Torque applied over center vs. off-center I have a tool that loosens a bolt with the torque applied over the center of the bolt (box end wrench).
I have a different tool that uses a socket which applies the torque (has the ratchet connection) 1.25" off-center to the same bolt.
How do I determine the difference in the torque required for each tool to loosen the same bolt?
Thank you
 A: There isn't a difference in the torque required to loosen each bolt, but rather a difference in the force you apply to the tool.
First, an equation: $\tau=F\times d$
Scenario 1: The Box End

Please forgive the crude, MS Paint drawing of your box end wrench, but hopefully you can see how this math works. The bolt is tightened to a certain holding torque, which we'll say is 1N-m. If your wrench were 1m long, you'd only need 1N of force, applied at the end and perpendicular to the handle, to loosen the bolt. As long as it's perpendicular, we can go ahead and turn the cross product into a simple multiplication, so $1$N-m$=1$N$\times$$1$m
Scenario 2: The Ratchet

Here we have the ratchet with an extension that increases the length of our lever arm by 50%, or 0.5m, in this case. You mention yours is 1.25", but the math is the same. Keeping the force at the end of the handle and perpendicular to the handle, the bolt's holding torque doesn't change, but we can calculate the force required to apply that torque.
$$F=\frac{\tau}{d}=\frac{1}{1.5}=\frac{2}{3}$$
So, now only $\frac{2}{3}$N is required to turn the bolt.
Bonus Scenario 3: How to properly use extensions with a torque wrench

So my drawing skills really need work, but in the automotive world, we use torque wrenches to properly tighted nuts and ensure that the fastened connection will handle the expected loads. To do this, we need a torque wrench. These are designed to alert the user when the appropriate torque has been applied, but they are calibrated for their exact length. If you use a crow's foot wrench you change the length of the lever arm, and the calibration is thrown off. You can either determine the length increase, and thus the torque decrease (as seen by the tool which doesn't know its length has changed), or you can be smarter and orient the tool in such a way that the line of action doesn't increase in magnitude, as drawn. In this drawing, the extension still exists, but it is parallel to the direction of the force, so from our equation at the top, it contributes no torque.
This video might show it a little more clearly, though I haven't watched it myself.
A: A pure torque (force couple) does not have a point of application. From the point of view of mechanics, the location of a torque does not enter into the equations anywhere.
Only the location of the line of action of a force is important because it imparts an equipollent torque. In this sense, forces are line vectors because they belong to a line, and torques are free vectors because they can be placed anywhere and it won't change the problem.
