A quantity increases in a manner such that the ratio of its values in any two consecutive years is constant. If the quantity doubles every 6 years, by what factor does it increase in two years?
Can someone provide a simple solution?
Here is a formula for exponential growth:
Final amount = original amount * multiplier^(number of changes).
Into the formula above, plug in the following values:
Original amount = 1.
Final amount = 2. (Since the original amount doubles every 6 years.)
2-year multiplier = x. (The factor by which the original amount increases in 2 years.)
Number of changes = 3. (Since over 6 years the original amount will be multiplied by x three times.)
2 = 1 * x^3
x = 2^(1/3).
Alternate approach:
Since the GMAT would provide answer choices, we could PLUG IN THE ANSWERS.
The answer choices would represent by what factor the original amount increases in 2 years.
Answer choices:
2^(1/6)
2^(1/4)
2^(1/3)
2^(1/2)
2^(2/3).
Let the original amount = 1.
When the correct answer choice is plugged in, the original amount will double to 2 in 6 years.
Answer choice D: 2^(1/2)
Amount after 2 years = 1 * 2^(1/2) = 2^(1/2)
Amount after 4 years = 2^(1/2) * 2^(1/2) = 2.
Since the amount is increasing TOO QUICKLY, the correct multiplier must be SMALLER.
Eliminate D and E.
Answer choice B: 2^(1/4)
Amount after 2 years = 1 * 2^(1/4) = 2^(1/4)
Amount after 4 years = 2^(1/4) * 2^(1/4) = 2^(1/2)
Amount after 6 years = 2^(1/2) * (2^1/4) = 2^(3/4).
Since the amount is increasing TOO SLOWLY, the correct multiplier must be GREATER.
Eliminate A and B.
The correct answer is
C.
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