Problem 40 - Algebra 2

Question

INCOME In \(2003,\) the population of Texas was about \(2.21 \times 10^{7}\) . The personal income for the state that year was about \(6.43 \times 10^{11}\) dollars. What was the average personal income?

Step-by-Step Solution

Answer

The average personal income was approximately \(2.91 \times 10^{4}\) dollars.

Step 1: Understand the Given Information

We are provided with two key pieces of information: the population of Texas in 2003, which is given as \(2.21 \times 10^{7}\), and the total personal income for Texas in that year, which is \(6.43 \times 10^{11}\) dollars. We need to find the average personal income per person.

Step 2: Set up the Division Problem

To find the average personal income per person, we need to divide the total personal income by the population. This gives us the expression \(\frac{6.43 \times 10^{11}}{2.21 \times 10^{7}}\).

Step 3: Divide the Numerical Parts

First, divide the numerical parts of the numbers: \(\frac{6.43}{2.21} \approx 2.91\).

Step 4: Divide the Powers of Ten

Next, divide the powers of ten by subtracting the exponents: \(\frac{10^{11}}{10^{7}} = 10^{11-7} = 10^{4}\).

Step 5: Calculate the Final Result

Combine the results from the previous steps: \(2.91 \times 10^{4}\). This means the average personal income is approximately \(2.91 \times 10^{4}\) dollars.

Key Concepts

Scientific Notation

Scientific notation is a convenient way to express very large or small numbers using powers of ten. It simplifies numbers by focusing on significant digits (the important part of a number). For example, instead of writing a large number like 64,300,000,000, we use scientific notation and write it as \(6.43 \times 10^{11}\). This makes it easier to do math with these numbers without dealing with a long string of digits every time.
To write a number in scientific notation:

• Identify the significant digits. For 64,300,000,000, these are 6.43.
• Determine how many places the decimal point has moved from the right of the first digit. In this case, it moves 11 places to the left.
• Express it in the form \(a \times 10^n\), where \(a\) is the significant digits, and \(n\) is the number of decimal places moved.

Using scientific notation helps keep calculations more manageable, especially when dealing with very large populations or figures like personal income, as in our exercise.

Division of Powers

When working with scientific notation, we'll often need to divide numbers that include powers of ten. This concept is known as division of powers. Dividing powers is a straightforward process where you divide the coefficients (the numbers) first and then handle the powers.
Here's how it works:

• First, divide the coefficients. For instance, using our exercise example \(\frac{6.43}{2.21} \approx 2.91\).
• Next, divide the powers of ten by subtracting the exponents. With \(\frac{10^{11}}{10^{7}}\), you perform the operation \(11 - 7\) to get \(10^{4}\).
• Finally, combine the two results: \(2.91 \times 10^{4}\).

By understanding division of powers, you can efficiently handle calculations involving large and small numbers, ensuring you end up with simplified results.

Average Calculation

The average calculation is a basic mathematical operation used to find the mean value of a set of numbers. In the context of our exercise, it helps determine the average personal income per person.
Average calculations involve:

• Summing all the values of interest. In this case, it's the total personal income, \(6.43 \times 10^{11}\) dollars.
• Counting how many values there are. This is represented as the population number, \(2.21 \times 10^{7}\) people.
• Dividing the total value by the number of values: \(\frac{6.43 \times 10^{11}}{2.21 \times 10^{7}}\).

The result \(2.91 \times 10^{4}\) dollars gives us the average income per person, making the enormous data manageable and comprehensible. Understanding average calculations is vital, as it frequently appears in real-life scenarios, such as budgeting, statistics, and policy planning.