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A certain established organization has exactly 4096 members. [#permalink]

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08 Jan 2008, 12:23

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A certain established organization has exactly 4096 members. A certain new organization has exactly 4 members. Every 5 months the membership of the established organization increases by 100 percent. Every 10 months the membership of the new organization increases by 700 percent. New members join the organizations only on the last day of each 5- or 10-month period. Assuming that no member leaves the organizations, after how many months will the two groups have exactly the same number of members?

4096*2^[m/5]=4*8^[m/10] where, m - the number of months. 2^12*2^[m/5]=2^2*2^(3*[m/10]) 2^(12+[m/5])=2^(2+3*[m/10]) 12+[m/5]=2+3*[m/10] 10+[m/5]=3*[m/10]

10+2x=3x (x=[m/10]) x=10 m=100.

100/5=20, 100/10=10. are integers.
_________________

A certain established organization has exactly 4096 members. A certain new organization has exactly 4 members. Every 5 months the membership of the established organization increases by 100 percent. Every 10 months the membership of the new organization increases by 700 percent. New members join the organizations only on the last day of each 5- or 10-month period. Assuming that no member leaves the organizations, after how many months will the two groups have exactly the same number of members?

(A) 20 (B) 40 (C) 50 (D) 80 (E) 100

For the established group the membership is doubling every 5 months. If you set x = to # of 5 month periods you could say that the membership is equal to original membership*2^x. However, since the other group is expanding every 10 months, and the answers were all in increments of 10, I set x equal to 10 months. Of course 10 months = two 5 month periods so the equation is 2^2x instead of 2^x. The 2^2 = 4 comes from the population quadrupling every 10 months. You could also have the equation set up as 4^x where x is the number of 10 month periods. You get the same results, I just wanted to keep everything in powers of 2.

The new organization is 8x bigger after 10 months (growing by 700 percent = x+7x = 8x). 2^3 = 8 so 2^3(x) = membership after x number of 10 month periods.

Try plugging in numbers and testing them on a calculator to see for yourself. Start with a membership size of 1 person and play around with different time periods to see how it works.

can someone pls explain how does 100 months come???

A certain established organization has exactly 4096 members. A certain new organization has exactly 4 members. Every 5 months the membership of the established organization increases by 100 percent. Every 10 months the membership of the new organization increases by 700 percent. New members join the organizations only on the last day of each 5- or 10-month period. Assuming that no member leaves the organizations, after how many months will the two groups have exactly the same number of members?

(A) 20 (B) 40 (C) 50 (D) 80 (E) 100

Every 5 months the membership of the established organization increases by 100 percent, means that every 5 months the membership of the established organization doubles.

Every 10 months the membership of the new organization increases by 700 percent, means that every 10 months the membership of the new organization increases 8 times.

Say these organizations will have the same number of members in x months.

The number of members in the established organization in x months will double x/5 times (since it doubles every 5 months), and become \(4096*2^{\frac{x}{5}}=2^{12}*2^{\frac{x}{5}}\);

The number of members in the new organization in x months will increase 8 times x/10 times (since it increases 8 times every 10 months), and become \(4*8^{\frac{x}{10}}=2^2*2^{\frac{3x}{10}}\).

Re: A certain established organization has exactly 4096 members. [#permalink]

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03 Jun 2015, 04:24

Bunuel wrote:

virajgawade wrote:

can someone pls explain how does 100 months come???

A certain established organization has exactly 4096 members. A certain new organization has exactly 4 members. Every 5 months the membership of the established organization increases by 100 percent. Every 10 months the membership of the new organization increases by 700 percent. New members join the organizations only on the last day of each 5- or 10-month period. Assuming that no member leaves the organizations, after how many months will the two groups have exactly the same number of members?

(A) 20 (B) 40 (C) 50 (D) 80 (E) 100

Every 5 months the membership of the established organization increases by 100 percent, means that every 5 months the membership of the established organization doubles.

Every 10 months the membership of the new organization increases by 700 percent, means that every 10 months the membership of the new organization increases 8 times.

