If a cyclist has to push his bike up a hill because he has run out of puff, it is because his lungs can't supply the oxygen his muscles demand and/or because his heart can't circulate his blood fast enough to deliver the oxygen to his muscles. In all cases it is because something in his transmission was badly chosen, usually too optimistic.
All of this relates through the cyclist's maximum heart rate to his cadence. His theoretical maximum heart rate as a first approximation is roughly 220 minus his age, but that notional maximum is a number only professional athletes under medical supervision should approach. We generally speak in percentages because everyone's maximum heart rate is different. Ninety percent is possible for young men but you don't want to be in the 80 percents because there lactic acids are formed which will undo your good work. So for most of us, and especially for middle-aged cyclists, 75 per cent of age-related maximum is a reasonable heart-rate if you wear a heart monitor and pay attention to its output, otherwise lower still. If you're a careless rider, pick something even lower.
After heart surgery and physiotherapy, I found that the easiest way to relate this is through the cadence I could maintain on the flat for several hours. (I'm never on the bike for more than a couple of hours without a break.) Heavy tourers and cyclists who want to cross a small country in a day could pick that long day's best maintainable average cadence.
But all this knowledge will avail the cyclist nothing if the
tooth counts on his sprocket or his chainring are optimistically chosen. The cyclist must start with a rationally and realistically chosen consistently achievable cadence and match the tooth counts on the gears to that cadence.
On this page--
http://coolmainpress.com/BICYCLINGHebieChainglider.htmlyou will find tables for
--60x622mm tyre at a cadence of 60rpm
--26x1.6in tyre at a cadence of 80rpm
--26x1.6in tyre at a cadence of 120rpm
--26inx50mm Schwalbe Dureme at a cadence of 65rpm
You can interpolate between the tables with only two facts: the circumference (standing in for the rolling length) of your chosen tyre and your cadence, if different from those used in the tables listed above. You don't want gears below your ability to balance the bike, and you don't want gears that force you to pedal faster than your reasonable cadence maximum. Coasting on the downhills is a wonderful opportunity to enjoy the surroundings.
After all that work, I found that on my preferred 622mm rims clad with 60mm Big Apples at my preferred cadence of 40rpm I liked the ex-factory default of 16T sprocket with a 38T chainring, super for hauling heavy easels and paint in my all-hills countryside in West Cork. Since fitting a torquey electric centre motor, I've moved up to a 42T chainring.
