Problem 66 - College Algebra

Question

The top three countries in oil production in the same year are Saudi Arabia, the United States, and Russia. In millions of barrels per day, the top three countries produced \(31.4 \%\) of the world's produced oil. Saudi Arabia and the United States combined for \(22.1 \%\) of the world's production, and Saudi Arabia produced \(2 \%\) more oil than Russia. What percent of the world oil production did Saudi Arabia, the United States, and Russia produce?

Step-by-Step Solution

Answer

Saudi Arabia, the United States, and Russia produced 31.4% of the world's oil.

Step 1: Define Variables

Let \( S \), \( U \), and \( R \) represent the percent of the world's oil production for Saudi Arabia, the United States, and Russia, respectively.

Step 2: Write Equations Based on Given Information

We know according to the problem:1. \( S + U + R = 31.4 \) (Equation representing total oil production by the three countries).2. \( S + U = 22.1 \) (Combined production of Saudi Arabia and the United States).3. \( S = R + 2 \) (Saudi Arabia produced 2% more than Russia).

Step 3: Substitute and Reduce

Substitute equation 3 into equation 1:\( (R + 2) + U + R = 31.4 \).Simplify to:\( 2R + U + 2 = 31.4 \).

Step 4: Use Known Production of Saudi Arabia and US

From equation 2:\( S + U = 22.1 \),Substitute \( S = R + 2 \):\( (R + 2) + U = 22.1 \)Simplify to:\( U = 20.1 - R \).

Step 5: Substitute Back to Solve for R

Substitute \( U = 20.1 - R \) into the modified equation from Step 3:\( 2R + (20.1 - R) + 2 = 31.4 \)Simplify to:\( R + 22.1 = 31.4 \).

Step 6: Solve for R

Solving \( R + 22.1 = 31.4 \):\( R = 31.4 - 22.1 \),\( R = 9.3 \).

Step 7: Solve for S and U

Now solve for \( S \):\( S = R + 2 = 9.3 + 2 = 11.3 \).Solve for \( U \):\( U = 20.1 - R = 20.1 - 9.3 = 10.8 \).

Step 8: Confirm the Total

Check the sum:\( S + U + R = 11.3 + 10.8 + 9.3 = 31.4 \).The total matches the given initial information, confirming our calculations are correct.

Key Concepts

Algebraic Equations

Algebraic equations form the backbone of solving numerical problems involving unknowns, like the oil production percentages of countries. In algebra, equations are mathematical statements that assert the equality of two expressions. They typically include variables (such as \(S\), \(U\), and \(R\) in our example) which represent unknown quantities. You need to manipulate these variables and expressions to solve the equations. The key is to isolate variables and replace terms to simplify calculations. In the given exercise, three equations were established based on the production data:

• \( S + U + R = 31.4 \): total oil production by the three countries.
• \( S + U = 22.1 \): combined production of Saudi Arabia and the United States.
• \( S = R + 2 \): showing that Saudi Arabia produced 2% more than Russia.

By solving these equations, one can uncover the oil production percentages for each country.

System of Equations

A system of equations involves multiple equations with multiple variables that are solved simultaneously. This exercise is a challenge involving a system of algebraic equations. By having three equations and three variables, our task is to find the values that satisfy all the given conditions at the same time.
In the given problem:

• Equation 1 gives the total oil production percentage for the three countries.
• Equation 2 relates the combined production values of Saudi Arabia and the United States.
• Equation 3 compares the production of Saudi Arabia to Russia.

To solve the system, we perform operations such as substitution and elimination. With these techniques, we reduce the complexity by tackling one variable at a time until all unknowns are discovered. This process exemplifies the beauty and utility of using algebra to solve real-world problems.

Problem Solving Steps

When tackling a problem involving a system of equations, it is essential to break it down into manageable steps. This technique not only simplifies the problem-solving process but also improves accuracy.
Let's break down the solution:
1. **Define Variables**: Establish variables to represent unknown values. Here, \(S\), \(U\), and \(R\) are assigned for the oil production percentages of Saudi Arabia, the United States, and Russia, respectively.
2. **Formulate Equations**: Develop equations based on the problem statement. These expressions represent relationships between variables:

• The sum of the percentages equals 31.4%.
• The combined percentage for two countries is 22.1%.
• Saudi Arabia's production is 2% more than Russia's.

3. **Substitute and Solve**: Use substitution to simplify and solve the system. This involves replacing one equation’s variable with another’s to make one variable explicit.4. **Verify Solution**: Always substitute back to verify that all original conditions are satisfied. Checking the sum ensures that the calculated percentages agree with the problem constraints.
By following these methodical steps, tackling even complex algebra problems becomes systematic and simple.