He question: A gasoline engine producing 150 Nm of torque at a rotational speed of 300 rad/s is used to drive a winch and lift a weight as shown in figure 6. The winch drum has a radius of 0.25 m and is driven from the engine via a 1:50 speed reduction gear. What mass could be raised with this setup? (Assume the winch is in rotational equilibrium, )
I can solve this but please give me a hint about how this speed reduction gear effects the torque and speed of the winch. I think as speed is being reduced, the torque should also be reduced. Because the greater the (net) torque, the greater the force right?

You need to share fig 6 with us. I suspect it shows how the 50:1 reduction gear drives the winch, and the radiuses of the winch drum versus that gear, right?

This is the figure. You are right about the reduction ratio.

A speed reduction gear has the effect of increasing the torque that the gear can provide. Here's why: to reduce the output speed of a motor by 50:1 requires a gear that is 50 times the radius of the engine drive gear. Since the force provided by the engine output gear = the force driving the reduction gear, the resulting torque on the output gear is 50 times greater. And the resulting speed of the output gear is 1/50 the input. So, as speed is being reduced the torque provided by the gear is increased.

I must say I don't like how the problem is presented, especially the figure. If the problem stated that the winch turns at 1/50th the speed of the engine then it would be clear. But note how the notation about the 50:1 reduction gear is associated with the gray-colored thing on the right - this is connected to the winch by a drive belt, which means that we need to know the details of the drive belt pulleys to understand how much torque is delivered to the winch, and the relative rotational speed of winch and engine. Note how the drive belt is wrapped around pulleys at each end (one end on the reduction gear output and the other end on a hub at the center of the winch). To determine the weight of mass M that can be lifted you must make an assumption about the radii of those two pulleys, because the torque that the winch provides is dependent on the ratio of the radii of those pulleys. From the figure it looks like those pulleys may be the same radius, but I don't like having to make that assumption.

Thank you very much for your answer. It is clear to me now.