Data Sufficiency

The number of antelope in a certain herd increases every year by a constant factor. If there are 500 antelope in the herd today, how many years will it take for the number of antelope to double?

(1) Ten years from now, there will be more than ten times the current number of antelope in the herd.

(2) If the herd were to grow in number at twice its current rate, there would be 980 antelope in the group in two years.

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The number of antelope in a certain herd increases every year by a constant factor. If there are 500 antelope in the herd today, how many years will it take for the number of antelope to double?

(1) Ten years from now, there will be more than ten times the current number of antelope in the herd.

(2) If the herd were to grow in number at twice its current rate, there would be 980 antelope in the group in two years.

In the original condition, the number of different animals in the herd, the number of antelopes after the herd doubled, constant factor that increases every year are all variables. Thus we need 3 equations to match the 3 variables. Since there are 1 equations each in (1) and (2), E is likely the answer. In actual calculation, even if we use both (1) &(2) we can't find out the constant factor as it states that the number is ten times higher after ten years, and in (2) we can't find out the number of the antelopes when the herd is doubled. Therefore the answer is E.


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Hi,
The answer to the question is B and not E

rommysingh wrote:The number of antelope in a certain herd increases every year by a constant factor. If there are 500 antelope in the herd today, how many years will it take for the number of antelope to double?

(1) Ten years from now, there will be more than ten times the current number of antelope in the herd.

(2) If the herd were to grow in number at twice its current rate, there would be 980 antelope in the group in two years.

Question stem, rephrased:
At what rate does the herd increase each year?

Statement 1:
Here, the herd increases by MORE THAN A FACTOR OF 10 every ten years, but the EXACT rate of increase is unknown.
INSUFFICIENT.

Statement 2:
Here, TWICE the current rate over 2 years = (number of antelopes after 2 years)/(current number of antelopes) = (980)/(500) = 98/50 = 49/25.
Since (49/25) = (7/5)(7/5) -- at TWICE the current rate -- the herd is multiplied by a factor of 7/5 each year.
Since 7/5 = 140/100 = 140%, at TWICE the current rate the herd increases by 40% each year.
Thus, at the ACTUAL current rate, the herd increases by 20% each year.
SUFFICIENT.

The correct answer is B.

Note:
We can conclude that statement 2 is sufficient without doing any actual math.
Since the value of twice the rate over 2 years is known, it must be possible to determine the actual rate of increase over 1 year.
SUFFICIENT.

Let's do this algebraically. Since the number increases by a constant factor each year, we can name that factor c. So this year we have 500 antelope, but next year we'll have 500 * c, and the year after that we'll have 500 * c * c.

As you can see, this means that n years from now, we'll have 500 * c� antelope in the herd. We want to know at what point 500 * c� ≥ 1000, or when c� ≥ 2.

S1::

In ten years, we have > 5000 antelope, so

500 * c¹� > 5000, or

c¹� > 10, or

c > (the tenth root of 10)

So we could have c = 1.3, in which case it will take a few years for the herd to double ... or we could have c = 47, in which case it will take one year or less. NOT SUFFICIENT

S2::

500 * (2c)² = 980

This allows us to solve for c, and once we have the value of c, we can solve the equation c� ≥ 2. On test day we DON'T want to solve this -- we've already learned that S2 is sufficient -- but if we had to ...

500 * 4c² = 980
4c² = 49/25
2c = 7/5

So if we increase at TWICE the normal rate, we're multiplying by (7/5) per year, or increasing by 40%. From here we can find that the normal rate is a 20% increase, which is like multiplying by (6/5). Now our equation is (6/5)� ≥ 2, or 6� ≥ 2*5�. Checking small values of n, we see that this is first true for n = 4, so we're done, at least assuming that we only want an integer approximation of n.