Chapter 7.4, Problem 82PPE

(a).

To determine

To find: The maoney owed by the government per person in $2000$ rounded to nearest dollar.

Answer to Problem 82PPE

The money owed by the government per person in $2000$ is $\$1.99\times {10}^4$ .

Explanation of Solution

Given information: The population of the United States is $282.4\text{million}$ and the government owed $\$5.63\text{trillion}$ money by creditors.

Calculation:

The government owed total amount $\$5.63\text{trillion}$ and the population of the United States is $282.4\text{million}$ .

The value of trillion and million on base ten terms is:

  $\begin{array}{l}1\text{million}={10}^6\\ 1\text{trillion}={10}^{12}\end{array}$

The money that the government owed from a person is:

  $\text{money}\text{owed}\text{per}\text{person}=\frac{5.63\text{trillion}}{282.4\text{million}}$

Now,

  $\begin{array}{c}\text{money}\text{owed}\text{per}\text{person}=\frac{5.63\times {10}^{12}}{282.4\times {10}^6}\\ =0.01994\times \frac{{10}^{12}}{{10}^6}\\ =\frac{1.994}{100}\times \frac{{10}^{12}}{{10}^6}\\ =\frac{1.994}{{10}^2}\times \frac{{10}^{12}}{{10}^6}\end{array}$

Apply the property of division of powers with same base in the above expression.

  $\begin{array}{c}\text{money}\text{owed}\text{per}\text{person}=1.994\times {10}^{12-6-2}\\ \approx 1.99\times {10}^4\end{array}$

Therefore, the money owed by the government per person in $2000$ is $\$1.99\times {10}^4$ .

(b).

To determine

To find: The money owed by the government per person in $2005$ rounded to nearest dollar.

Answer to Problem 82PPE

The money owed by the government per person in $2005$ is $\$2.66\times {10}^4$ .

Explanation of Solution

Given information: The population of the United States in $2005$ is $296.9\text{million}$ and the money owed by the government is $\$7.91\text{trillion}$ .

Calculation:

The value of trillion and million on base ten terms is:

  $\begin{array}{l}1\text{million}={10}^6\\ 1\text{trillion}={10}^{12}\end{array}$

The money that the government owed from a person is:

  $\text{money}\text{owed}\text{per}\text{person}=\frac{\$7.91\text{trillion}}{296.9\text{million}}$

Substitute ${10}^{12}$ for trillion and ${10}^6$ for million in the above expression.

  $\begin{array}{c}\text{money}\text{owed}\text{per}\text{person}=\frac{7.91\times {10}^{12}}{296.9\times {10}^6}\\ \approx 0.02664\times \frac{{10}^{12}}{{10}^6}\\ \approx \frac{2.66}{100}\times \frac{{10}^{12}}{{10}^6}\\ \approx \frac{2.66}{{10}^2}\times \frac{{10}^{12}}{{10}^6}\end{array}$

Apply the property of division of powers with same base in the above expression.

  $\begin{array}{c}\text{money}\text{owed}\text{per}\text{person}\approx \text{2}\text{.66}\times {10}^{12-6-2}\\ \approx 2.66\times {10}^4\end{array}$

Therefore, the money owed by the government per person in $2005$ is $\$2.66\times {10}^4$ .

(c).

To determine

To find: The increase in percent of the average amount owed per person from the year $2000$ to $2005$ .

Answer to Problem 82PPE

The increase in percent of the average amount owed per person from the year $2000$ to $2005$ is $34\%$ .

Explanation of Solution

Given information: The average amount owed per person from 2000 to 2005.

Calculation:

As calculated in part (a) and part (b), the price owed by the government per person in $2000$ is $\$1.99\times {10}^4$ and in $2005$ is $\$2.66\times {10}^4$ .

The increase in price from the year $2000$ to $2005$ is:

  $\$2.66\times {10}^4-\$1.99\times {10}^4=\$0.67\times {10}^4$

The formula to determine percent increase is:

  $\%\text{increase}=\frac{\text{increase}\text{in}\text{price}}{\text{original}}\times 100\%$

Substitute $\$0.67\times {10}^4$ for increase in price and $\$1.99\times {10}^4$ for original in the above expression.

  $\begin{array}{c}\%\text{increase}=\frac{\$0.67\times {10}^4}{\$1.99\times {10}^4}\times 100\%\\ \approx 0.3366\times 100\%\\ \approx 33.6\%\\ \approx 34\%\end{array}$

Therefore, the increase in percent of the average amount owed per person from the year $2000$ to $2005$ is $34\%$ .