The problem is that in bypass engines, the fan blades are much longer than the turbine blades that drive them.
Both compressors and turbines should rotate as fast as possible, without shock waves occurring at the tip - so linear tip speed has an upper limit, meaning the compressor with longer blades must turn much slower than the turbine driving it.
This is what the geared turbofan does, reduces the rotational velocity relative to that of the turbine so that both can rotate at their optimum angular velocity.
The forces involved and the required reliability make it not an easy design task, as noted in this answer , Does it improve fuel efficiency?
It does, when both the fan and the compressor can rotate at their optimum speed, and when the gearbox does not add much friction.
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The gearbox is located between the front fan and the rest of the engine.
It lets the front fan spin at a lower rate than the main shaft.
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The advantage of this setup is that the front fan can have longer blades to cover a larger cross-section area, while the low-pressure compressor and turbine blades in the core can spin faster to improve fuel efficiency.
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The purpose of the reduction gearbox is to improve engine efficiency.
Normally in a turbofan engine
the low speed turbine and fan are connected by a direct drive turbine shaft that requires the low turbine and fan to run at the same speed.
In a geared engine, the gearbox allows both the fan and turbine to run at their optimum speeds.
In this case, the turbine can run faster with fewer stages and airfoils, which increases efficiency.
The fan can run at a slower speed but at a larger diameter to push larger amounts of air at a slower velocity.
A simple formula for propulsive efficiency is: $$ N =
\frac{2}{1 +
V_e / V_o} $$ Where $V_e$ is the exhaust velocity of the engine and $V_o$
is the velocity of the aircraft or inlet of the engine.
As $V_e$ lowers to the point where it equals the speed of the aircraft,
efficiency N approaches 100%.
The larger slow moving fan in a geared engine allows large mass flow with low velocity which improves efficiency.
The low velocity also makes the fan quieter.
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To increase the propulsive efficiency (total efficiency is the product of gas turbine thermodynamic and propulsive efficiency)
we need to accelerate more mass flow less fast: this requires larger fan diameters.
See e.g. this answer which describes how the bypass air provides thrust.
The formula: $$ F = \frac{\text{d}}{\text{d}t} p = m \frac{\text{d}}{\text{d}t}
v = m \cdot a $$ shows that increasing the mass flow $m$ for the same amount of thrust requires less acceleration $a$ .
This can potentially improve the fuel economy, but there is a limit, increasing frontal area will also cause the drag to increase.
As said in this answer , tip speed is a limiting factor, so we need to reduce the spool speed.
The problem with decreasing the spool speed is that the work generated in the low pressure turbine (which is coupled by a shaft to the fan) needs a large diameter and more stages to get enough power from the gas to power the fan (let's not get too technical in discussing blade loading).
A large low turbine diameter is not very effective for the bypass flow of the engine (how do you get the bypass air around it), therefore the spool speed of the low pressure turbine is increased to be able to use a smaller diameter.
This mismatch in spool speed requires a gearing solution.
Note that very large amounts of power are being transmitted through the gear box, even with very high efficiencies this requires a lot of heat to be dissipated.
Getting rid of the waste heat and the large loading has proven to be a very difficult design task.
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The preceding explanations are good.
But - interesting add-on is the notion that the gearing of a turbine is hardly a new concept
- it's been around for decades in the form of turboprop engines.
The Allison T56 is an example of a geared Turboprop engine.
Simplistically, the difference between earlier turboprop designs and the more recent GTF design(s) - are the difference between a ducted fan and unducted propeller.
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I've read that the P&W PW1000G engine has a reduction gearbox. What is it and does it have any relation to the fan or compressor speed? Does it improve fuel efficiency?
What does a reduction gearbox do in a turbofan engine?
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<geared-turbofan>
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This is what the geared turbofan does, reduces the rotational velocity relative to that of the turbine so that both can rotate at their optimum angular velocity. The advantage of this setup is that the front fan can have longer blades to cover a larger cross-section area, while the low-pressure compressor and turbine blades in the core can spin faster to improve fuel efficiency.    It allows the front fan and the main shaft to operate each at their optimum speed and angular velocity, thus being more efficient.
It allows the front fan and the main shaft to operate each at their optimum speed and angular velocity, thus being more efficient.