## June 15, 2009

### Maths Challenge #9 – Ring a–Ring a–Ringroad

The M25 motorway completely encircles London, and in Britain we drive on the left. So if you travel clockwise around the M25 you stay on the outside carriageway, whereas travelling anti-clockwise keeps you on the inside carriageway, which is shorter. But how much shorter?

The total length of the M25 is 188km (117 miles), so the advantage of being on the inside carriageway outght to be quite a lot – shouldn’t it?

Suppose that two cars travel around the M25, staying in the outside lane – one going clockwise and one going anti-clockwise. Also suppose that the distance between these two lanes is always 10 metres (to make it specific).

How much further does the clockwise van travel than the anti-clockwise one? You should also assume that the roads all lie in a flat plane.

### 9 comments by 2 or more people

1. #### Simon Whitehouse

If the M25 is a perfect circle, which it isn’t, then

C = πd

so, if you add 20m to the diameter then you will add 20π metres to the circumference. Which is approximately 62.83 metres.

The actual circumference of the M25 is irrelevant to the answer.

15 Jun 2009, 12:06

2. #### Duncan

I assumed a perfect circle as a simplification, worked out what the radius would be (28 647.9m), subtracted 10m and worked out the new circumference, and then calculated the difference. I also got 62.8m but Simon’s solution is much neater (and quicker)

15 Jun 2009, 12:16

3. #### joe

If the M25 was square, you’d be driving an extra 80m, 20m on each side. So the answer is presumably somewhere between 62.8 and 80, depending on the exact shape of the M25. Unless the M25 is triangular or an irregular shape in which case it’d probably be something totally different.

15 Jun 2009, 12:37

4. I point at Simon’s answer, deforming the M25 to a circle… But I think you’ve made a mistake. The distance between the lanes is only 10m, not 20m. Although seeing as 3 people didn’t spot this then I wonder at my own reasoning and imagine I’ll probably look a this tomorrow and slap my head.
Assuming that the circumference of 118km is the distance if one were to travel along the crash barrier in the middle of the mway, and that the lanes are 5m either side of this line (10m apart), then you can play with algebra and do as Simon did.
So the extra distance is 10*pi = 31.4m ? But yes as he says it doesn’t matter that the M25 is 118km.

15 Jun 2009, 18:28

5. Dammit! And I spotted my mistake within seconds of posting the comment. I requested deletion but the page “cannot be found” !! Delete it please! And this one. I hate it when I can’t spot diameter/radius subtleties :(

15 Jun 2009, 18:31

6. #### Sue

We had a postcard at the very beginning of the new year from friends who were on holiday in New Zealand. They travelled out there on Christmas day because they don’t have any family and apparently it’s much cheaper to fly then. On the postcard they’d written “The M25 on Christmas day is to be recommended!”

16 Jun 2009, 07:27

7. #### Eleanor Lovell

The difference is 20$\pi$ metres, or roughly 63 metres, for roads on the flat. It doesn’t depend on the length of the motorway, or how wiggly it is, provided the curvature is gradual enough for “10 metres distance between lanes” to be unambiguous.

As usual, congrats to @siwhitehouse : ) and others who got it right!

16 Jun 2009, 14:38

8. #### Limo Hire Nottingham

Overall it makes a lot of difference, in a car it may not, now lets look at how long it would take if we walked on the inside or the outside

16 Jun 2009, 17:14

9. #### Formal Dresses

We should drive on the shorter side during rush hour traffic. Switching may be a good way for the country to save on foreign fuel dependency.

06 Jul 2009, 19:59

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