Problem 65 - College Algebra

Question

The top three countries in oil consumption in a certain year are as follows: the United States, Japan, and China. In millions of barrels per day, the three top countries consumed \(39.8 \%\) of the world's consumed oil. The United States consumed \(0.7 \%\) more than four times China's consumption. The United States consumed \(5 \%\) more than triple Japan's consumption. What percent of the world oil consumption did the United States, Japan, and China consume? \(^{[27]}\)

Step-by-Step Solution

Answer

The United States, Japan, and China consumed 39.8% of the world's oil.

Step 1: Identify Variables

Let's denote China's oil consumption as \( C \), Japan's oil consumption as \( J \), and the United States' oil consumption as \( U \).

Step 2: Relate US and China's Consumption

According to the problem, the United States consumed \(0.7\%\) more than four times China's consumption. Therefore, this can be expressed as \( U = 4C + 0.7\% \).

Step 3: Relate US and Japan's Consumption

Similarly, the United States consumed \(5\%\) more than triple of Japan's consumption, which can be expressed as \( U = 3J + 5\% \).

Step 4: Express Total Consumption

The total consumption by the United States, Japan, and China is given as \(39.8\%\) of the world's oil. Therefore, we have \( U + J + C = 39.8\% \).

Step 5: Solve for US Consumption

By using \( U = 4C + 0.7\% \) and \( U = 3J + 5\% \), equate the two expressions: \( 4C + 0.7\% = 3J + 5\% \). Simplifying, we get \( 4C - 3J = 4.3\% \).

Step 6: Express Japan and China in terms of US

From the above equation \(4C = 3J + 4.3\%\), and substituting back into \( U = 3J + 5\% \), relate \( C \) and \( J \) as \( C = \frac{J + 1.43}{4} \).

Step 7: Use Total Percentage Equation

Substitute the above relation back into the total equation: \( U + J + C = 39.8\% \). Since \( U = 4C + 0.7\% \), solve for one variable and substitute to find all consumption percentages.

Step 8: Calculate Individual Percentages

Assume \( C = x\% \), then \( U = 4x + 0.7\% \), and solve \( U + J + C = 39.8\% \) to find their values.

Step 9: Conclusion

After finding the values, verify the conditions: \( U = 4C + 0.7\% \), \( U = 3J + 5\% \), and \( U + J + C = 39.8\% \) to ensure solution consistency.

Key Concepts

Systems of equations

When solving algebra problems, especially those involving percentages and multiple conditions, systems of equations become essential. A system of equations is a set of equations with multiple variables. Our task is to find a solution that satisfies all conditions of the problem simultaneously.
Each equation in the system represents a constraint or a relationship between the variables. In the given exercise, we have three main equations:

• The United States' consumption in terms of China's: \( U = 4C + 0.7\% \).
• The United States' consumption in terms of Japan's: \( U = 3J + 5\% \).
• The total consumption of all three countries: \( U + J + C = 39.8\% \).

These equations together form our system of equations. By solving them, we seek values for the consumptions that meet all the given conditions.

Variable identification

In algebra problems, especially those dealing with systems, correctly identifying variables is crucial. Variables represent unknown quantities that, once solved, will give the solution to the problem.
In this example, we identified the following variables:

• \( C \): China's oil consumption.
• \( J \): Japan's oil consumption.
• \( U \): The United States' oil consumption.

Naming these variables early on helps in expressing complex conditions more simply, such as in the equations relating the consumptions of one country to another. It also allows for a structured approach in substituting and solving within a system of equations.

Problem-solving steps

Effective problem-solving in algebra follows a logical sequence. Let's break down the steps demonstrated in our exercise:

• Step 1: Identify and define the variables. We started by naming our unknowns as \( C \), \( J \), and \( U \).
• Step 2: Translate the problem conditions into equations. The statements in the problem become mathematical expressions.
• Step 3: Setup and simplify the system of equations. We equated and rearranged the equations to isolate variables where possible.
• Step 4: Solve the system by substitution or other methods. Input the variables in different forms to achieve simplification aimed at finding their values.
• Step 5: Verify solutions ensure they satisfy all original conditions. This checks if the relationships hold mathematically.

Following these steps helps avoid confusion and guides to a solution that meets all given criteria.

Percentage calculations

When an exercise includes percentages, it's important to understand how to translate these into equations. This involves expressing percentages as decimals and integrating them into expressions.
For instance, if the United States consumes 0.7% more than four times China's consumption, it becomes:

• \( U = 4C + 0.007 \). This is an example of converting percentage directly into a decimal for equation formulation.

Likewise, when total consumption equals 39.8% of the world's oil, it becomes part of the total equation:

• \( U + J + C = 0.398 \).

Calculating percentages in equations involves precise interpretations and conversions to ensure the equations accurately reflect the problem's conditions.