Chapter 8, Problem 9TPS

Sub part (a):

To determine

How many great ideas will occur in the world of 1800.

Answer to Problem 9TPS

300 ideas when population is 300 million.

6,000 ideas when population is 6 billion.

Explanation of Solution

The relation between the population size and the number of great ideas is given. It is that only 1 idea out of every one million. Here, the total population is given as 300 million. Thus, by dividing total population with one million, the number of great ideas can be calculated as follows:

$\begin{array}{c}\text{Number of great ideas}=\frac{\text{Total population}}{\text{1,000,000}}\\ =\frac{300,000,000}{1,000,000}\\ =300\end{array}$

Thus, when the population is 300 million, the number of great ideas is 300. When the population is 6 billion, the same can be calculated as follows:

$\begin{array}{c}\text{Number of great ideas}=\frac{\text{Total population}}{\text{1,000,000}}\\ =\frac{6,000,000,000}{1,000,000}\\ =6,000\end{array}$

 Therefore, when the population is 6 billion, the number of great ideas is 6,000.

Sub part (b):

To determine

How many great ideas will occur in the world of 1800 and today.

Answer to Problem 9TPS

60 ideas in 1800 and 1,200 great ideas in this year.

Explanation of Solution

The relation between the population size and the number of great ideas is given. It is that only 1 idea out of every one million. Here, the total population is given as 300 million. However, only 20 percent of the population is able to invent new ideas. Thus, by dividing total population with one million, the number of great ideas can be calculated as follows:

$\begin{array}{c}\text{Number of great ideas}=\frac{\text{Total population}}{\text{1,000,000}}\\ =\frac{\left(300,000,000\times 0.20\right)}{1,000,000}\\ =\frac{60,000,000}{1,000,000}\\ =60\end{array}$

Thus, in the year 1800, the number of great ideas is 60. When the population is 6 billion, the same can be calculated as follows:

$\begin{array}{c}\text{Number of great ideas}=\frac{\text{Total population}}{\text{1,000,000}}\\ =\frac{\left(6,000,000,000\times 0.20\right)}{1,000,000}\\ =\frac{1,200,000,000}{1,000,000}\\ =1,200\end{array}$

Therefore, when the population is 6 billion (this year), the number of great ideas is 1,200.

Sub part (c):

To determine

How many great ideas will occur in India and China.

Answer to Problem 9TPS

1,000 ideas in each country.

Explanation of Solution

The relation between the population size and the number of great ideas is given. It is that only 1 idea out of every one million. The total population by the two countries is 2 billion, which is approximately 1 billion each. Thus, the number of great ideas from each country can be calculated as follows:

$\begin{array}{c}\text{Number of great ideas}=\frac{\text{Total population}}{\text{1,000,000}}\\ =\frac{1,000,000,000}{1,000,000}\\ =1,000\end{array}$

Therefore, when the population is 1 billion, the number of great ideas is 1,000. Since both countries have equal population, there will be 1,000 ideas from each.

Sub part (d):

To determine

How many great ideas will occur in the world when the population cuts into half this year.

Answer to Problem 9TPS

600 ideas in this year.

Explanation of Solution

The relation between the population size and the number of great ideas is given. It is that only 1 idea out of every one million. When the population is 6 billion, only 20 percent of the population invents new ideas. Thus, the number of great ideas can be calculated as follows:

$\begin{array}{c}\text{Number of great ideas}=\frac{\text{Total population}}{\text{1,000,000}}\\ =\frac{\left(6,000,000,000\times 0.20\right)}{1,000,000}\\ =\frac{1,200,000,000}{1,000,000}\\ =1,200\end{array}$

Therefore, when the population is 6 billion (this year), the number of great ideas is 1,200. When the population becomes half, then the number of ideas invented can be calculated as follows:

$\begin{array}{c}\text{Number of great ideas}=\frac{\text{Total population}}{\text{1,000,000}}\\ =\frac{\left(6,000,000,000\times 0.50\right)\times 0.20}{1,000,000}\\ =\frac{3,000,000,000\times 0.20}{1,000,000}\\ =\frac{600,000,000}{1,000,000}\\ =600\end{array}$

Thus, when population today becomes half of the 6 billion, the number of great ideas becomes 600.