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The number of water lilies on a certain lake doubles every [#permalink]
21 Nov 2012, 21:18

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Difficulty:

95% (hard)

Question Stats:

40% (02:20) correct
60% (01:39) wrong based on 192 sessions

The number of water lilies on a certain lake doubles every two days. If there is exactly one water lily on the lake, it takes 60 days for the lake to be fully covered with water lilies. In how many days will the lake be fully covered with lilies, if initially there were two water lilies on it?

Re: The number of water lilies on a certain lake doubles every [#permalink]
21 Nov 2012, 21:29

1

This post received KUDOS

rainbooow wrote:

The number of water lilies on a certain lake doubles every two days. If there is exactly one water lily on the lake, it takes 60 days for the lake to be fully covered with water lilies. In how many days will the lake be fully covered with lilies, if initially there were two water lilies on it?

(A) 15

(B) 28

(C) 30

(D) 58

(E) 59

My approach doesn't work Please, share your ideas!

Starting from 1 Water Lilly it takes 60 days. If there are already two present, it can be taken as the first day is over. It will take 59 more days.

Re: The number of water lilies on a certain lake doubles every [#permalink]
22 Nov 2012, 04:18

2

This post received KUDOS

Expert's post

nawaab wrote:

rainbooow wrote:

The number of water lilies on a certain lake doubles every two days. If there is exactly one water lily on the lake, it takes 60 days for the lake to be fully covered with water lilies. In how many days will the lake be fully covered with lilies, if initially there were two water lilies on it?

(A) 15

(B) 28

(C) 30

(D) 58

(E) 59

My approach doesn't work Please, share your ideas!

Starting from 1 Water Lilly it takes 60 days. If there are already two present, it can be taken as the first day is over. It will take 59 more days.

Notice that we are told that "the number of water lilies on a certain lake doubles every two days", thus if initially there were two water lilies instead of one, we can consider that two days are over and therefore only 58 days are left.

Re: The number of water lilies on a certain lake doubles every [#permalink]
30 Nov 2012, 06:57

I understand the logic, but am not able to solve it algebraically.

since the series is in the geometric progression with the common ration (r) = 2, initial condition can be rewritten as:

a(n) = 1.2^60-1 {a(n = a.r^n-1)} === which gives us total number of lillies in the pool ==>2^59.....no this is equated when the the pool starts with 2 lillies...==> 2^59 = 2.2^n-1 ===>n=59..

Re: The number of water lilies on a certain lake doubles every [#permalink]
30 Nov 2012, 08:53

Expert's post

pavanpuneet wrote:

I understand the logic, but am not able to solve it algebraically.

since the series is in the geometric progression with the common ration (r) = 2, initial condition can be rewritten as:

a(n) = 1.2^60-1 {a(n = a.r^n-1)} === which gives us total number of lillies in the pool ==>2^59.....no this is equated when the the pool starts with 2 lillies...==> 2^59 = 2.2^n-1 ===>n=59..

Where am I going wrong?

We are told that "the number of water lilies on a certain lake doubles every TWO days".

If there are two lilies, then in order to cover the lake they would need to double one time less than in case with 1 lily. Since lilies double every two days, then 60-2=58 days are needed. _________________

Re: The number of water lilies on a certain lake doubles every [#permalink]
09 Mar 2014, 04:46

Bunuel or anyone,

Please confirm if my approach is correct.

Sum of lillies for 30 days using Geo Series: a= 1+2+2^2+2^3..2^30 --(1) 2a = 2+2^2...2^31 -- (2) Subtract 1 from 2 a=2^31 - 1 (Total lillies in the pond)

Now let x be number of times, both lillies expanded at once lilly 1 -> a=1+2+2^2...2^x sum of lilly 1 using Geo series described above = 2^x+1 - 1 lilly 2 -> a=1+2+2^2....2^x sum of lilly 2 using Geo series described above = 2^x+1 - 1 --> sum of lilly 1 + sum of lilly 2 = 2^31 -1 so 2(2^x+1 -1) = 2^31 - 1 2^x+2 - 2 = 2^31 -1 approximately 2^x+2 = 2^31 x+2 = 31, x= 29 times ....so 58 days as lillies doubles evry 2 days _________________

Re: The number of water lilies on a certain lake doubles every [#permalink]
09 Mar 2014, 06:21

Expert's post

maaadhu wrote:

Bunuel or anyone,

Please confirm if my approach is correct.

