Problem 79 - Prealgebra

Question

Crude Inventory. US commercial crude oil inventories decreased by \(3.8\) million barrels in the week ending June \(19 .\) If there were \(353.9\) million barrels the following week, what were crude oil inventories before the decline? rttnews.com \(06 / 24 / 09\)

Step-by-Step Solution

Answer

The crude oil inventories before the decline were 357.7 million barrels.

Step 1: Understand the Problem

We need to find the initial number of crude oil barrels before the decline. We know that the inventory decreased and that the inventory at the end of the week is given.

Step 2: Represent the Problem Mathematically

Let \( x \) represent the initial number of barrels before the decrease. We know that the barrels decreased by 3.8 million, so the equation is: \[ x - 3.8 = 353.9 \]

Step 3: Solve the Equation

We need to solve for \( x \). To do that, add 3.8 to both sides of the equation to isolate \( x \): \[ x = 353.9 + 3.8 \]

Step 4: Calculate the Initial Inventory

Calculate the sum: \[ x = 353.9 + 3.8 = 357.7 \] Hence, the initial number of barrels was 357.7 million.

Key Concepts

Problem Solving

In the realm of algebra, problem-solving is akin to assembling a jigsaw puzzle where each piece represents a part of the story. In our exercise, we are tasked with unveiling how many crude oil barrels existed before a known reduction. Here’s how we break it down effectively:

• First, we identify the goal, which is to find how many barrels were initially stocked.
• We gather all given facts, such as the decrease in barrels and the resulting quantity.
• Construct a plan or method to reach our answer, starting with representing the scenario mathematically.

Analyzing and comprehending the problem thoroughly are key to finding a clear path to the solution. Think critically about each given piece of information and how it relates to what's being asked.

This mode of critical thinking and strategic planning is a valuable skill in algebra, enhancing your ability to tackle problems efficiently and accurately.

Equation Solving

Solving equations is a foundational element in algebra. It begins with understanding the structure of an equation and the variables involved. Our given problem uses an equation to relate the initial stock of barrels to the reduction and the stock after the decline.

We establish the equation: - Representing the initial stock by a variable, say \( x \), and understanding what needs to be isolated- Translating words into mathematical terms, such as realizing the reduction translates to subtraction: \[ x - 3.8 = 353.9 \] To solve the equation, we focus on isolating the variable \( x \), usually achieved by performing inverse operations:

- For our equation: Adding 3.8 to both sides helps cancel the subtraction effect and isolates \( x \) - Then, simplify to find \( x \), the initial quantity of barrels.
This process teaches the art of maintaining a balance, ensuring that operations are equal on both sides to derive the correct solution.

Mathematical Representation

Mathematical representation is a critical skill in algebra, allowing real-world problems to be translated into equations and expressions. In our exercise, the decline in crude oil stock is embodied in the equation \( x - 3.8 = 353.9 \). Here's why representation is so important:

• It simplifies complex scenarios, using symbols and operations to clarify the relationships between quantities.
• By assigning variables like \( x \), it becomes easier to manipulate and solve for unknowns.

The key steps in forming a mathematical representation include understanding the operations involved (such as addition or subtraction) and accurately depicting these within the equation. The variable \( x \) acts as a placeholder for the initial number of barrels, setting the stage for subsequent problem-solving and equation-solving steps.

This powerful tool not only aids in solving algebra problems but also equips you to analyze and engage with mathematical models across various real-world contexts.