Problem 116 - College Algebra

Question

a. In 2009, the United States government collected $2.20 trillion in taxes. Express this number in scientifi c notation. b. In 2009, the population of the United States was approximately 308 million. Express this number in scientifi c notation. c. Use your scientifi c notation answers from parts (a) and (b) to answer this question: If the total 2009 tax collections were evenly divided among all Americans, how much would each citizen pay? Express the answer in decimal notation, rounded to the nearest dollar.

Step-by-Step Solution

Answer

Each American citizen would pay approximately $7,143.

Step 1: Convert Tax Collection into Scientific Notation

The United States government collected $2.20 trillion dollars in taxes. In scientific notation, this would be written as $2.2 * 10^{12}$, because trillion means \(10^{12}\) or 1,000,000,000,000.

Step 2: Convert Population into Scientific Notation

The population of United States was approximately 308 million. The number 308 million in scientific notation would be \(3.08 * 10^8\), because million means \(10^{6}\) or 1,000,000.

Step 3: Calculate Tax Per Citizen

Tax collections divided evenly among all American citizens would be given by dividing the amount collected by the number of citizens. In other words, \(2.2 * 10^{12}\) divided by \(3.08 * 10^8\). This simplifies to \(0.7143 * 10^4 = 7,143\) (rounded to the nearest dollar)

Key Concepts

Expressing Numbers in Scientific Notation

When you're dealing with very large or very small numbers, scientific notation offers a simplified method to express them. Here's how to put a number into scientific notation: Identify the most significant digit of the number, move the decimal point just to the right of this digit, count how many places you moved the decimal point to determine the exponent of 10, and then multiply this 'base' by 10 raised to the appropriate power.

For instance, the U.S. tax collection number, \(2.20 trillion, or \)2,200,000,000,000 in regular form, becomes more manageable when we express it as \(2.2 \times 10^{12}\). To convert this, we find the first non-zero digit (2), place the decimal point immediately after it, and then count the digits to the right to determine the exponent of the power of 10. The exponent in this case is 12 because we jump 12 digits from the first 2 to the end of the trillion figure.

Division with Scientific Notation

Dividing numbers in scientific notation may seem daunting, but it's a matter of simplifying the process into two main steps: dividing the decimal parts (the coefficients) and then dividing the powers of ten. To divide powers of ten, you subtract the exponent of the divisor from the exponent of the dividend.

Let's take the tax division example. Here we had \(2.2 \times 10^{12}\) dollars and we needed to divide it by 308 million people, represented as \(3.08 \times 10^8\) in scientific notation. The division of the coefficients (2.2/3.08) gives us approximately 0.7143. Then, subtract the exponents of 10 (12 - 8 = 4). The result is the coefficient (0.7143) multiplied by \(10^4\), or \(0.7143 \times 10^4\), which gives each person's tax share.

Decimal Notation

After the division problem is solved in scientific notation, it's often necessary to express the result in decimal notation, especially if the number has real-world meaning, like currency. Decimal notation is simply writing out the number in full, using a decimal point where necessary. This is more commonly the number format we use in everyday life.

To finish our example, the individual tax share calculated earlier as \(0.7143 \times 10^4\) can be expressed in decimal notation. To do this, simply move the decimal point 4 places to the right (as indicated by the \(10^4\)), giving us 7143. It's now clear and understandable that each citizen would be contributing $7,143 in taxes, a figure that's directly usable and can be easily related to without the complexities of scientific notation.