Jones-Ross formula

Are we there yet? Naismith re-imagined

No one loved maths at school. I hated it, but then I was rubbish at it. Some will doubtless say they loved it, but I think they mean they could do it well. Loved it? Really? Quite a few years later, I remember hearing the booming voice of an assessor on a Mountain Leader course asking a participant how long the next navigational leg would take based on Naismith timing calculations, and I watched the poor candidate whither before my eyes, willing the ground to eat them up at that very moment. We’ve all been there. “How long until we get to the end / pub / summit / car?”. I know I’m quite guilty of standard responses of “just another 5 minutes”, or “it’ll be about half an hour”. If maths was never fun at school, it’s certainly even less fun up a mountain.

But we can do better then that, and we don’t need to be scared of complex formulas, as smart phones can do all that maths trickery for us these days. Poor old Naismith gets blamed for quite a lot of people being completely turned off from planning routes, especially when a know-it-all chips in with suggestions that you really should be factoring in the Tranter / Aitken / Langmuir / Scarf corrections. Honestly, it would send the best of us to sleep, and I’d be first in the queue.

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William W Naismith, is I fear, a very misunderstood guy. His ‘rule’ was made back in the mists of time in 1892, based on a walking trip on the Crianlarich mountains of Ben More, Cruach Ardrain and Stop Binnein. Based on that trip, his formula stated that a walker should allow one hour for every three miles, plus an extra hour for every 2000 feet of ascent. Already we’ve run into a problem here, because we don’t have any modern maps in miles and feet. The UK went metric 55 years ago in 1965.

To convert the original Naismith figures, they roughly equate to 1hr / 5km, plus 1hr / 600m. Conveniently the metric conversion means an extra 10mins per 100m gained. Let’s quickly put that into practise on an example. If you’re planning a 10km walk, with 500m height gain, that would work out as follows;

Distance = 10km. Based on 1hr / 5km, this is 10km / 5, so 2 hours walking
Vertical = 500m. At 10 mins / 100m, this is 500 / 100 = 5. So 5 x 10mins = 50 mins
Total = 2 hours for horizontal, plus 50mins for the vertical, so 2hr 50mins in total.

Without doubt, Naismith sadly gets a poor press. There’s many a keyboard warrior furiously saying how he got it wrong for their walk, but most fail to realise his aim was to estimate time for a whole days walk, not a shorter section of it, let alone a single navigational leg. Other warriors type back things like “I go to the hills for a physical workout, not a mathematical one”. Naismith wasn’t trying to provide an answer for every route, condition, and terrain, but a generic tool to approximately calculate the length of a day out. And, nearly one hundred and thirty years later, it still does a pretty good job, in certain contexts.

Naismith works well on straightforward terrain, such as the Ranger path up Snowdon, yet for the scramble along Crib Goch, it’s necessary to double the timing suggested by the rule. That’s the single biggest limitation of the Naismith rule, in that it applies to hill-walking rather than technical terrain. His rule also assumes that you are reasonably fit, but there are no corrections built into his simple formula for the effects of adverse weather, carrying heavy loads, getting fatigued, complex navigation (night or poor visibility), taking rests, or more tricky ground underfoot such as bogs, steep slopes, screes, or thick vegetation. No one, even Carol Vorderman or Rachel Riley, wants a matrix for all those variables.

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The summary is that Naismith timings are generally quite ambitions, and only really apply to good paths on non-technical mountain routes. However the rule still lingers around today, and is actually enshrined in UK statute law, for the definition of trekking in the Adventure Activities Licensing Regulations (1996 & 2004). That has led to people using Naismith as a foundation, and making adaptions to it. There are four key ‘corrections’, that are briefly summarised as follows;

1) Tranter – this table allowed corrections to adjust for your fitness level and fatigue, as well as conditions underfoot, adverse weather, and carrying heavy loads (a whopping 20kg). Fitness was judged by the time a walker ascended 1000 feet (300m) over a distance of ó mile (800m). Naysayers criticise this method as it relies on a fitness test that is geared to short distance speed, rather than steady stamina, suggesting it means Usain Bolt is faster than Mo Farrah on a long distance, which is evidently wrong. As one reviewer suggested “One day when I was feeling particularly Aspergic I took to using Tranter’s variations. The feeling went away by lunchtime”. My maths allergic brain is already in meltdown.

2) Aitken (1977) – this method adjusted Naismith for the surface and ascent speed. It assumes 1hr / 3 miles (5km) for easy paths, tracks and roads. For other surfaces this speed is reduced to 1hr / 2.5 miles (4km). On both speeds Aitken added 1hr for every 2000ft (600m) of ascent.

