Compare the light gathering power of a 24 cm telescope to one that is only 4 cm.

The so-called "light-gathering power" simply means how much AREA the main lens or mirror has.  All the light that shines on THAT area gets focused onto the same tiny area in the telescope.  

For example, if the astronomer is going to LOOK through the telescope for a while, then all the light that hits the main lens or mirror gets focused down to a beam small enough to fit through the pupil of his eye !  Even if the main lens or mirror is several FEET across.

So the light gathering power is just proportional to the AREA of the main lens or mirror.

The AREA of a circle is  A = π R² , and  R  is the radius of the circle.  The important thing to notice here is that little  ²  next to the radius.  That tells us that whatever you do to change the radius, the effect on the area is the SQUARE of that number.

If you double the radius, you get (2²) = 4 times as much area.  If you triple the radius, you get  (3²) = 9 times as much area.

In this question, the radius of the big scope is (24/4) = 6 times the size of the little scope.  So the light gathering power is (6²) = 36 times as much as the little one.

compass can be used day and night, cloudy or sunny