Question

If Avogadro's number of pennies is divided equally among the 321 million men, women, and children in the United States, how many dollars would each receive? How does this compare with the gross domestic product (GDP) of the United States, which was \(\$ 17.419\) trillion in \(2015 ?\) (The GDP is the total market value of the nation's goods and services.)

Short answer

Each person would receive approximately \(1.875 \times 10^{15}\) dollars. The total amount, \(6.018\times10^{23}\) dollars, is significantly larger than the 2015 US GDP of \(17.419 \times 10^{12}\) dollars. The ratio between the total amount and the GDP is approximately \(3.45 \times 10^{10}\), which means the total amount is approximately 34.5 billion times larger than the 2015 US GDP.

Step-by-step solution

Find the number of pennies per person

Divide Avogadro's number of pennies (\(6.022\times10^{23}\)) by the number of people in the United States (321 million): \[ \text{Pennies per person} = \frac{6.022\times10^{23}}{3.21\times10^8} \]

Convert pennies to dollars

Since there are 100 pennies in a dollar, we can convert the number of pennies per person to dollars per person by dividing the above result by 100: \[ \text{Dollars per person} = \frac{\text{Pennies per person}}{100} \]

Calculate the total amount in dollars

Now, to find the total amount in dollars, multiply the dollars per person by the number of people (321 million): \[ \text{Total amount in dollars} = \text{Dollars per person} \times 3.21\times10^8 \]

Compare the total amount with the US GDP in 2015

Finally, compare the total amount calculated in step 3 with the given GDP of the United States in 2015 which is \(17.419 \times 10^{12}\) dollars: \[ \text{Ratio} = \frac{\text{Total amount in dollars}}{\text{GDP of the United States in 2015}} \]

Key concepts

gross domestic product (GDP)

The Gross Domestic Product (GDP) is a significant economic measure that reflects the total market value of all goods and services produced within a country during a specific period, usually a year. In essence, GDP is an economic snapshot that gives an indication of a country's financial health.

• GDP includes all private and public consumption, government outlays, investments, and exports minus imports that occur within a defined territory.
• A higher GDP figure in comparison to previous periods can indicate economic growth.
• The GDP of the United States in 2015, as referenced in the exercise, was $17.419 trillion, showing the vast scale of the US economy.

Understanding GDP is crucial as it influences economic policy decisions and can indicate the prowess of an economy on the global stage. When comparing GDP to monetary distributions like in this exercise, it helps illustrate the magnitude of national wealth and economic capacity.

conversion of units

Conversion of units is an essential mathematical skill used to express quantities in different forms, making comparisons simpler or calculations more compliant with specific needs.

• In the exercise, we converted pennies to dollars because it is more practical to discuss large sums in terms of dollars rather than pennies.
• To convert these units, remember that 100 pennies equate to 1 dollar. So, dividing the number of pennies by 100 gives the equivalent amount in dollars.
• This conversion step is crucial in helping comprehend the scale of values and facilitates easier communication, interpretation, and economic analysis.

Unit conversion helps maintain consistency in data measurement, ensuring there are little to no misunderstandings when comparing or accumulating different datasets.

basic arithmetic operations

Basic arithmetic operations are foundational for solving mathematical problems. These operations include addition, subtraction, multiplication, and division.

• In the exercise, division was used to determine the number of pennies each person would receive, highlighting how division distributes a quantity evenly among individuals.
• Subsequently, multiplication was used to find the total dollar amount when converted from pennies and then distributed over the population.
• These operations are not only mathematical rules but also tools of reasoning critical for practical problem-solving in everyday scenarios.

Mastering these operations is essential as they are building blocks for more complex calculations such as those found in statistics, economics, and other quantitative sciences. It leads to greater economic understanding and numeracy required for analyzing data and interpreting results like comparing with GDP.