# How can I find the total number of rabbits if they reproduce at a constant rate? [closed]

The problem is as follows:

Roger adopts two female rabbits and two male rabbits. Three months later each female rabbit gives birth three female rabbits and three male rabbits. Assuming that if the same happens every three months. How many rabbits maximum would Roger have after a year from the adoption of his pets?.

The alternatives given in my book are as follows:

$$\begin{array}{ll} 1.&\textrm{486 rabbits}\\ 2.&\textrm{2048 rabbits}\\ 3.&\textrm{512 rabbits}\\ 4.&\textrm{1024 rabbits}\\ \end{array}$$

How exactly can I find the number of rabbits in this situation?. I don't know if the fibonacci sequence applies in this situation. Or can this be solved using a just ordinary equation?. What sort of rationale should be used here?. Since I'm slow learner. Can someone explain me this step by step? please.

• I'm afraid we distinguish quite sharply around here between "mathematical puzzles" and "mathematics problems" -- see the link in the off-topic notice -- and this is definitely the latter. – Gareth McCaughan Mar 17 '20 at 10:13
• Also, I get the strong impression from many of the questions you've posted here that they're problems you've been set by someone else (a teacher?). PSE really isn't meant to be a please-do-my-homework service! – Gareth McCaughan Mar 17 '20 at 10:16
• @GarethMcCaughan I'm aware of the purpose of this stack community. I'm sorry for any impression which may have arised due this. But these are riddles which I have not understood how to solve them and not a homework. Regarding the mathematical problems I will find other places to ask. – Chris Steinbeck Bell Mar 17 '20 at 23:56

As the number of rabbits added is thrice that of the original, the population of rabbits is quadrupled every 3 months. As such the growth can be modelled by $$4^{(n/3) + 1}$$, where n is the time in months after the first adoption.

This can also be done iteratively as such:

0 months: 2F, 2M
3 months: 2F, 2M + 2(3F + 3M) = 8F, 8M
6 months: 8F, 8M + 8(3F + 3M) = 32F + 32M
9 months: 32F + 32M + 32(3F + 3M) = 128F + 128M
12 months(1 year): 128F + 128M + 128(3F + 3M) = 512F + 512M

... Where F represents the female rabbits and M represents the male rabbits.

There should be 1024 rabbits after a year, unless I'm missing something crucial.

• Why in the third month did you multiply this? $2(3F + 3M)$? Is it because we're adding the rabbits from the first generation to the second one?. How did you obtained the formula for the growth? I mean how did you obtained $4^{(n/3)+1}$. Can you please explain this?. I'm stuck at that part. – Chris Steinbeck Bell Mar 17 '20 at 23:54
• I'm still stuck at why in the third generation it is $8$ and in the fourth generation it is $32$ why is it?. Can you help me with that part as well?. – Chris Steinbeck Bell Mar 18 '20 at 0:00
• Is it because that ideally all females will produce three aditional males and three aditional females? and those new females will join the ones in the previous generation to produce again three males and three females?. – Chris Steinbeck Bell Mar 18 '20 at 0:02
• @ChrisSteinbeckBell Yes, the new female rabbits will be of course be able to produce 3 new female rabbits and 3 new male rabbits, thus adding on to the number of female rabbits as well. Thus, it can be said the number of female rabbits is quadrupled every quarter. – shanylong Mar 18 '20 at 2:24
• But how did you obtained the growth formula to be $4^{(n/3)+1}$? This is where I am stuck, can you help me there please? – Chris Steinbeck Bell Mar 18 '20 at 2:34

How many rabbit maximum would Roger have after a year from the adoption of his pets?

4. 1024 rabbits

How exactly can I find the number of rabbits in this situation?

For this situation, I would just write it down. There are only four steps to take (4 quarters in a year), thus not much work to write it down. The growing rate is exatcly the same for the female as for the male (f stand for female and m for male):

1. $$2(f+m) = 4$$
2. $$2(f+m) + 2(f+m)*3 = 16$$
3. $$8(f+m) + 8(f+m)*3 = 64$$
4. $$32(f+m) + 32(f+m)*3 = 256$$
5. $$128(f+m) + 128(f+m)*3 = 1024$$

Thus 512 females and 512 males, which makes a total of 1024 rabbits.

Also a pattern can be found in the list above:

1. $$2^2$$
2. $$2^4$$
3. $$2^6$$

Each step the total amount of female rabbit is multiplied by 4 ($$=2^2$$), thus the following formula can be applied:

Total amount of rabbits = $$2^2 * 2^{2q} = 2^{2+2q}$$, where q is the amount of quarters.

• Why in the third month is it $8(f+m)+8(f+m)\times 3$?. Why is it eight?. – Chris Steinbeck Bell Mar 17 '20 at 23:59