Say these organizations will have the same number of members in x months.

The number of members in the established organization in x months will double x/5 times (since it doubles every 5 months), and become \(4096*2^{\frac{x}{5}}=2^{12}*2^{\frac{x}{5}}\);

The number of members in the new organization in x months will increase 8 times x/10 times (since it increases 8 times every 10 months), and become \(4*8^{\frac{x}{10}}=2^2*2^{\frac{3x}{10}}\).

Can the above problem be solved using the concept of geometric progression ?

I tried but couldn't make it. Formula used : nth term of GP = ar^(n-1) 4096 (2)^n-1=4(8)^m-1 n: number of 5 month periods m: number of 10 month periods

A certain established organization has exactly 4096 members. A certain new organization has exactly 4 members. Every 5 months the membership of the established organization increases by 100 percent. Every 10 months the membership of the new organization increases by 700 percent. New members join the organizations only on the last day of each 5- or 10-month period. Assuming that no member leaves the organizations, after how many months will the two groups have exactly the same number of members?

(A) 20 (B) 40 (C) 50 (D) 80 (E) 100

Established Organization growth = 100% i.e. x becomes x+(100/100)x = 2x in 5 months New Organization growth = 700% i.e. x becomes x+(700/100)x = 8x in 10 months

Which means the population of established organization become 4 times of previous population every 10 months and the population of New organization become 8 times of previous population every 10 months

Let's say after a cycles of 10 months the population of both organization becomes equal

Population of New Organization after 'y' cycles of 10 months = 4 x (8^y) = (2^2) x [2^(3y)] = [2^(2+3y)]

Population of Established Organization after 'y' cycles of 10 months = 4096 x (4^y) = (2^12) x [2^(2y)] = [2^(12+2y)]

i.e. [2^(2+3y)] = [2^(12+2y)]

i.e. 2+3y = 12+2y

i.e. y = 10

i.e. 10 cycles of 10 months i.e. 10x10 = 100 months

Prosper!!! GMATinsight Bhoopendra Singh and Dr.Sushma Jha e-mail: info@GMATinsight.com I Call us : +91-9999687183 / 9891333772 Online One-on-One Skype based classes and Classroom Coaching in South and West Delhi http://www.GMATinsight.com/testimonials.html

can someone pls explain how does 100 months come???

A certain established organization has exactly 4096 members. A certain new organization has exactly 4 members. Every 5 months the membership of the established organization increases by 100 percent. Every 10 months the membership of the new organization increases by 700 percent. New members join the organizations only on the last day of each 5- or 10-month period. Assuming that no member leaves the organizations, after how many months will the two groups have exactly the same number of members?

(A) 20 (B) 40 (C) 50 (D) 80 (E) 100

Every 5 months the membership of the established organization increases by 100 percent, means that every 5 months the membership of the established organization doubles.

Every 10 months the membership of the new organization increases by 700 percent, means that every 10 months the membership of the new organization increases 8 times.

Say these organizations will have the same number of members in x months.

The number of members in the established organization in x months will double x/5 times (since it doubles every 5 months), and become \(4096*2^{\frac{x}{5}}=2^{12}*2^{\frac{x}{5}}\);

The number of members in the new organization in x months will increase 8 times x/10 times (since it increases 8 times every 10 months), and become \(4*8^{\frac{x}{10}}=2^2*2^{\frac{3x}{10}}\).

Can the above problem be solved using the concept of geometric progression ?

I tried but couldn't make it. Formula used : nth term of GP = ar^(n-1) 4096 (2)^n-1=4(8)^m-1 n: number of 5 month periods m: number of 10 month periods

so, n=2m

not getting where am i going wrong

can you please help ?

