Treebeard's Homepage : Stumpers

Treebeard's Stumper Answer
15 March 2002

Gearing Up For a Ride

Winter is turning into spring. It's a perfect time to tune up your bike and go for a ride! The most confusing part of biking for kids is surely the gears. My old mountain bike has 3 chain wheel gears in front that turn with the pedals and have 28, 38, and 48 teeth. I also have 7 freewheel gears in back with 13, 15, 17, 20, 23, 26, and 30 teeth. That makes 3 x 7 = 21 combinations. How does this work? What is the shifting pattern from low (for climbing) to high (for cruising)? Which gear combination is in the middle? Would even more gears make biking easier?

This is a live image map. Click on a bike part name to get some good practical info from UtahMountainBiking.com. Bicycles are efficient and elegant machines, but they are only "simple" for an engineer. Kids find the gears complicated, but I'm more confused by the hidden bearings in the headset and the bottom bracket. By the way, this bike is way too clean to be mine! The real stumper is to understand your own bike. I put a paper clip on one gear tooth of each cog and counted my way around one by one. Then find your calculator and figure the gear ratios.


My bike's low granny gear uses the small cog in front and the large rear cog for a gear ratio of 28 / 30 = 0.93, so every full pedal is less than one full turn of my wheels. That's power for climbing. My highest gear has a ratio of 48 / 13 = 3.7, so I go four times farther with every pedal, good for cruising. But in between, all successive shifts involve using both front and rear shifters in a complex pattern. It's easy enough to find a gear that works, but for competitive biking, those complicated shifts might make a split-second difference. Keep reading for the details.

Notes:

It's easy to figure the gear ratios on your bicycle. Count the number of teeth on the front chainrings and the rear freewheel cogs. If your front chainring has 48 teeth and the rear freewheel cog has 24 teeth, then every time you pedal around once, the chain moves around 48 teeth. That means your 24-tooth rear cog - and your rear wheel - goes around twice. That would provide a ratio of 1-to-2 or 1:2.00. If your rear cog has 12 teeth, the ratio would be 1-to-4 or 1:4.00, so you would go farther with every pedal. A higher ratio means your effort is applied over a greater distance, which is good for cruising but not for climbing. The gear ratios aren't as neat on my bike, but it works the same way.

I started to figure my gear ratios with a calculator, but then I found Barry Masterson's handy Gear Inch & Shifting Pattern Calculator web site that does it all at once. The hard part is counting the teeth on each gear cog, but you only have to do it once.

First I submitted the data for my bike:

The result is a confusing table of data, but it's simpler than it looks. The important numbers are the gear combination (CRxFW) and the gear ratio (RT). All 21 gear combinations are shown in order from low (for climbing) to high (for cruising):
 Wheel Diameter: 26.00 inches
 Gears: 28/38/48   13-15-17-20-23-26-30
----------------------------------------------------------------------------------
 SP  CRxFW    GI      GIdf      DI      diff      DF        PRPM      RT      MPH
----------------------------------------------------------------------------------
 1>  28x30   24.27   15.38%    76.24   11.73    6' 4.24"   831.10   1:0.93    4.33
 2>  28x26   28.00   13.04%    87.96   11.47    7' 3.96"   720.29   1:1.08    5.00
 3>  28x23   31.65    4.05%    99.44    4.02    8' 3.44"   637.18   1:1.22    5.65
 4>  38x30   32.93   10.53%   103.46   10.89    8' 7.46"   612.39   1:1.27    5.88
 5>  28x20   36.40    4.40%   114.35    5.03    9' 6.35"   554.07   1:1.40    6.50
 6>  38x26   38.00    9.47%   119.38   11.31    9'11.38"   530.74   1:1.46    6.78
 7>  48x30   41.60    2.94%   130.69    3.84   10'10.69"   484.81   1:1.60    7.43
 8>  28x17   42.82    0.31%   134.53    0.42   11' 2.53"   470.96   1:1.65    7.64
 9>  38x23   42.96   11.74%   134.95   15.84   11' 2.95"   469.50   1:1.65    7.67
10>  48x26   48.00    1.11%   150.80    1.68   12' 6.80"   420.17   1:1.85    8.57
11>  28x15   48.53    1.79%   152.47    2.72   12' 8.47"   415.55   1:1.87    8.66
12>  38x20   49.40    9.84%   155.19   15.27   12'11.19"   408.26   1:1.90    8.82
13>  48x23   54.26    3.21%   170.47    5.46   14' 2.47"   371.69   1:2.09    9.69
14>  28x13   56.00    3.78%   175.93    6.65   14' 7.93"   360.15   1:2.15   10.00
15>  38x17   58.12    7.37%   182.58   13.45   15' 2.58"   347.02   1:2.24   10.37
16>  48x20   62.40    5.56%   196.04   10.89   16' 4.04"   323.21   1:2.40   11.14
17>  38x15   65.87   11.46%   206.93   23.70   17' 2.93"   306.20   1:2.53   11.76
18>  48x17   73.41    3.53%   230.63    8.13   19' 2.63"   274.73   1:2.82   13.10
19>  38x13   76.00    9.47%   238.76   22.62   19'10.76"   265.37   1:2.92   13.57
20>  48x15   83.20   15.38%   261.38   40.21   21' 9.38"   242.41   1:3.20   14.85
21>  48x13   96.00    0.00%   301.59    0.00   25' 1.59"   210.08   1:3.69   17.14
----------------------------------------------------------------------------------
 SP  CRxFW    GI      GIdf      DI      diff      DF        PRPM      RT      MPH
----------------------------------------------------------------------------------

Here's what it all means, and how it's calculated. Try to figure out the formulas!
(CR=front chainring teeth, FW=rear freewheel teeth, WD=wheel diameter.)

