# Cycling’s rules of two

**Modern Cyclist’s “senior” training guru Garth Coppin turns bike races into math lessons as he tries to conqueror his age-old bugbear, the hill.**

In my early years of entering cycling races, I realised that climbing hills wasn’t my strong point. While I could legitimately put it down to my age, my weight or the extent of my training (or more honestly – the lack of it), I decided otherwise and felt there must be some way to improve my hill climbing.

I was overjoyed to then find a cycling magazine which offered to reveal the secret to better hill climbing. I bought the magazine and the article said when approaching a hill I should ride to the front of the peloton and then as I went up the hill, I could slide from the front of the group to the back.

Having tried this technique a number of times I realised it didn’t work – or at least it didn’t work for me, probably largely for the reasons I had initially, conveniently ignored.

Having decided that the conventional methods to climb hills better on a bicycle didn’t work, and still ignoring the obvious reasons, I eventually decided there must be another approach – a mental one.

Not the power of positive thinking, although I am sure that could help, but mental arithmetic. I am sure most of us did that at school. Now being an accountant, there was the danger of this becoming a highly complex issue, but what if the maths centred on the number two? Surely even if we are physically tired, our minds could cope with that?

Let’s start with a simple but practical issue. You are cycling along at an average speed of 25 kilometres an hour and you start to go gently uphill and slow down to 20 kilometres an hour. You know from past experience that the uphill and downhill are much the same length. What speed must you go downhill to keep your average speed at 25 kilometres an hour? You might think that with your speed being 5 kilometres per hour slower uphill that you need to go 5 kilometres an hour faster downhill (in other words, 30 kilometres an hour). If you thought that, you would be wrong, so is it 35 or even 40 kilometres an hour? Actually the answer is 33.3 kilometres an hour.

How do I work this out? There are a number of steps involved.

Step one. In rough terms, work out how much slower you are riding than your desired average speed as a percentage. In this case, you are going five kilometres an hour slower than you would like. So 10% of 25 is two-and-a-half, so five is 20% (two-and-a-half times two). Is that easy enough so far?

Step two. Take this percentage and change it to a fraction, with the number one on top. In this case 20% is 20/100 which is the same as a fifth.

Step three. Deduct two from the bottom number in step two, so a fifth becomes a third (three equals five minus two).

Step four. Increase your desired average speed by a third. Divide 25 by three gives you 8.3 which is added to 25 to get 33.3 kilometres per hour. Hey presto - that is how fast you need to go downhill.

Now if you were going instead at 15 kilometres an hour up the hill, what would your average speed downhill have to be?

Step one. Fifteen is 25 less 10. If five is 20% then 10 is 40%.

Step two. Forty over 100 is two fifths or one over two-and-a-half.

Step three. Two-and-a-half less two is a half, so the number is one over a half, or two.

Step four. Increase 25 twice; if you increase it once you get 50, so to increase it twice you get 75. Yes you need to go 75 kilometres an hour downhill to keep an average speed of 25 kilometres an hour!

If you work it out, once you are riding at less than half your speed (in this case 12.5 kilometres an hour) you will never get to an average speed of 25 kilometres an hour by the time you reach the bottom of the hill – you will need to keep above the above speed for longer than the end of the hill.

Realising this is more of an incentive to go up hills faster than the advice given in the magazine – or more correctly not to go too slow up hills became of the impact on the average speed.

The other beauty of this approach is that it works for all cyclists, but probably even better for the average cyclist. For the top cyclists the approach outlined in the magazine probably works well for them and they might need to more mental maths because they probably want to be more precise.

It is for that reason that the examples given above use nice round numbers, which make the mental maths easier to do. For the average cyclist they would probably be happy knowing an approximate speed rather than a precise one.

For those who battle with this mental maths, we can make it easier, using the table below. This table is based on using 25 kilometres an hour, but works with other average speeds as well.

Speed below average speed |
Converted to 1/x |
Change to 1/(x-2) |
Downhill speed |

40% |
1/2.5 |
1/0.5 = 2 |
75 |

30% |
1/3.33 |
1/1.33 =3/4 |
43.75 |

20% |
1/5 |
1/3 |
33.3 |

10% |
1/10 |
1/8 |
28.1 |

5% |
1/20 |
1/18 |
26.4 |

The closer the current speed is to the average speed, the smaller the number in the third column (for example, at current speed being only five percent below the average speed, the number is 1/18). At this point the maths can become difficult, but at that speed the reason to do the calculation becomes less.

The beauty of this system is that it can also work the other way round, namely if you are going down a hill at a certain speed, what speed do you need to go up the hill on the other side of the same length to keep your average speed? The only change that is needed in the calculation is that instead of subtracting two, you add two.

Let’s look at an example. You are going 50 kilometres an hour downhill and want to keep an average speed of 25 kilometres an hour by the time you reach the top of the next hill. What average speed do you need to go up the hill?

Step one. Fifty is twice 25, so you are going 100% faster than the average.

Step two. 100% equals one over one.

Step three. One over one becomes one over three after adding two to the bottom number.

Step four. 25 less a third is 16.7 kilometres an hour.

The number two can also be used in another way. If you are riding, say, at 20 kilometres an hour into a wind and on the way back you ride at 30 kilometres an hour, what is your average speed?

Step one. Calculate the midpoint of the two speeds, which in this case is 25 kilometres an hour.

Step two. Calculate the percentage difference between the two speeds and the midpoint. Five (25 less 20) as a percentage of 25 is 20%.

Step three. Divide the amount in step two by 10, so 20/10 equals two.

Step four is where two comes in. Square the number in step three. Two squared is four.

Step five. Apply the number in step four as a percentage to the midpoint in step one and deduct this from the midpoint. Four percent of 25 kilometres an hour is one kilometre an hour, so the average speed is 24 (25 minus one) kilometres an hour.

While this can work easily in multiples of 10%, the percentages ending in five don’t have to be too daunting when squaring a number. The easy way to square numbers ending in five is to take the number and drop the five, then multiply the remaining number by the next highest number and add point-two-five. So 25% squared is two (25 without the five) multiplied by three (two plus one) equalling six and adding point two five equals 6.25%.

The table below will help those who want to make the calculation easier.

% difference between low speed and midpoint |
Divide by 10 |
Square the number |
Apply number as a % to midpoint of 25 |
Average speed based on midpoint of 25 |

5% |
0.5 |
0.25 (0*1+.25) |
0.1 |
24.9 |

10% |
1 |
2 |
0.2 |
24.8 |

15% |
1.5 |
2.25 (1*2+.25) |
0.6 |
24.4 |

20% |
2 |
4 |
1.0 |
24.0 |

30% |
3 |
6 |
2.2 |
22.8 |

40% |
4 |
16 |
4.0 |
21.0 |

50% |
5 |
25 |
6.2 |
18.8 |

60% |
6 |
36 |
9.0 |
16.0 |

This table shows the bigger the difference between the two speeds, the lower the average speed is in relation to the midpoint of the two speeds. While working out the average speed can be difficult, for an average cyclist a rough calculation should be sufficient - for 36% take a third of 25 which is eight point three, which is close enough; for 25% take a quarter, for four percent and six percent, take one over 20 or five percent.

So if mathematics is being used in the design of bicycles, why shouldn’t it also be used by cyclists themselves? Perhaps it could encourage them as they plod up steep hills.

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