I need some help with the math in this question

Question

Before hand a little explanation: I am a math idiot. It's been a long time since I did algebra, trinometry, etc. in school, and I have no idea how to teach myself to answer questions like the two below. I am looking for someone who can maybe detail the steps so I can follow and study my way from the questions to finding the correct answer:

Question 1: How much more light does a 1 m telescope gather than a 10 cm telescope? How many magnitudes fainter should be detectable by a 1m telescope than by a 10 cm telescope?

Question 2: Determine the size of a sunspot, in kilometres, given that it has an angjular diameter of 1". You may assume the sun is 30' in diameter at a distance of 1 AU.

Answer 1

There are 100 cm in a meter. So a 1 m telescope is 10x the diameter of a 10 cm telescope. Light gathering is by area, so you need to square the diameter ratio - 10 x 10 = 100x.  Magnitudes are logarithmic.  5 magnitudes is exactly 100x.  So no math is needed - the answer is 5 magnitudes.  However, most telescopes are reflectors, and you have to subtract off the area of the secondary mirrors.  So in real life, this is just an approximation.

Thirty minutes of arc is 30 * 60 = 1800 seconds of arc.  So the sunspot is 1/1800 of the diameter of the Sun.  From Wikipedia, the Sun's radius is 6.955×10^8 m = 6.955×10^5 km.  So the diameter is 1,391,000 km.  1/1800 of that is 772 km.

You don't need to know how far away the Sun is, unless you want to compute the diameter of the Sun in AU.  You'd still have to convert AU to km.  To do that, you'd have to look something up.

I'm getting a Deja Vu feeling.  Hasn't this been answered before?