3) Langmuir (1984) – he took a different tack, to adjust the speeds for descent, allowing for people going slightly faster on easy slopes (5-12 degrees), and slower on steeper slopes (>12 degrees). His formula was to subtract 10m / 300m descent on easy slopes, and to add 10m for the same descent on steeper slopes. Langmuir also tweaked Naismith timings for the typical slowest person in a group, to 1hr / 4km, plus 1hr / 450m of ascent.

4) Scarf (2008) – this was a leap forward, in that it allowed calculations for any speed, not just one as in all other ‘corrections’, and for the first time this allowed for mountain runners not just hill walkers. He introduced a Naismith ratio of 7.62 units of distance being equivalent to 1 unit of ascent. The formula produces an equivalent flat distance, which can be divided by the flat speed of that individual. To give a worked example, take a 20km route with 2000m vertical, the equivalent flat distance would be the horizontal 20km, plus vertical distance (2000 vertical metres / 1000 = 2) multiplied by 7.62. So, the equation is 20 + (2 x 7.62) = 35.24 km. Divide this by a flat average moving speed of 5km/h, and this route would take (35.24 / 5) = 7hr 3mins. The huge advantage of the Scarf method is that it can be adjusted for any individuals chosen speed on the flat.

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Very much hoping that you’re still awake at this point, and if it didn’t make too much sense so far, don’t worry! Here’s where it gets easier. The long and short of it, is that rather then correct Naismith’s rule, which evidently isn’t working for many people, there was clearly a need for a completely new system of calculation. That system needed to incorporate the range of speeds that was offered in the Scarf system, as well as to allow for different ratios of ascent to descent speed for various mountain user groups. While researching and writing a new guidebook on the Tour du Mont Blanc (Vertebrate Publishing, April 2020), a new innovative system was devised for calculating timings on legs of a mountain journey; the Jones-Ross formula.

Four typical mountain user groups were identified; walkers, trekkers, fastpackers, and mountain runners. It’s up to the user to decide what category they identify with best, but the characteristics are the most important factor. A mountain runner will typically descend twice as fast as they climb, whereas a walker ascends only very slightly slower than they descend. The figures used were based on timing data from various legs, whilst guiding the Tour du Mont Blanc over 50 times. Obviously this is along well established trails, with few difficulties. Timings are as follows;

users

These figures aren’t set in stone, and as a mountain user, it’s over to you to tweak them to adjust for your speed, and qualities as an ascender and descender. The speed isn’t just a measure of fitness, but also a result of your abilities and qualities of movement over simple mountain terrain. It assumes fairly established trails rather than open mountain terrain, normal conditions without fresh snow or poor visibility, and no degradation in speed due to fatigue. If you find yourself moving slower, it’s easy to tweak the workings to cater for that.

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The actual Jones-Ross formula is quite straightforward;

Time = (Distance / Flat Speed) + Adjustment for ascent + Adjustment for descent

This is calculated as follows…

formula

For the complete 169km Tour du Mont Blanc, the Jones-Ross formula equates to totals of Walkers taking 82hr35 (over 10 days), Trekkers 66hr15 (over 8 days), Fastpackers 49hr15 (over 6 days), and Mountain Runners 32hr45 (over 4 days).

It’s not been mentioned before, as hopefully you’ll have realised why it’s so important to have a good calculation method for timing sections of your journey in the mountains. Use it to increase your enjoyment, comfort, and safety; i) The formula allows you to plan realistic length days, and any associated hut or overnight accommodation on multi-day trails. ii) It improves your navigational accuracy, and therefore safety, with less potential for any mountain rescue call outs, due to being in the wrong place at the wrong time, or being overdue for your return / finish. iii) You develop a customised itinerary for your strengths and weaknesses, instead of being told one single time that applies to all.

So there you have it, a new formula for calculating times of various legs of mountain trails, with the ability to adjust speeds, both ascent and descent capabilities. It’s not for me to dictate what these figures are, but a tool for you to adapt to your / group performance. Over to you!

Kingsley Jones
UIMLA International Mountain Leader

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Article notes

  • Formula nerds – The formula was devised by Kingsley Jones (@KingsleyJones) and and Stephen Ross (@LakelandMM), so you know where to send hate mail for making you do some maths. Sorry for lying at the start of this article that maths was never fun. Now you know it just could be!
  • Buy the book – Tour du Mont Blanc, the first guidebook to be published, with the timings calculated using the Jones-Ross formula, was released in April 2020 by Vertebrate Publishing. You can buy a copy direct from the publishers (link), signed copies from Icicle (link), or from Amazon (link). here’s also an accompanying map for the TMB, printed on a convenient 1:40k scale (link).

TMB book and map

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