Problem 102 - Algebra and Trigonometry

Question

National Debt As of July 2010 , the population of the United States was \(3.070 \times 10^{8},\) and the national debt was \(1.320 \times 10^{13}\) dollars. How much was each person's share of the debt?

Step-by-Step Solution

Answer

Each person's share of the debt was 42,900 dollars.

Step 1: Understanding the Problem

We have the population of the United States as of July 2010, which is given as \(3.070 \times 10^{8}\). The national debt at that time is \(1.320 \times 10^{13}\) dollars. The problem asks us to find each person's share of the national debt, meaning we need to divide the total debt by the total population.

Step 2: Calculating Person's Share

To find each person's share of the debt, divide the total national debt by the total population:\[\text{Person's share} = \frac{1.320 \times 10^{13}}{3.070 \times 10^{8}}\]

Step 3: Simplifying the Division

Simplify the expression by dividing the coefficients and subtracting the exponents in the powers of ten:\[\text{Person's share} = \frac{1.320}{3.070} \times 10^{13-8}\]Calculate the division of the coefficients:\[\frac{1.320}{3.070} \approx 0.429\]Now compute the power of ten:\[10^{13-8} = 10^5\]

Step 4: Final Calculation

Multiply the simplified coefficient by the power of ten:\[0.429 \times 10^5 = 4.29 \times 10^4\]This means each person's share of the debt is \(4.29 \times 10^4\) dollars, or 42,900 dollars.

Key Concepts

Understanding Scientific Notation

When dealing with very large or very small numbers in mathematics, scientific notation is a powerful tool to simplify and work with these numbers. The format of scientific notation is a number between 1 and 10, known as the coefficient, multiplied by a power of ten. This helps to clearly express and compute values without writing out too many zeros.
For instance, the national debt given as \(1.320 \times 10^{13}\) dollars represents a very large number—1.320 followed by 13 zeros. Similarly, the population, \(3.070 \times 10^{8}\), is a large number appropriate for the context, allowing easier calculation and comparison.
In practical scenarios such as calculating the share of national debt per person, scientific notation allows us to handle division efficiently by managing the powers of 10 separately from the coefficients.

Linear Equations in Problem Solving

Linear equations are fundamental in solving numerous everyday problems, including financial calculations. When faced with a problem that requires division, like determining an individual share, envisioning it as a linear equation can simplify complex arithmetic.
In our example, the linear equation takes the form of dividing the total debt by the number of people: \(\text{Share per person} = \frac{\text{Total Debt}}{\text{Population}}\). This division translates into a proportionate distribution of the debt amongst the total population, illustrating a real-world application of linear equations.
By expressing the equation and understanding the role of each element—total debt and population—we bridge mathematical concepts with tangible real-world scenarios. Problem-solving through linear equations helps us break down larger tasks into manageable computations.

Problem Solving in Mathematics

Mathematical problem-solving is a structured approach to finding solutions by understanding the problem, devising a plan, carrying out the plan, and reviewing the solution. Breaking down the initial problem into smaller steps, like calculating each person's debt share, is typical in effective problem-solving.
In this problem, solving involves:

• Understanding the variables: Recognizing the national debt and the population as core components.
• Planning: Using scientific notation to simplify the division.
• Executing: Performing the arithmetic steps, such as dividing coefficients and managing exponents.
• Reviewing: Recalculating if necessary to check the soundness of the solution.

By approaching mathematical problems methodically, we increase accuracy and efficiency, ensuring reliable results. This strategy of problem-solving can be adapted to a broad array of scenarios, reinforcing the power and versatility of mathematics in everyday contexts.