Problem Solving

If each day a tree loses half of its remaining leaves, what fraction of its leaves will it have lost after 5 days ?

1/32

1/16

9/10

15/16

31/32


What is the logic behind this? I picked 80 and cut half for 5 days, btu didin't work. [spoiler]OA 31/32[/spoiler]

imo a)
with 80 your no. we will have 2.5 lost on 5th day
and fraction is 2.5/80=1/32

days actual lost remaining
1 x x/2 x/2
2 x/2 x/4 x/4
3 x/4 x/8 x/8
4 x/8 x/16 x/16
5 x/16 x/32 x/32

hence 1/32 is the req fraction

This can be approached in a more simple way.

Assume tree leaf = 1 (Intial stage)

Half of the remaining leaves is lost per day --> 1/2
so for 5 days --> 1/2^5 = 1/32

Now, subtract lost leaves from the initial stage of the tree to get the current number of leaves.
1 - 1/32 = 31/32

papgust wrote:This can be approached in a more simple way.

Assume tree leaf = 1 (Intial stage)

Half of the remaining leaves is lost per day --> 1/2
so for 5 days --> 1/2^5 = 1/32

Now, subtract lost leaves from the initial stage of the tree to get the current number of leaves.
1 - 1/32 = 31/32

But what is the difference between choosing 1 or any other number ? if i choose any other number it wil not work. this why I am too confused.

You can choose any number that is comfortable for you. For me, it was comfortable using 1 which is helpful in the calculation of this problem i've shown

papgust wrote:You can choose any number that is comfortable for you. For me, it was comfortable using 1 which is helpful in the calculation of this problem i've shown

If you can,please whow how to do it with 80

heshamelaziry wrote:

papgust wrote:You can choose any number that is comfortable for you. For me, it was comfortable using 1 which is helpful in the calculation of this problem i've shown

If you can,please whow how to do it with 80

You don't want to pick 80 here. Before you pick a number, reason with the problem a little bit. When picking numbers you want to pick numbers that are permissible and manageabe. So, you always want to think about what kind of numbers will be manageable given the specific conditions of the problem. Here, whatever number we pick, we are going to be dividing it by 2 over and over again. So, be sure to pick a number that can divide 2 over and over again nicely. Ideally, you want to pick a power of 2. Since we will be dividing it by 2 five times, you can pick something like 2^5 or 2^6.

If you pick 2^6 = 64, then original number of leaves is 64. We half it once, giving us 32, half it again giving us 16, and then half it three more times arriving at 2. So when we start with 64 leaves, we are left with 2. Therefore, we have lost 62. So, the proportion lost is 62/64 = 31/32.

Thank you Testluv.