A co-worker and myself are having a friendly bike / run contest. What is a good ratio for miles ridden vs. miles run? I know it depends on pace. But I'm curious what a generally accepted range of ratios might be.
If you want to do a competition which gives a (good runner, mediocre cyclist) even chances against a (good cyclist, mediocre runner) then you might use the same ratio as the Ironman triathlon.
It has a 112-mile (180.25 km) bike and a marathon 26.2-mile (42.2 km) run which puts the ratio of distance cycled/distance run at roughly 17/4.
Based on questions in comments, let me clarify.
I think 17/4 is probably well on the low side for a long race. I've done a double metric century with 3km of climbing and ridden the next day. I doubt I could run the day after a marathon on anything but a gentle downhill slope.
For shorter distances though, it seems more reasonable. If I push myself for 10km (6.2 miles), I expect it would take more out of me than if I hammered for 42km (26 miles) miles, but that's because I'm a better cyclist. I think the hammering would take just as much extra out of some runner friends of mine.
Weak cyclists can benefit from drafting a strong cyclist in a way that weak runners can't, but I think I'm strong enough to drop a strong runner/weak cyclist so that isn't a factor. I would definitely try to have the ride first though, because otherwise the run might take enough out of me that I would have more trouble dropping a tail.
The short answer to your question is "probably between 2.5 and 3.5 miles; the faster the runner the closer to 2.5 while the slower the runner the closer to 3.5." The long answer is quite long.
As you already noted in your question, the full answer depends on pace; but it also depends on the surface being run or ridden on, whether it is up hill, down hill, or flat, whether it is windy or calm, and how "aero" the cyclist is. That said, if we are talking about a road bike on a firm flat surface under calm conditions, it is possible to make some estimates of the “equivalent” distance. The simple answer is given in the chart below, which relates the distance one must cycle in a given amount of time to equal the effort in running a mile at a given pace.
In this chart, the x-axis is the run pace for a mile and the y-axis shows the distance one must cycle in the same amount of time. For example, if you can run a mile in 4 minutes, an equivalent cycling distance if you were riding "in the drops" would be to ride about 2 miles in the same 4 minutes (that is, to average 30 mph). If you ran a mile at a 10 minute/mile pace, it would take you 10 minutes to run a mile and an equivalent cycling distance would be to ride about 3.5 miles in the same 10 minutes (that is, to average about 21 mph).
How was the chart calculated? In running, the power needed to run X meters/second is approximately X watts/kg of body weight (cf. Hall, Figueroa, Fernhall, and Kanaley, 2004 or McArdle, Katch & Katch, 2007), while the power needed to ride at Y meters/second varies with Y^3 (cf. Wilson and Papadopoulos, 2007). Thus, if we know the amount of time it takes to run a mile, the weight of the individual and the bike, and some other simplifying assumptions we can make the conversion. An exact estimate of running power will depend on the individual's gross metabolic efficiency, the runner's energetic economy (that is, how economical in an energy sense is the individual's running style), and the runner's resting metabolic rate, but it is commonly observed that for a normal-sized individual without unusual abilities or disabilities, energy expenditure while running on a firm flat surface will range between about 0.8 to 1.1 kcal/kg/km. Thus, a common rule of thumb is that running requires about 1 kcal/kg/km. There are 4.2 joules per kilocalorie but gross metabolic efficiency usually lies in the range of 19-25%; if we assume a value of 23.8% for GME, we can derive the happy simplification that (running speed in meters per second)*(body weight in kg) is a reasonable estimate of power in watts; or speed in meters/second ~ watts/kg.
For this chart, I first calculated the speed in meters per second for various running paces along the x-axis, from a 4 minute mile (= ~6.7 m/s) to a 10 minute mile (= ~2.7 m/s). By the running rule of thumb, this requires roughly 6.7 watts/kg down to about 2.7 watts/kg. I then presumed “average” values for the coefficient of rolling resistance (Crr) of .005, flat ground, no wind, and a CdA (drag area) of the cyclist of 0.25, 0.33, and 0.42 sq. meters (typical of a road bike rider "on aerobars", "in the drops", and "on the hoods") and calculated the cycling speed for a cyclist able to produce between 6.7 watts/kg and 2.7 watts/kg. The running pace defines a total amount of time needed to run a mile so the question becomes "how far must a cyclist producing this much power ride in the same amount of elapsed time?" Knowing the speed and the elapsed time lets us calculate the distance. That is what is shown on the y-axis. Note that for the fastest runners, the equivalent cycling distance approaches 2 miles while for the slowest runners competing against the most aerodynamic cyclists, the equivalent cycling distance approaches 4 miles. Thus, for a friendly race between you and your co-worker, if neither of you is a world-class athlete, a reasonable range is 2.5x - 3.5x the running distance to handicap a race so that you both finish in approximately the same time.
If the cyclist were a bit more aerodynamic (for example, if the cyclist used aerobar extensions and subsequent CdA was below 0.25 m^2) the curve would move upward and the rider would have to ride farther in the same amount of time (that is, faster) to match the runner's energy expenditure. If the cyclist were a bit less aerodynamic (for example, if the cyclist's position was more upright, or was wearing loose and flappy clothing) the rider would be expending energy at a faster rate so the curve would move down.
