Doubling and Halving
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The Concept of Doubling and Halving in Real-Life Scenarios
Doubling and halving computations are applicable in real-life situations. Chessboard versus wheat grains is a typical case involving halving and doubling. Perhaps, mathematicians apply geometric progression concepts to determine grains in a given area of the chessboard. Radioactivity is another phenomenon involving halving and doubling. Therefore, halving and doubling are concepts often used in chessboard versus wheat grains and radioactivity.
Starters
1. Chessboard Versus Wheat Grains: Case 1
Table 1: Chessboard Square Number Versus Grains.
| Square No | Grains in that Square No. | Total Grains |
| 1 | 1 | 1 |
| 2 | 2 | 3 |
| 3 | 4 | 7 |
| 20 | ? | ? |
Grains in square number, as in Table 1, are a geometrical progression with a common ratio (i.e., r=2) and the first term (i.e., a=1). To find the 
-term, we use the expression 
. Having an interest in the total grains, the 
and 
terms are established and added, as illustrated in Exhibit 1.
2. Ebola Cases
Ebola cases form a geometric progression with the common ration (i.e., r=2) and the first term (i.e., a=3,300). Assuming the cases as of 
October 2014 is the baseline, it will only change after 21 days. Between October 2014 and 
December 2014, there are 57 days. Dividing 57days by 21 days, we obtain 2. Therefore, as of December 2014, Ebola cases remain at the second term of the geometric series (i.e., 66,000 Ebola cases), as shown in Exhibit 2. Similarly, the number of days between October 2014 and of October 2015 is 422 days, culminating in of term (i.e., 
) when divided by 21 days.
3. Wealth-Based Calculation
Using the conversion calculator, 1¢=0.01 dollars. Hence, 100¢=$1. The geometric progression equation can be used to determine the position of the term at which the doubling investment (i.e., r=4 and 2) will be equal. For the first progression a=$1, while the second series has its first term a, as $500. The position of the term is executed in Exhibit 3. From the calculation, 
. Arguably, case 1 involving a daily doubling of wealth is the best since there is an ease in accumulating such less amount which will exceed the large amount accumulated within any other day in the long run.
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Group Work
4. Radioactivity
Let N and 
to represent initial and final masses, respectively. Given the material has a half-life( 
) and duration (t=9000 years).
5. Diaper
Let N (i.e., 0.001) and 
to represent initial and final amounts, respectively. Given the material has a half-life( 
). Starting from 0.001, the doubling concept is applied till 1 is obtained as illustrated in Table 1.
Table 1: Application of Doubling.
| 1. | 0.001 |
| 2. | 0.002 |
| 3. | 0.004 |
| 4. | 0.008 |
| 5. | 0.016 |
| 6. | 0.032 |
| 7. | 0.064 |
| 8. | 0.128 |
| 9. | 0.256 |
| 11. | 0.512 |
| 12. | 1.024 |
6. Marijuana Users
Assume N and 
to represent initial and final masses, respectively. And the material has a half-life( 
) and duration (
).
7. Wheat Grains Versus Chessboard: Case 2
Using a=1, r=2, and the geometric progression for determining the -term in question 1 (i.e., -term= ), the 
term is calculated as follows.
8. Bottle Versus Bacteria
Using 11.00 am as the baseline time, by 11.01, the bacteria shall have divided in two. Considering a bottle as one whole unit, the bacteria undergo 50 divisions as of 11.50 am, and spread sheet is applied to half the 1 unit 50 times, yielding 1.78E-15.
In conclusion, the concept of doubling and halving has been applied in this context to calculate the number of grains and the total number of grains. These two concepts are further used in determining Ebola cases and final units of radioactive elements.