The bear alarm at Grizzly's Peak ski resort sounds an

BTGmoderatorDC wrote:The bear alarm at Grizzly's Peak ski resort sounds an average of once every thirty days, but the alarm is so sensitively calibrated that it sounds an average of ten false alarms for every undetected bear. Despite this, the alarm only sounds for three out of four bears that actually appear at the resort. Approximately how many bears appear at the resort each year?

A) 1
B) 3
C) 4
D) 10
E) 13

The alarm sounds for 3 of EVERY 4 BEARS that actually appear,
If 4 bears appear, then the number of alarms = 3.
Since the alarm sounds only 3 times for 4 bears, 1 of these 4 bears is undetected.
Since there are 10 false alarms for every 1 undetected bear, the number of false alarms = 10.
Thus, for every 4 bears that actually appear, the total number of alarms = (3 alarms for bears that actually appear) + (10 false alarms) = 13 alarms.
Thus, the bear appearance rate = 4 bears for every 13 alarms.

The alarm sounds every 30 days, implying a total 12 alarms every year.
Since there are 12 alarms at a rate of 4 bears for every 13 alarms, we get:
12 alarms * (4 bears)/(13 alarms) = 48/13 alarms ≈ 4 bears.

The correct answer is C.

BTGmoderatorDC wrote:The bear alarm at Grizzly's Peak ski resort sounds an average of once every thirty days, but the alarm is so sensitively calibrated that it sounds an average of ten false alarms for every undetected bear. Despite this, the alarm only sounds for three out of four bears that actually appear at the resort. Approximately how many bears appear at the resort each year?

A) 1
B) 3
C) 4
D) 10
E) 13

This exercise is not GMAT-like: VERY confusing wording!

We know there are (on average) 12 alarm sounds per year, not all of them occurred when there was a "bear there"... sometimes the bear was not there and the alarm sounded ("false alarm").
So far, so good...

Now... when the question stem says "ten false alarms for every undetected bear", we must (I believe) understand this as the following:

The alarm sounds 10 times with "no bear there" for each time there is no sound (alarm failed) when there WAS a bear there...

Finally, we know that for every 4 times "bear is there", only 3 alarm sounds...

With all that, I created the following "grid" (double matrix). Please take a moment to check the info above was properly put below:

\[\begin{array}{*{20}{c}}
{}&{{\text{alarm}}\,\,{\text{sounds}}}&{{\text{alarm}}\,\,{\text{does}}\,\,{\text{not}}\,\,{\text{sound}}}&{{\text{total}}} \\
{{\text{bear}}\,\,{\text{there}}}&{3k}&{m\,\,\,\,\boxed{ = k}}&{4k} \\
{{\text{bear}}\,\,{\text{not}}\,\,{\text{there}}}&{10m}&{}&{} \\
{{\text{total}}}&{12\,\,\,\,\boxed{ = 13k}}&{}&{}
\end{array}\]

Our FOCUS is 4k, and by the grid we know that 12 = 13k, hence:
\[? = 4k = 4\left( {\frac{{12}}{{13}}} \right) = \frac{{39 + 9}}{{13}} = 3\frac{9}{{13}} \cong \boxed4\]

Regards,
fskilnik.