Please explain gear ratios to me

[GeoffBroom]

I understand the word ratio and I understand the word gear because I have a degree in both physics and Mathematics but I feel a fool when I see gear ratios written down.  How does a 22 tooth ring and a 34 tooth sprochet give a ratio of 16.8 ( I think on a 26 inch wheel) and a 44 ring and 11 sprocket give a ratio of 104 ?
Also what is a 100" gear ratio.  I always thought a ratio was dimensionless.  And where do percentages come in to the equation?

I really want to know so I can compare my bike with those I read about in articles.
Please help an old fool.

[julk]

The distance travelled by a chain driven fixed sprocket rear wheel for one revolution of the chain ring =
pi  x  wheel diameter  x  no. chain ring teeth  /  no. sprocket teeth.

The 'gear ratios' you give refer to a 26" wheel and leave pi out of the calculation.
The 'gear ratios' calculated in this way are useful to evaluate any combination of derailleur cogs you might want to try on your bike or to understand the gearing available from a hub gear system.

16.8" is a very low gear.

104" is quite a high gear.

I hope this helps.

[Wild]

The way in which ratios are worked out is quite simple.  In most cases cyclists will however, talk in gear inches.

To work out the ratio simply divide the number of teeth selected on the front sprocket by the number of teeth being used on the rear sprocket.  So for example if you are riding on a front sprocket with 42 teeth, and are using a rear sprocket of say 15 teeth, then the ratio is 42 divided by 15 which would give a ratio of 2.8

If you wanted to express this in gear inches, then the diameter of the wheel comes into the equation.  So keeping the same selected cogs stated above, and riding on 26" wheels, the equation would be:
42 divided by 15 x26.  The gear inches would be 72.8"

If you consider the road bikes in the Tour De France as another example, then a rider racing on a front sprocket with 52 teeth, and a rear sprocket of 11 teeth on 27" wheels would be pushing massive gear inches, 52 divided by 11 x27 which equals 127.6 gear"  Fit men are they not.

I hope this helps.

[Wild]

Hi Geoff, Just to add to my earlier posting.  If you look at the example that you used in your posting, then this is how the figures work:  You are actually stating gear inches.
22 tooth Front Ring Divided by 34 tooth rear sprocket x 26" dia wheel.
Therefore 22 divided by 34 x 26 = 16.8" gear.  Obviously really easy for steep climbs.  If you want, simply count all the teeth on the individual sprockets on your bike and work out all the gear inches.  You will normally find that there are overlaps when selecting the front ring to use in relation to those at the back, but this is common.
The 'Rohloff' rear hub used on many of the Thorn cycles at SJS cycles, are geared to give a small change in gear inches between each gear, and therefore efficient cycling
Cheers.  Dave.

[ians]

So - a gear of 104" means that the wheel will travel 104" for one revolution.


ff

[Joatamon]

The Gear Inch figure is the diameter of the wheel you would be pedalling if the pedals were connected directly to the axle.  I think.  Here's a good explanation:

http://www.ctc.org.uk/DesktopDefault.aspx?TabID=3521

The Thorn PDF "Living With a Rohloff" includes a table showing typical ratios of derailleur bikes as compared with the Rohloff - very interesting.  Have a look.

[Wild]

I think that the term gear inches can be confusing when talked about.  The term inches (") refers to the distance travelled following one full revolution of the pedals in a given gearing.

So as an example, using a front ring of 42 teeth with a rear sproket of say 21 teeth on 26" wheels, the distance travelled would be:
42 divided by 21 x 26 = 52".  52" being the distance travelled by the wheel after one revolution of the pedals.  I think thats how it works anyway.

[stutho]

quote: The term inches (") refers to the distance travelled following one full revolution

Sorry Wild but that isn't correct – not your fault however.  Whoever invented the term “gear inches” needs to go back to school!

