Before I get into this let's define the difference between "Single Speed" and "Fixed Gear", because sometimes people get these two terms confused because both setups commit you to use only one gear.
Most people are familiar with the "Single Speed". This setup uses a freewheel (pictured above) allows you to freewheel or coast. For example you can ride a bicycle down a hill without having to pedal down it.
Less familiar to people is the "Fixed Gear". With this setup the gear (pictured above) screws directly on the hub (It's fixed to the hub) and there is not a freewheeling mechanisim. For example if you go down a hill, your pedaling down the hill. as long as the rear wheel is turning, the cranks are turning. You can not coast.
Both single speed and Fixed have their Pros and Cons, but you have to ride both these bicycles differently than multi-speed bicycles because you can't shift gears when things get tough. You have to develope stratagies using your momentum to get over the top of all kinds of hills. You have to develope a flow so to speak.
There are basically 3 types of hills, Gradual hill, Rolling hills, and steep hills.
A gradual hill, is a hill that for the most part you ride seated from the bottom to the top.
Rolling hills are the meat and potatoes of the single speed/fixed gear experience. You use the momentum of the descent of the previous hill to propel you over the top of the next hill. Right at the beginning of the climb you will start losing speed/momentum. You have to mentally calculate when and how much power to the pedals. sometime you gradualy apply power, and sometime you need to stomp. The calculations will include current fitness, speed, and length climb.
Steep hills are just steep hills. These hill require brute force to finish. If the hill get too steep you can always tack or switchback up the hill, provided you don't have on coming traffic.
Beware though, a rolling hill can become a steep hill if you choose a gear that is too large. Conversly a steep hill will become a rolling hill the more fit, and single speed/fixed gear savy you become.
So how do you pick a gear? You pick a speed you want to average, at a cadence you find comfortable. The last time I rode 100 miles I wanted to average over 20mph on my fixed gear. I found 48x16 (or 3:1...I could have also ridden 45x15, 42x14, or 51x17 with the same 3:1 result) at a pedal cadence of 85 rpm, produces 20.1 mph. This configuration sounds a little big, but I know I'm going to make a few stops so I'm going to have to ride faster than 20mph in order to make up for lost time at a food stop. I also have to think about the hills on the route. I was familiar with 48x16, and I've been able to ride up any hill in the area without too much trouble. I rode the 100 miles in 4 hours, 52 minutes. I remember having 2 hills that were trouble. I think I might have been able to push 50x16.
I'm thinking about doing a 128 mile ride in the near future. My fitness it's what it was, and I'd like to average 17.5 mph. I found a 42x16
at a cadence of 85rpm produces 17.6 mph.
So why do I look at 85 rpm when people say optimum cadence is between 90 to 100 rpm. Like I say, you have to ride these bicycle differently. At the bottom of some hills you might be pedaling at 190- 200 rpm, and some hills you might be slogging away at 30 rpm to get over. In my experience basing the target speed at 80 rpm works well.
So how do you calulate speed? Find the circumfernce of your wheel, for example the diameter of a 700x23c tire is 672mm, or 2111mm circumference. (672 x3.1415 =2111) convert to feet (6.926'). Now multiply that times your gear ratio to get your travel per crank revolution. for example 48x16 or 3 x6.926' =20.78'. To get MPH for a certain cadence, for example 85rpm. Multiply 20.78' x 85rpm /5280...convert from feet to miles, 5280 feet in a mile, = .33 now you have miles per minute, now multiply that times 60 to get miles per hour...20.07mph.
(672 x3.1415) /25.4) /12) *3) *85) /5280) *60 =20.07mph
(2111.08) /25.4) /12) *3) *85) /5280) *60 =20.07
(83.11) /12) *3) *85) /5280) *60 =20.07
(6.926) *3) *85) /5280) *60 =20.07
(20.778) *85) /5280) *60 =20.07
(1766.166) /5280 *60 =20.07