Turtle is one of my reference guys...agree and kinda sorta disagree on this one...
Correct in that gear ratios do NOT change the input torque....it mulitiplies that input torque via the ratio between the input gear (pinion) and driven gear (ring gear).
Multiplication can be zero (1:1)...to less (2:1)...to more (1:3.73)...etc, etc.
Here is a site that explains it well
How do gears multiply torque?quora.com wrote:

Click For Full-Size Image.
Assume the smaller gear is driven by a motor producing X units of power. This energy is used to rotate smaller gear. To turn the gear faster you'll need more power. Smaller gear in turn rotates the bigger (driven) gear.
If smaller gear rotates 4 times per second, bigger gear will rotate only once per second. {because, smaller gear (9 teeth) has to rotate 4 times to traverse all the 36 teeth of bigger gear.}
Now,
Power is constant (since energy is conserved)**
Bigger gear rotates 4 times slower than smaller gear {i.e. Angular speed is 1/4th the smaller gear}
We can easily see,
Power = Constant
Angular Speed1 = 4*Angular Speed2
Torque1*Angular Speed1 = Torque2*Angular Speed2
Hence, Torque2 = 4*Torque1 {i.e. Bigger gear produces 4 times bigger torque. Torque is multiplied.}
If you get an opportunity to observe gearbox of an automobile, do observe the sizes of 1st, 2nd and 3rd stage gears. You'll observe the 1st is the biggest gear and then size reduces successively. This is because initially car needs maximum torque and bigger gear would result is larger torque multiplication
Forget about the rear gear box...AKA differential, for a moment and look at the gear box ahead of the diff
It has an even higher numeric (AKA...lower gear ratio) than the rear gear box...why ?
The exact same ICE power source for this discussion and use 200 ft/lbs as an easy reference to noodle in this discussion
Say the gear box bolted to the ICE via the bell housing will have 200 ft/lbs on the input shaft.
Quick search got this:
- Ford F150 10 speed is 4.69
- RAM 8 speed is 4.71
- GM 6 speed is 4.03
So all about the same ratio's and if you look at their second gear box (diff) ratio's...most are in the 3.xx range
So they all have about the same gear box ratio from the transmission (gear box #1) to the differential (gear box #2)
Hold that and now to how a gear box ratio works...simply put...gears either or both change the direction of rotation to transmit torque...and multiplies that torque (minus losses)
Changes direction from straight on, in line with input shaft...to off set. Most common is 90*
So, with the exact same ICE in first...the torque multiplication in 1st has higher torque by multiplication...say 6 times the base input shaft torque. That is what gets it moving...but will run out of steam as the ICE reaches it's red line or whatever the computer says to shift into a higher gear
2nd would be that higher gear and has a smaller multiplication. So less torque, but with the ability to spin the output shaft faster, but with less torque multiplication
3rd, 4th, 5th, etc....are using less and less torque multiplication for more RPMs, but less torque...as the mass is no moving and does NOT need to get it moving from a standstill..
All of that is then modified by the rear gear box's (diff) ratio, but is singular gear ratio...