Sum of lillies for 30 days using Geo Series: a= 1+2+2^2+2^3..2^30 --(1) 2a = 2+2^2...2^31 -- (2) Subtract 1 from 2 a=2^31 - 1 (Total lillies in the pond)

Now let x be number of times, both lillies expanded at once lilly 1 -> a=1+2+2^2...2^x sum of lilly 1 using Geo series described above = 2^x+1 - 1 lilly 2 -> a=1+2+2^2....2^x sum of lilly 2 using Geo series described above = 2^x+1 - 1 --> sum of lilly 1 + sum of lilly 2 = 2^31 -1 so 2(2^x+1 -1) = 2^31 - 1 2^x+2 - 2 = 2^31 -1 approximately 2^x+2 = 2^31 x+2 = 31, x= 29 times ....so 58 days as lillies doubles evry 2 days

No, that's not correct. Neat algebraic manipulations though...

Notice that the total number of lilies is not 1+2+2^2+2^3..2^30, it's 2^30.

Initially = 1; After 2 days = 2, not 1+2; After 4 days = 2^2 = 4, not 1+2+4. ... After 60 days = 2^30, not 1+2+2^2+2^3+...+2^30.

Similarly, if initially there are 2 lilies, then the total number would be 2*2^x.

Re: The number of water lilies on a certain lake doubles every [#permalink]
09 Mar 2014, 18:38

Bunuel wrote:

maaadhu wrote:

Bunuel or anyone,

Please confirm if my approach is correct.

Sum of lillies for 30 days using Geo Series: a= 1+2+2^2+2^3..2^30 --(1) 2a = 2+2^2...2^31 -- (2) Subtract 1 from 2 a=2^31 - 1 (Total lillies in the pond)

Now let x be number of times, both lillies expanded at once lilly 1 -> a=1+2+2^2...2^x sum of lilly 1 using Geo series described above = 2^x+1 - 1 lilly 2 -> a=1+2+2^2....2^x sum of lilly 2 using Geo series described above = 2^x+1 - 1 --> sum of lilly 1 + sum of lilly 2 = 2^31 -1 so 2(2^x+1 -1) = 2^31 - 1 2^x+2 - 2 = 2^31 -1 approximately 2^x+2 = 2^31 x+2 = 31, x= 29 times ....so 58 days as lillies doubles evry 2 days

No, that's not correct. Neat algebraic manipulations though...

Notice that the total number of lilies is not 1+2+2^2+2^3..2^30, it's 2^30.

Initially = 1; After 2 days = 2, not 1+2; After 4 days = 2^2 = 4, not 1+2+4. ... After 60 days = 2^30, not 1+2+2^2+2^3+...+2^30.

Similarly, if initially there are 2 lilies, then the total number would be 2*2^x.

Re: The number of water lilies on a certain lake doubles every [#permalink]
10 Jun 2014, 12:36

That is pretty easy one. Full = 60 days, knowing that the number of lilies doubles each 2 days we can deduce that the half of the lake was full at 58 days. Taking initial information that we have 2 lilies at day 1 we can just simply multiply 2 lilies by 1/2 of the lake which means that the lake will be full at 58 days.

Re: The number of water lilies on a certain lake doubles every [#permalink]
21 Jun 2014, 04:22

I thought this way : 30 days will take to complete the pond with lillies count as 2^30 (since it takes 2 days to double hence will take 30 days of 60) on first day - 2^0 =1 lilly on day 2 - 2^1 = 2 on day 3 - 2^2 = 4 so on ... now since the there are two lillies already it will take 2^30/ 2^1 = 2^29 ...this will take complete 2* 29 days i.e 58 days

Re: The number of water lilies on a certain lake doubles every [#permalink]
16 Sep 2014, 03:08

1

This post was BOOKMARKED

I am going by this formula : y(t) = y(0) x K^t where y(t) = desired value after t period y(0) = initial value k = multiplier (or the factor by which the value increases every t period) t = time period

Given - # of lilies doubles every two days ==> t= 2 days k^t = k^2 = 2 ==> k = sqrt(2) Now, it takes 60 days for a lake to be fully covered with water lilies starting from 1 lily so, y(0) = 1 t = 60 days y(t) i.e no. of lilies after 60 days y(t) = 1 x sqrt(2)^60

now, we have the final count of lilies after 60 days if we start from 1 lily. we can calculate the time period if we start from 2 lilies ( the # of lilies after 60 days will not change as the multiplier is constant)

1 x sqrt(2)^60 = 2 x sqrt(2)^t

Solving this will give t= 58 days.

I hope it helps.

gmatclubot

Re: The number of water lilies on a certain lake doubles every
[#permalink]
16 Sep 2014, 03:08