Q. A certain established organization has exactly 4096 members. A certain new organization has exactly 4 members. Every 5 members the membership of the established organization increased by 100 percent. Every 10 months the membership of new organization increases by 700 percent. New members join the organization only on the last day of each 5- or -10 months period. assuming that no member leaves 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
Percentage Problem
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Increasing a quantity by 100% is the same as multiplying that quantity by 2 (adding the whole amount on top of what's already there). So we can say that every 5 months the established organization doubles its membership.
To see how we can then get an expression for the members of the organization after some number of months, it can be helpful to consider numbers first.
After 10 months, the old organization will have doubled its membership twice (10/5 times). So it would have 4096*2*2 members. Or 4096 * 2^2 times.
After 20 months, the old organization will have doubled its membership four times (or 20/5 times). So it would have 4096*2*2*2*2 members. Or 4096*2^4 members.
After M months, the old organization will have doubled its membership M/5 times. So it would have 4096*2^(M/5) members.
Similarly, Increasing a quantity by 700% is the same as multiplying that quantity by 8. So we can say that every 10 months the new organization multiplies it's membership by 8.
Algebraically, we can say that after M months, the new organization will have 4*8^(M/10) members.
We can now set theses quantities equal to get an equation:
4096*2^(M/5) = 4*8^(M/10)
To solve equations in which the variable is in an exponent, we can make the bases the same. Luckly, all our bases are powers of 2. So replacing 4096 with 2^12, and 4 with 2^2 and 8 with 2^3 we get:
(2^12)*2^(M/5) = (2^2)*(2^3)^(M/10)
Using the exponent rules for powers (when a quantity is raised to an exponent, and that whole quantity is raised to another exponent, we can multiply the exponents), we get:
(2^12)*2^(M/5) = (2^2)*2^(3*M/10)
Using the exponent rule for multiplication (when two terms with the same base are mutiplued, add the exponents), we get:
2^(12 + M/5) = 2^(2 + 3M/10)
So we've got 2 raised to on thing equal to two raised to another thing. That means the two exponents must be the same:
12 + M/5 = 2 + 3M/10 ---> multiplying by 10
120 + 2M = 20 + 3M
100 = M
To see how we can then get an expression for the members of the organization after some number of months, it can be helpful to consider numbers first.
After 10 months, the old organization will have doubled its membership twice (10/5 times). So it would have 4096*2*2 members. Or 4096 * 2^2 times.
After 20 months, the old organization will have doubled its membership four times (or 20/5 times). So it would have 4096*2*2*2*2 members. Or 4096*2^4 members.
After M months, the old organization will have doubled its membership M/5 times. So it would have 4096*2^(M/5) members.
Similarly, Increasing a quantity by 700% is the same as multiplying that quantity by 8. So we can say that every 10 months the new organization multiplies it's membership by 8.
Algebraically, we can say that after M months, the new organization will have 4*8^(M/10) members.
We can now set theses quantities equal to get an equation:
4096*2^(M/5) = 4*8^(M/10)
To solve equations in which the variable is in an exponent, we can make the bases the same. Luckly, all our bases are powers of 2. So replacing 4096 with 2^12, and 4 with 2^2 and 8 with 2^3 we get:
(2^12)*2^(M/5) = (2^2)*(2^3)^(M/10)
Using the exponent rules for powers (when a quantity is raised to an exponent, and that whole quantity is raised to another exponent, we can multiply the exponents), we get:
(2^12)*2^(M/5) = (2^2)*2^(3*M/10)
Using the exponent rule for multiplication (when two terms with the same base are mutiplued, add the exponents), we get:
2^(12 + M/5) = 2^(2 + 3M/10)
So we've got 2 raised to on thing equal to two raised to another thing. That means the two exponents must be the same:
12 + M/5 = 2 + 3M/10 ---> multiplying by 10
120 + 2M = 20 + 3M
100 = M
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We know that the members of the established organization increase by 100% in a 5-month period, thus they become 200% of the initial member. Or, they become 400% of the initial members in a 10-month period. 400% of a number = 4 times of a number.Joy Shaha wrote:Q. 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 new organization increases by 700 percent. New members join the organizations only on the last day of each 5 or 10 months 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
There are was a typo in the question. Corrected.
Similarly, we know that the members of the new organization increase by 700% in a 10-month period, thus they become 800% of the initial member. 800% of a number = 8 times of a number.
Say the number of members of the both the organizations become equal in n numbers of 10-months periods.
Thus, 4096*4^n = 4*8^n
=> 1024*4^n = (2*4)^n
=> 1024 = 2^n
=> 2^10 = 2^n
=> n = 10 numbers of 10-months periods = 100 months.
Relevant book: Manhattan Review GMAT Word Problems
Answer: E
-Jay
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