A rough validation of this approximation is that, anecdotally, people who both run and cycle say that running 10 km in about 42 minutes is roughly about as hard as riding 40 km in an hour. That's a running pace of about 6:45 per mile, and the chart equates that to cycling about 2.75 miles in 6:45, or about 24.5 mph -- which is 39 km/h. Our rule of thumb for equivalence is "between 2.5 and 3.5 times as far; closer to 2.5 times if the runner is fast and closer to 3.5 for a slower runner." Running a mile at a 6:45 pace is moderately fast, so we would expect the equivalent multiplier for cycling distance to be closer to 2.5 than to 3.5 -- as you can see, the predicted multiplier is 2.75 so the rule of thumb appears to work. Further evidence can be gathered from duathlons or triathlons. Below you can see data from the 2010 Ironman World Championship in Kailua-Kona, Hawaii. The Ironman comprises a 2.3-mile ocean swim, a 112-mile bike ride, and a 26.2-mile marathon. The scatterplot matrix below shows the swim, bike, and run times (in hours) for each of the finishers.
If we ignore the swim leg and concentrate only on the relationship between the run and the bike, we can prorate them to equivalent 10k run times and 40k bike times, as is done here:
What these latter two charts clearly show is that equivalent bike distance will depend heavily on run pace but also on the individual. There is a great deal of "scatter" in the scatterplot, which indicates that while the rule of thumb is reasonable it is not precise.
The problem here is that the effort spent is differently.
A "good" 45 y.o. runner capable of qualifying for Boston Marathon might do a 3:30 marathon (26.2) while a "good" biker on reasonably flat course might cover around 90 miles on a bike in the same time (87.5 miles @ 25 mph). The marathoner would likely be very much done for the day while the biker would likely still have some left in the tank.
Take it to one extreme, run a 100m sprint as hard as you can. Have your buddy do a 400m lap at the velodrome. The biker will recover MUCH faster. At the other extreme (running/biking for 12 hours+) things are even murkier. As HeltonBiker points out, the ration is not linear (I doubt it's even quadratic). There is also a level of fitness at issue here. Because your are, in essence, multiplying effort small differences between levels of fitness will be blown way out of proportion.
For your contest, you could semi-level the playing field and make it about semi-equivalent effort spent. My $0.02 would be to get heart rate monitors (I would even step it up and get GPS devices) that take into account type of activity, heart rate, etc to generate # of calories burned. Person that burns the most over a specified period wins.
If you are looking to do a single race rather than a longer contest, then you should look at average times for equivalent races. The various triathlon distances would be a good start, but I would discount the run times by 5-10% as the runner will not be doing a swim and bike first!
While the idea of comparing the distances of the cycling and running portions of the Ironman triathlon might seem appealing at first, you'll quickly see that the times of those distances don't correlate at all. The cycling portion of the Ironman typically takes participants about twice as long as the running portion, varying somewhat between competitors but always hovering around twice as long for cycling.
It might be better to compare world records set over an equal amount of time. As it happens, there are world records held for the maximum distance covered in an hour for both cycling and running. Currently, the hour record for running is just a smidge over 21 km. The cycling hour record is a little bit trickier due to the fact that it has been set on a number of different types of bicycles, many of which are illegal in competition today. However, the hour record on a standard drop-bar road bike with a steel frame, rounded tubes, and wire spokes (much like the one most of us might ride on the road today) is held by none other than the great Eddy Merckx and is a bit over 49 km. This suggests an approximately 5:2 ratio between cycling and running.
However, when comparing calories burnt over an hour of cycling verses an hour of running, you don't get the same approximate 5:2 ratio. It's a little closer to 2:1. This ratio actually does appear to be nearly linear, at least until you approach professional level speeds where the calorie calculator I was using topped out. NB: I was calculating the calories burnt by a 190 lb man. The values and ratios come out wildly different as body weight approaches the very thin and the very heavy.
So I guess it really depends on the metric you'd like to use for comparison.
I am aged 50+ and cycled about 10,000 km (from a 36 km total daily commute) in the last 15 months.
Only very recently (after all that) have I found that I have enough strength in my legs and 'feet' and heart, to run any continuous distance at all.
So for my body, actual running (on my toes and not my heels) was unquantifiably harder (and required a year of cycling just as training, to get started on running).
You could just try it once to get baseline values. E.g. you both go and do your thing for an hour, then come back with the results. Then you can use that as your ratio.
To prevent one another from gaming the calibration you could keep a running average that you both try to beat.
Of course the bike will always win for the reasons stated above, and because on a bike you can increase efficiency by spending money, whereas a runner cannot.
A more fair contest would be to compete in each others' sports and compare results fairly. E.g. "I beat you by 15% when you biked, and you beat me by 12.3% when I jogged." Or whatever.
In Bulgaria we have an annual 100-km race in Vitosha mountain. Speed walkers/runners set off at midnight, while cyclists start at 6 AM. The terrain asphalt and mountain trails, with slopes up and down. This is considered fair.
I, personally, pity the runners, for reasons mentioned in other answers here - walking 100 miles - heroic torture, cycling 100 miles - pleasure.