Problem 67 - College Algebra

Question

The top three sources of oil imports for the United States in the same year were Saudi Arabia, Mexico, and Canada. The three top countries accounted for \(47 \%\) of oil imports. The United States imported \(1.8 \%\) more from Saudi Arabia than they did from Mexico, and \(1.7 \%\) more from Saudi Arabia than they did from Canada. What percent of the United States oil imports were from these three countries?

Step-by-Step Solution

Answer

Mexico: 14.5%, Saudi Arabia: 16.3%, Canada: 16.2%

Step 1: Set Up Variables

Let's assign variables to each country's oil import percentage. Let \( M \) represent the percentage of oil imports from Mexico. Then, from the given information, the oil import percentage from Saudi Arabia \( S \) is \( M + 1.8\% \), and from Canada \( C \) is \( M + 1.6\% \). We know the sum of these percentages is \( 47\% \).

Step 2: Write the Equation

The equation for the total percentage of oil imports from the three countries is:\[M + (M + 1.8\%) + (M + 1.7\%) = 47\%\]

Step 3: Solve for M

Now simplify and solve the equation:\[M + M + 1.8\% + M + 1.7\% = 47\%\]Combine like terms:\[3M + 3.5\% = 47\%\]Subtract \(3.5\%\) from both sides:\[3M = 43.5\%\]Divide both sides by 3:\[M = 14.5\%\]

Step 4: Find Percentages for Each Country

Now substitute \( M = 14.5\% \) back to find \( S \) and \( C \):- Saudi Arabia: \( S = M + 1.8\% = 14.5\% + 1.8\% = 16.3\% \)- Canada: \( C = M + 1.7\% = 14.5\% + 1.7\% = 16.2\% \)

Key Concepts

Percentage Calculation

When it comes to percentages, understanding how to calculate them is a key skill in algebra problem-solving. Percentages represent parts per hundred and often describe proportions, like in this oil import problem. In the exercise provided, we're dealing with percentage calculations to understand contributions from different countries to the total oil imports.

To add or compare percentages, just like regular numbers you perform arithmetic operations. For example:

• Adding: To find the total percentage of oil imports from all three countries, we add Mexico's percentage with the increments from Saudi Arabia and Canada.
• Subtracting: In the given problem, there's a step where percentages are subtracted to isolate the variable 'M'.

The basic arithmetic with percentages helps streamline the process of equation solving. It ensures that each country's contribution is accurately reflected in the overall calculation.

Variable Assignment

Variable assignment is a fundamental aspect of solving algebraic problems. In our oil import problem, it's crucial to define variables to make sense of the information.

We start by assigning a variable:

• M stands for the percentage of oil imports from Mexico, the baseline value.

Using the relationships provided:

• For Saudi Arabia, we derive the formula \( S = M + 1.8\% \).
• For Canada, we use \( C = M + 1.7\% \).

Assigning variables systematically simplifies tracking and manipulating algebraic expressions. It helps not only to organize the information but also makes solving equations more manageable.

Equation Solving

Solving equations is the heart of algebra problem solving. Once variables are set up to represent the different oil percentages, they are combined into an equation that represents the total percent from the three countries. The steps are as follows:

1. **Set Up the Equation**: Using available data, we establish the equation:
\[M + (M + 1.8\%) + (M + 1.7\%) = 47\%\]
2. **Combine Like Terms**: Simplify it to:
\[3M + 3.5\% = 47\%\]
3. **Isolate the Variable**: Subtract 3.5% from both sides, leaving:
\[3M = 43.5\%\]
4. **Solve for the Variable**: Divide by 3 to find:
\[M = 14.5\%\]
Equations allow us to represent relationships between known and unknown values. Solving these equations involves isolating the variable to find its value, ensuring a precise solution to any problem.