Manhattan Quant Guides

Hi,
Let's say the quantities are t1,t2,t3,...
Given that t2/t1 =t3/t2 =t4/t3 =.... = r.
So, t1,t2,t3.. are in a geometric series.
Given that t7/t1 = 2
So, r^6 = 2
We need to find t3/t1 = r^2.
So, r^2 = 2^(1/3)

Frankenstein wrote:Hi,
Let's say the quantities are t1,t2,t3,...
Given that t2/t1 =t3/t2 =t4/t3 =.... = r.
So, t1,t2,t3.. are in a geometric series.
Given that t7/t1 = 2
So, r^6 = 2
We need to find t3/t1 = r^2.
So, r^2 = 2^(1/3)

Hi!

could you pls explain what "t7/t1 = 2" means? and why "r^2 = 2^(1/3)" ?? Can somebody explain how to solve this problem in other words?

Thanks in advance!

Frankenstein wrote:Hi,
Let's say the quantities are t1,t2,t3,...
Given that t2/t1 =t3/t2 =t4/t3 =.... = r.
So, t1,t2,t3.. are in a geometric series.
Given that t7/t1 = 2
So, r^6 = 2
We need to find t3/t1 = r^2.
So, r^2 = 2^(1/3)

Hi!

could you pls explain what "t7/t1 = 2" means? and why "r^2 = 2^(1/3)" ?? Can somebody explain how to solve this problem in other words?

Thanks in advance!

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.

We could also say that if it is multiplied by a factor of x each year, that

x� = 2

Taking the sixth root of each side, we find that x = �√2, so it goes up by the sixth root of 2 each year. Our question wants two years, so we'd just square this and be done.