Growth Of The Rat Population
Problem Statement
Two rats, one male and one female, have made their home on an abandoned island. Given ideal conditions can we estimate the number of offspring produced by the rats in one year? The ideal conditions were:
- The original female gives birth to six young on January 1. She produces another litter of six rats every 40 days thereafter as long as she lives.
- Each female rat born on the island will produce her first litter of six young 120 days after her birth. She will produce a new litter of six rate every 40 days thereafter.
- Every litter has three males and three females.
- The rats have no natural enemies on the island and plenty of food. This means no rats die in the first year.
Using these ideal conditions we must find how many rats will live on the island by the following January 1, including the original pair.
Process
This problem, in my opinion, was a problem that just needed some organization. With proper organization, this problem became extremely easy and simple to understand and solve. To organize my data I started by labeling days in a year by 40 day intervals. For example I started labeling by the first day of the year (January 1) then labeled after 40 days (February 10) and so on till I reached the following year which ended at December 27. In each day there were different sections, for example:
Jan 1 = 8 rats (4 males, 4 females)
Babies - 3 males, 3 females
Adults - 1 male, 1 female
Feb 10 = 14 rats (7 males, 7 females)
Babies - 6 males, 6 females
Adults - 1 male, 1 female
........
I followed this set till I arrived at December 27.
Solution
By the end of the year I arrived at 1808 rats. I followed the above steps and also found a little pattern to help me move along faster. First here is the remaining information:
Mar 22 = 20 rats (10 males, 10 females)
Babies - 9 males, 9 females
Adults - 1 male, 1 female
May 1 = 44 rats (22 males, 22 females)
Babies - 18 males, 18 females
Adults - 4 males, 4 females
Jun 10 = 86 rats (43 males, 43 females)
Babies - 36 males, 36 females
Adults - 7 males, 7 females
Aug 29 = 278 rats (139 males, 139 females)
Babies - 117 males, 117 females
Adults - 22 males, 22 females
Oct 8 = 536 rats (268 males, 268 females)
Babies - 225 males, 225 females
Adults - 43 males, 43 females
Nov 17 = 974 rats (487 males, 487 females)
Babies - 414 males, 414 females
Adults - 73 males, 73 females
Dec 27 = 1808 rats (904 males, 904 females)
Babies - 765 males, 765 females
Adults - 139 males, 139 females
To help move along the lines quicker I was given a little trick by a friend to help me out, this trick only helped me starting from May 1, after the 120 days when the first babies start reproducing. So on Jan 1 you have 3 baby females and 4 in total with the mommy. On Mar 22 there were 20 rats in total. So to solve how many rats were there on May 1 you go three intervals back or 120 days back (January 1) and multiply the total number of female (4 female) rats by 6 (the number of baby rats produced by a female) and then add the product to the number of rats 40 days ago (20 rats). On May 1 there were 44 rats on the island. Since each mother produces 3 male and 3 female rats the number of female and male rats will always be equal and even. So I divided the total of rats by 2 to achieve the number of male and female rats. I followed these steps till I reached December 27 to solve my answer.
Self-assessment
The reason I am confident about my answer is because we were given the answer doing our class time. The purpose of this problem of the week was to figure out and understand how to get to the answer as opposed to just solving it and not understanding why or how you got the answer. I am confident that I have a correct method but there is always more than one way or one solution to get you final answer.
Two rats, one male and one female, have made their home on an abandoned island. Given ideal conditions can we estimate the number of offspring produced by the rats in one year? The ideal conditions were:
- The original female gives birth to six young on January 1. She produces another litter of six rats every 40 days thereafter as long as she lives.
- Each female rat born on the island will produce her first litter of six young 120 days after her birth. She will produce a new litter of six rate every 40 days thereafter.
- Every litter has three males and three females.
- The rats have no natural enemies on the island and plenty of food. This means no rats die in the first year.
Using these ideal conditions we must find how many rats will live on the island by the following January 1, including the original pair.
Process
This problem, in my opinion, was a problem that just needed some organization. With proper organization, this problem became extremely easy and simple to understand and solve. To organize my data I started by labeling days in a year by 40 day intervals. For example I started labeling by the first day of the year (January 1) then labeled after 40 days (February 10) and so on till I reached the following year which ended at December 27. In each day there were different sections, for example:
Jan 1 = 8 rats (4 males, 4 females)
Babies - 3 males, 3 females
Adults - 1 male, 1 female
Feb 10 = 14 rats (7 males, 7 females)
Babies - 6 males, 6 females
Adults - 1 male, 1 female
........
I followed this set till I arrived at December 27.
Solution
By the end of the year I arrived at 1808 rats. I followed the above steps and also found a little pattern to help me move along faster. First here is the remaining information:
Mar 22 = 20 rats (10 males, 10 females)
Babies - 9 males, 9 females
Adults - 1 male, 1 female
May 1 = 44 rats (22 males, 22 females)
Babies - 18 males, 18 females
Adults - 4 males, 4 females
Jun 10 = 86 rats (43 males, 43 females)
Babies - 36 males, 36 females
Adults - 7 males, 7 females
Aug 29 = 278 rats (139 males, 139 females)
Babies - 117 males, 117 females
Adults - 22 males, 22 females
Oct 8 = 536 rats (268 males, 268 females)
Babies - 225 males, 225 females
Adults - 43 males, 43 females
Nov 17 = 974 rats (487 males, 487 females)
Babies - 414 males, 414 females
Adults - 73 males, 73 females
Dec 27 = 1808 rats (904 males, 904 females)
Babies - 765 males, 765 females
Adults - 139 males, 139 females
To help move along the lines quicker I was given a little trick by a friend to help me out, this trick only helped me starting from May 1, after the 120 days when the first babies start reproducing. So on Jan 1 you have 3 baby females and 4 in total with the mommy. On Mar 22 there were 20 rats in total. So to solve how many rats were there on May 1 you go three intervals back or 120 days back (January 1) and multiply the total number of female (4 female) rats by 6 (the number of baby rats produced by a female) and then add the product to the number of rats 40 days ago (20 rats). On May 1 there were 44 rats on the island. Since each mother produces 3 male and 3 female rats the number of female and male rats will always be equal and even. So I divided the total of rats by 2 to achieve the number of male and female rats. I followed these steps till I reached December 27 to solve my answer.
Self-assessment
The reason I am confident about my answer is because we were given the answer doing our class time. The purpose of this problem of the week was to figure out and understand how to get to the answer as opposed to just solving it and not understanding why or how you got the answer. I am confident that I have a correct method but there is always more than one way or one solution to get you final answer.