Hi Radhika,

Please check the below solution as you want it to be

You need to understand here that the the terms are increasing by common ratio of 2 and 8 for Established and New organization respectively the period of increase are different so you need to avoid taking (-1) in the power because (n-1) will lead to a different number of 5 months period than no. of (m-1) periods of 10 months

After n periods of 5 months the Established organization will have population = 4096 (2)^n and After m periods of 10 months the New organization will have population = 4 (8)^m

4096 (2)^n=4(8)^m n: number of 5 month periods m: number of 10 month periods

so, n=2m

4096 (2)^(2m)=4(8)^(m)

i.e. (2^12)[2^(2m)] = (2^2)[2^(3m)] ------ [Please note that 8 = 2^3 therefore 8^(m) = 2^3(m) = 2^(3m)]

i.e. [2^(12+2m)] = [2^(2+3m)]

i.e. (12+2m) = (2+3m)

i.e. 12+2m = 3m+2

i.e. m=10 _________________

Prosper!!! GMATinsight Bhoopendra Singh and Dr.Sushma Jha e-mail: info@GMATinsight.com I Call us : +91-9999687183 / 9891333772 Online One-on-One Skype based classes and Classroom Coaching in South and West Delhi http://www.GMATinsight.com/testimonials.html

Could you please elaborate on why we cant solve this using G.P and why we shouldn't take n-1 in the power ? Here is how i solved the prob :

Old organization = 2^12 * 2^0 (initial value); r = 2^2 New organization = 2^2*2^0 (initial value) ; r = 2^3

as per G.P, equating the two GPs to get n

2^12*[2^(2n-2)]=2^2[2^(3n-3)] I get n = 11. Not sure what the issue is in this method.

TIA!

As per your solution

Formula used : nth term of GP = ar^(n-1)

a= 4096 for Established organization and r=2^2

@n=1, The Total Employees = 4096 @n=2, The Total Employees = 4096*2^2(1) [After FIRST cycle of 10 months] @n=3, The Total Employees = 4096*2^2(2) [After SECOND cycle of 10 months] @n=4, The Total Employees = 4096*2^2(3) [After THIRD cycle of 10 months] --- --- --- @n=11, The Total Employees = 4096*2^2(n-1) [After TENTH cycle of 10 months] i.e.Total Month passes = 10*10 = 100

So n=11 tells that 11th term equates the two numbers (no. of employees for two organization) but 11th term comes after 10 cycles of 10 months hence 100 months

I hope this helps!!!
_________________

Prosper!!! GMATinsight Bhoopendra Singh and Dr.Sushma Jha e-mail: info@GMATinsight.com I Call us : +91-9999687183 / 9891333772 Online One-on-One Skype based classes and Classroom Coaching in South and West Delhi http://www.GMATinsight.com/testimonials.html

Re: A certain established organization has exactly 4096 members. [#permalink]

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17 May 2016, 04:45

Bunuel wrote:

virajgawade wrote:

can someone pls explain how does 100 months come???

A certain established organization has exactly 4096 members. A certain new organization has exactly 4 members. Every 5 months the membership of the established organization increases by 100 percent. Every 10 months the membership of the new organization increases by 700 percent. New members join the organizations only on the last day of each 5- or 10-month period. Assuming that no member leaves the organizations, after how many months will the two groups have exactly the same number of members?

(A) 20 (B) 40 (C) 50 (D) 80 (E) 100

Every 5 months the membership of the established organization increases by 100 percent, means that every 5 months the membership of the established organization doubles.

Every 10 months the membership of the new organization increases by 700 percent, means that every 10 months the membership of the new organization increases 8 times.

Say these organizations will have the same number of members in x months.

The number of members in the established organization in x months will double x/5 times (since it doubles every 5 months), and become \(4096*2^{\frac{x}{5}}=2^{12}*2^{\frac{x}{5}}\);

The number of members in the new organization in x months will increase 8 times x/10 times (since it increases 8 times every 10 months), and become \(4*8^{\frac{x}{10}}=2^2*2^{\frac{3x}{10}}\).

Bunuel I have a query re the question above and a few similar questions where you seem to have used the formula for GP without subtracting 1 from the power of r, ar^n (instead ar^n-1).

Can you please explain the reasoning behind this.?