SP     Shifting Pattern from 1 (low, for climbing) to 21 (high, for cruising).
CRxFW   Each ChainRing x FreeWheel combination (front x rear)
GI   Gear Inches is the traditional way to measure bike gears, calculated as Gear Ratio x Wheel Diameter. It's a biker's way to compare gears combinations on bikes with different size wheels. It's also the size of the front wheel on an equivalent old fashioned Penny-Farthing or high-wheeler bike. (Does this have something to do with torque and moment arm and mechanical advantage?)

GI = WD x (CR / FW)
GIdf   Percent increase between this GI value and the next GI value.
DI   Distance traveled in inches with a single pedal rotation

DI = GI x pi
diff   Difference in inches between this DI value and the next DI value.
DF   Distance traveled in feet with a single pedal rotation

DF = DI / 12
PRPM   Pedal Rotations Per Mile

PRPM = 5280 / DF
RT   Gear Ratio

RT = CR / FW
MPH   Miles Per Hour in this gear with a cadence of 60 rpm. Cadence is a bikers' term for how fast you're pumping. The ideal is to always keep peddling at the same rate and use the gears to adjust the terrain. My own cadence is far from ideal!

MPH = (DI x 3600) / (12 x 5280)

Really, all these numbers just show the same basic gear ratio data in different ways. In my 12th gear (for example), I'm on the middle 38-tooth cog of my front chainwheel and the middle 20-tooth cog of my rear freewheel. The gear ratio is 38 / 20 = 1.90. When I pedal around once, the chain will move around 38 teeth, so my rear 20-tooth cog goes around almost twice, rotating the rear wheel 1.90 times. I'll travel about 155 inches on my 26 inch wheels, or almost 13 feet with every rotation of my pedals. I'll have to pedal 408 times to go a mile in this gear, and I'll go about 9 miles per hour if I keep pedalling at that cadence of one turn per second (or 60 rpm).

That's mostly trivia for hard-core bikers. More important is that this gear is less than 2% different from my next lower gear, but almost 10% different from my next higher gear. It's not worth shifting down for such a small advantage, but it will make a big difference if I shift up.

My truck has a 5 speed stick shift, with the simple shifting pattern written on the shifter knob. It's more complicated with my bike. Here is my bike's shifting pattern, with the gear cogs numbered 1-3 and 1-7 from inside-to-outside like they are on my shifters:

My Bike's
Shifting Chart
Small
1 - 28
Medium
2 - 38
Large
3 - 48
1 - 30147
2 - 262610
3 - 233913
4 - 2051216
5 - 1781518
6 - 15111720
7 - 13141921

I expected my bikes' shifting pattern to be complicated, but not this complicated! Except for the extremes, all successive shifts involve using both front and rear shifters in a complex pattern. But ten of my shifts give less than a 5% advantage, and several combinations are virtually identical. And I shouldn't use the inner-outer combinations like 28-13 and 48-30 that can grind down my cogs and chain. If I had more gear combinations, there would be even more repeats. More gears are not better unless they are the right gears!

It's interesting that my gear in the middle has a ratio of {0.93 + (3.69 - 0.93) /2} = 2.31, which is between my 15th and 16th gear combinations, near the high end. Sure, I need more control when climbing hills.

Usually I think of my front gears as 1-low (for hills), 2-medium (for flats), and 3-high (for downhill). Then I use the rear shifter to adjust until I run out of room and need to change the front. Sometimes on steep hills I wish I had a lower front gear, but by then it's easier to walk and push. I tried shifting both gears at once in the mathematical order, but I can't do it very well so it's not worth it for me. In fact, racing bikes have (or used to have) the shifters down on the bike frame which would make it impossible to shift both front and rear gears at once. Maybe that's why racing bikes usually only have two front chainwheel gears. Road racers have fewer gear combinations, but they can shift fast in a meaningful pattern to the right gear using just one shift at a time in a crossover pattern like this:

Racing Bike
Shifting Chart
Small Large
Large1-
22-
33-
445
5-6
6-7
Small-8

Competitive bikers must spend a lot of time thinking about the ideal gear combinations for their style of riding. I recieved this email from Cynthia about a conversation with a recent inductee into the Mountain Bike Hall of Fame in Crested Butte, Colorado:

I asked him how consciously and constantly serious bike riders think about their gear ratios, and he said that at every moment, the cyclist knows precisely what gear he is in. Precisely. They even use certain ratios as part of their strategies and sometimes try to shift very subtly so that the opponent doesn't know. A guy will say, "I went up that hill in a 42x19," and a knowledgeable cyclist will understand a hundred things about the climb and the way it was handled just by those numbers. (Gosh, I never realized this!) He said for example, someone will say, "I did that descent in a 53x13," and that instantly tells you it must have been a very steep hill, maybe a tailwind also, and definitely the guy was moving fast. It's like a whole numerical language I never understood. And they skip around, playing the gears like a musical instrument, intuitively knowing exactly what they are doing. Now I understand [his] frustration with me when we ride together -- he will take one look and tell me I'm in a very stupid gear. It also helps me to understand something he told me about bike racing: Assuming you have the right fitness and body for it, "It's 90% between the ears, and 10% being able to withstand pain a little bit longer than the next guy."
Fortunately, the rest of us can go for a bike ride in the country on a beautiful spring day and usually find a gear that is good enough!

Here are some links for further Web research on bike gears:

Back to Stumper


Last modified .

Copyright © 2002 by Marc Kummel / mkummel@rain.org