## Concentric Circles Puzzle

This morning I happened across this tweet from David Marain (@dmarain)

As you may know, I have a penchant for puzzles, and find it hard to leave them unsolved, so I thought about it and came up with an answer. I thought I would jot down my thought process here.

For those who can’t see the picture, the puzzle is:

*A point is chosen at random inside the larger of two concentric circles. The probability it lies outside the smaller one is 0.84. What is the ratio of the larger radius to the smaller radius?*

It’s a lovely little puzzle that combines a bit of geometric thinking with probability theory, so do have a little go first.

You done? Good.

My thinking started as: “that 0.84 must be equal to the area of the big circle – the small circle all over the area of the big circle.”

I used a as the area of the big circle, b as the area of the small circle and formed the following equation:

*(a-b)/a = 0.84*

With a little rearranging I got:

*0.16a = b*

So a ratio of areas that is

*1:0.16 (a:b)*

Or

*6.25:1*

Which is equivalent to

*625:100*

Which simplifies to

*25:4*

As we are looking for the ratio of radii, we need to square root each side, which gives a ratio of

*5:2*

A nice little solution to a lovely puzzle. Thanks for sharing David.

*Nb: no photo of envelope workings as I did it mentally.*

A great little puzzle! I went straight for areas and cancelled out pi, so you end up with a^2 : b^2 = 0.16 : 1 (a being small circle radius, b large circle), square root for a : b = 0.4 : 1 = 2 : 5

A nice maths break whilst on a day out at Gullivers Kingdome :-). Thank you

Lovely solution. My brain never seems to.instinctively chose the most concise method!

Is the relationship between the ratio of lengths on similar shapes and the ratio of areas on similar shapes not taught as part of scale factors? I’m sure it is in the Collins IGCSE book.

I took if for granted that if the ratio for the areas is 100:16 , the ratio for lengths is found by taking the square root of both sides, i.e. 10:4 = 5:2

Aye, it is. But my default setting is to overcomplicate these things. All valid solutions leading to the same answer.