line graphs. They typically have one or two variables with no exponents or complex terms.

In this exercise, we encounter a linear equation:

• The equation given was 5.8x + x = 102, representing the total vegetable production of both gardens.
• This is a straightforward linear equation because it involves a simple relationship between 'x' and total production.
• To solve, combine like terms to get 6.8x = 102, then find x by isolating the variable through division.

Solving linear equations is a fundamental skill in algebra. It provides a direct path to determining unknowns in problems, as shown when we solved for 'x', the amount Garden B produced.