Gear Inches  = Diameter of wheel * chain ring / rear sprocket.  
It is proportional to distance travelled per pedal rev but is NOT equal to it (for that you need to multiply by Pi).  Note the term "Gear Inches" is bad mathematics - A gear is dimensionless.

Development Meters = Diameter of wheel * Pi *chain ring / rear sprocket.  
This IS the distance travelled per a pedal rev.  As a professional engineer this system makes far more sense to me unfortunately we don’t use it much in the UK.

[Wild]

Hi Stutho, I think you are no doubt right, and i'm wrong.  I wondered after i had posted the info on gear inches as to whether or not i'd made a mistake.

But look at your formula compared to mine.  Using the figures that i gave as an example and using your formula this would read.

Gear inches = Dia of wheel x chain ring divided by rear sprocket

Therefore 26 x 42 divided by 21 = 52.  This is the answer that my formula gave too.  Do both work.  What do you think?

[GeoffBroom]

Thanks everyone for your input. It seems that a little pi spoilt the pudding for me.  Just shows doesn't it.

[stutho]

Hi Wild, Yes your formula is correct (it is identical to mine!) BUT gear inches don't equal travelled distance - hence my previous post.  Gear inches are only any good for comparing gears not for calculating speed or distance travelled.

[Vangelis]

I wanted to get a general idea of the gearing of diverse bikes e.g.Brompton with hub/sprocket combination and 16' wheeel
Thorn cyclosportive with derailer 30 gears, MTB with Rohloff, Raven tandem with Rohloff and 26' wheel etc.

I decided to transform it all to distance traveled on a single pedal revolution. ( chainring/sprocket(*ratio for internal hubs)*wheel circumference)
This distance * cadence = speed!

This gives a fairly good idea about the behaviour of your gear ratios!

It makes it easier to compare a would be gear ratio, to something you have already experienced.

It works for me...

[graham]

In most of the rest of the world, they use 'development' which is the distance travelled by the bike per revolution of the pedal.

In good old blighty, we still think a 'safety bicycle' is a passing fad and are waiting for the 'penny farthing' to come back into fashion. A penny farthing has the crank directly attached to the big front wheel which is the driven one (of course). The bigger the wheel the further you go per turn of the pedal. So your gear size expressed in inches is the equivalent wheel size assuming you were still riding a proper bike (penny farthing) instead of one of those silly things where you sit between the wheels and are connected to the drive via a chain.

As previously stated, multiply the gear size in inches by Pi (3.142) and then convert to metric to get how far you'll go per turn of the pedals. Further simple calculations will give you your speed in KM/h or MPH for whatever gear you're thinking of.

As another guide (from memory so I could be wrong) a gear of 70" would mean you're going at about 18mph if you're pedalling at about 90rpm. We stop pedalling our 104" top gear at something over 30mph and in bottom gear (20") you'd be quicker walking.

[freddered]

Stutho summed it up perfectly.

It's easier if we don't discuss "Why Gear inches?" and just use them for direct comparison when selecting a new bike or gear setup.

Teeth on Front Chainring <DIVIDED BY> Teeth on rear Sprocket <MULTIPLIED BY> Diameter of Wheel in inches.

Example 48/12*27" = 108" (A reasonably high Top Gear)

Example 62/10*17" (Moulton)= 105" (Similar but achieved with small wheels and large Chainring).


At the other (low) extreme (MTB-type gearing)

Example 24/34*26" = 18"

Most people will find 110" top gear adequate for quick downhills and an 18" bottom gear as low as is practical to pedal without wobbling off.  In 18" gear you will most likely be doing <3mph with your legs spinning like a crazy person.

My Rohloff Hub has top of 94" (I spin-out at around 35mph) and a bottom of 18" (I can carry luggage up steep hills).

Because I knew what inch gearing I used on my other tourer it was easy for me to select the correct gearing for my Rohloff (Chainring and sprocket combination)using the chart provided on the Thorn Website.