I solved this as follows -

ar^n-1 Established company => 4096 * 2^2n-1 = 2^12 * 2^2n-1 New company => 4*2^n-1 = 2^2 * 8^n-1

Now as these are both equal, 2^12*2^2n-1 = 2^2 * 2^3(n-1)

Hence we get n=12 Therefore the number of months = 12*10 = 120.

Re: A certain established organization has exactly 4096 members. [#permalink]

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30 May 2016, 01:07

eschn3am wrote:

JCLEONES wrote:

A certain established organization has exactly 4096 members. A certain new organization has exactly 4 members. Every 5 months the membership of the established organization increases by 100 percent. Every 10 months the membership of the new organization increases by 700 percent. New members join the organizations only on the last day of each 5- or 10-month period. Assuming that no member leaves the organizations, after how many months will the two groups have exactly the same number of members?

(A) 20 (B) 40 (C) 50 (D) 80 (E) 100

For the established group the membership is doubling every 5 months. If you set x = to # of 5 month periods you could say that the membership is equal to original membership*2^x. However, since the other group is expanding every 10 months, and the answers were all in increments of 10, I set x equal to 10 months. Of course 10 months = two 5 month periods so the equation is 2^2x instead of 2^x. The 2^2 = 4 comes from the population quadrupling every 10 months. You could also have the equation set up as 4^x where x is the number of 10 month periods. You get the same results, I just wanted to keep everything in powers of 2.

The new organization is 8x bigger after 10 months (growing by 700 percent = x+7x = 8x). 2^3 = 8 so 2^3(x) = membership after x number of 10 month periods.

Try plugging in numbers and testing them on a calculator to see for yourself. Start with a membership size of 1 person and play around with different time periods to see how it works.

I fail to understand how your are considering x = Time period in months and then raising it as a power(Can you do that?). I understood the following with your explanation.

>> Upon prime factorisation we get 2^12. Then as it doubles every 5 months, we can say population after 5 months is 2^12 x 2. Hence after 10 months is 2^12 x 2 x 2 = 2^12 x 2^2. (We are considering 2^12 as the months are mentioned in 10 month period for the 2nd organisation)

>>2nd Organization - Population after 10 months increases by 700%, therefore total increase is 100% + 700% = 800% Therefore 2^2 x 2^3.

I have no idea how to take it ahead from here. Could you please guide me?

Re: A certain established organization has exactly 4096 members. [#permalink]

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31 May 2017, 07:50

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A certain established organization has exactly 4096 members. [#permalink]

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19 Aug 2017, 20:47

Bunuel wrote:

virajgawade wrote:

can someone pls explain how does 100 months come???

A certain established organization has exactly 4096 members. A certain new organization has exactly 4 members. Every 5 months the membership of the established organization increases by 100 percent. Every 10 months the membership of the new organization increases by 700 percent. New members join the organizations only on the last day of each 5- or 10-month period. Assuming that no member leaves the organizations, after how many months will the two groups have exactly the same number of members?

(A) 20 (B) 40 (C) 50 (D) 80 (E) 100

Every 5 months the membership of the established organization increases by 100 percent, means that every 5 months the membership of the established organization doubles.

Every 10 months the membership of the new organization increases by 700 percent, means that every 10 months the membership of the new organization increases 8 times.

Say these organizations will have the same number of members in x months.

The number of members in the established organization in x months will double x/5 times (since it doubles every 5 months), and become \(4096*2^{\frac{x}{5}}=2^{12}*2^{\frac{x}{5}}\);

The number of members in the new organization in x months will increase 8 times x/10 times (since it increases 8 times every 10 months), and become \(4*8^{\frac{x}{10}}=2^2*2^{\frac{3x}{10}}\).

Hi Bunuel, In above solution, i couldn't understand below red mark. Could you please let me know how and why did you take x/5 & x/10.

The number of members in the established organization in x months will double x/5 times (since it doubles every 5 months), and become \(4096*2^{\frac{x}{5}}=2^{12}*2^{\frac{x}{5}}\);

The number of members in the new organization in x months will increase 8 times x/10 times (since it increases 8 times every 10 months), and become \(4*8^{\frac{x}{10}}=2^2*2^{\frac{3x}{10}}\).